Abstract
Electrodepolymer interfaces dictate many of the properties of thin films such as capacitance, the electric field experienced by polymers, and charge transport. However, structure and dynamics of charged polymers near electrodes remain poorly understood, especially in the high concentration limit representative of the melts. To develop an understanding of electric field–induced transformations of electrodepolymer interfaces, we have studied electrified interfaces of an imidazoliumbased polymerized ionic liquid (PolyIL) using combinations of broadband dielectric spectroscopy, specular neutron reflectivity, and simulations based on the Rayleigh’s dissipation function formalism. Overall, we obtained the camelshaped dependence of the capacitance on applied voltage, which originated from the responses of an adsorbed polymer layer to applied voltages. This work provides additional insights related to the effects of molecular weight in affecting structure and properties of electrodepolymer interfaces, which are essential for designing nextgeneration energy storage and harvesting devices.
INTRODUCTION
The electrical double layer (EDL) (1–7) is a universal feature of electrified interfaces in all ionic materials. Spontaneous formation of the double layer is one of the primary mechanisms for storing electrical energy in devices such as supercapacitors (8). Understanding correlations between structure and properties of EDLs is vital for achieving control over device characteristics such as capacitance of electrochemical storage devices as well as charging and discharging rates of batteries. In the last two decades, special attention has been devoted to EDLs in highly concentrated electrolytes such as room temperature ionic liquids (ILs) (6, 7, 9–14) because of their relevance for a number of applications. Specifically, a number of theoretical, simulation, and experimental reports (6, 7, 9–14) have focused on understanding the structural changes of EDLs in an applied electric field and their connections with charge storage properties. As a result of such an extensive research, an interplay of electrostatics and crowding effects, resulting from finite size of ions, has been identified to be responsible for the anatomy of the EDLs in ILs. The EDLs in ILs have been shown to exhibit the layering of ions (11) and nonmonotonic dependence of the screening length on the concentration of ions in solutions of ILs (14).
Experimentally, the shortrange, surfaceinduced ordering of ions within ∼1 nm from surfaces has been confirmed with xray scattering (15). In contrast, recent surface force apparatus measurements (16) of ILs demonstrated the presence of an EDL of width equal to approximately 35 ionic diameters per surface, leading to a representation of ILs as very weakly dissociated electrolyte solutions (16). On the basis of the structure of the EDL, different shapes of capacitance curves as a function of applied voltages have been obtained for ILs. These shapes include bellshaped, Ushaped, and camelshaped curves (6, 7) in contrast to the capacitancevoltage curves observed in dilute electrolyte solutions, which are primarily Ushaped (1–5). Kornyshev (6) suggested an analytical model that takes into account contributions from steric exclusions between the ions, lattice saturation effects, and electrostatics, which explains the three shapes of capacitancevoltage curves. In addition, molecular simulations and experiments (17–19) showed that factors such as the chemistry of the electrolyte (7), electronic polarizabilities of the surface and electrolyte (20), the electrode surface topography (21, 22), porosity (23–25), and temperature (26) can markedly influence the EDL structure and dependence of the capacitance on the applied voltage for ILs.
In contrast, little is known about the capacitance of polymer electrolytes, specifically in the polymeric analogs of ILs known as polymerized ionic liquids (PolyILs). PolyILs (27, 28) are promising as solventfree organic electrolytes for application in batteries, solar cells, electroactuators, and supercapacitors. They have tunable mechanical properties and high stability, and they are nonflammable. Because of their structural organization with one ion mobile and another attached to the polymer chain, several phenomena can be expected. When a bias voltage is applied across an IL, it is generally accepted that the cations migrate toward the negative electrode and the anions migrate toward the positively charged electrode, resulting in an EDL layer of similar structures on both electrodes. However, it is not clear what would be the structure of an EDL in PolyILs where one ion has limited mobility due to its covalent attachment to the polymer chain. However, some hints can be found from molecular dynamics (MD) simulations of ILs where cations and anions have different sizes (7, 21). It has been shown that increasing the size of the ion (bead) can effectively change the shape of the capacitance curve from a bell to a camel shape and change the symmetry of the curve. In PolyILs, where all monomers are connected, the change in the shape should be potentially amplified, and different kinetics of EDL layer formation is expected on the cathode and anode because of the different mobilities of the ions, which could potentially lead to improved charge storage properties.
In this work, we have investigated electrodepolymer interfaces of a positively charged PolyIL–poly(Nvinyl ethyl imidazolium) with bis(trifluoromethane)sulfonimide as counterions. We have used combinations of broadband dielectric spectroscopy (BDS) and modeling of electrode polarization phenomena by the Rayleighian approach (29–31) to extract capacitance of electrodepolymer interfaces. These combined studies resulted in capacitancevoltage curves for the PolyIL films. Furthermore, specular neutron reflectivity (32–35) is used to study structure of polymers adsorbed on the electrode in the absence of an electric field. The model based on the Rayleighian approach (29–31) was used to understand the effects of polymer chain length on the capacitancevoltage curves, which is important for designing systems with improved storage properties.
RESULTS AND DISCUSSION
Representative BDS spectra measured at different frequencies and applied DC voltages are shown in Fig. 1A. The experimental protocol for varying the applied voltage is shown schematically in Fig. 1B. The spectra contain three different regions related to the rattling of ions in a Coulombic cage at high frequencies, followed by a DC plateau and the electrode polarization at lower frequencies. As seen from comparisons of different spectra acquired at different DC bias (cf. Fig. 1A), the noticeable changes in the spectra are observed only in the region of electrode polarization at lower frequencies. The electrode polarization process becomes noticeable when ions accumulate on the electrodes, and part of the spectra representing the electrode polarization carries information about the capacitancevoltage relations.
Determining practically important thicknesses of adsorbed and diffuse layers along with capacitance from impedance spectroscopy is typically (36) done by fitting with equivalent electrical circuitry models. Although useful, the physical interpretation of extracted quantities using equivalent circuits poses some serious challenges and remains troublesome. In contrast, we have used an electrode polarization model for extracting capacitance at the steady state from the BDS spectra. The model was developed using the Rayleighian approach in our previous work (31). In the model, each electrode was assumed to have a layer of lowdielectric material in contact with polymeric film of uniform static dielectric constant (cf. Fig. 1C). The model was used to interpret kinetics of charging in similar PolyILs using BDS measurements (31), and excellent agreements with the pulse field gradient–nuclear magnetic resonance (PFGNMR)–based measurements (for diffusion constants of counterions) were obtained. The measured BDS spectra of PolyILs were fitted by eq. A5 in the Supplementary Materials. An example of the fitting is presented in Fig. 1D. From the fitting, the apparent thickness of the lowdielectric layer (λ_{s}) and the length scale of the mutual diffusion (L_{m}) were extracted. Physically, L_{m} represents a length scale below which electrostatics and crowding effects are substantial. L_{m} is proportional to the Debye screening length (see eq. A10 in the Supplementary Materials). However, it can be greater than the screening length owing to the crowding effects included in the calculations. Both λ_{s} and L_{m} are plotted in Fig. 2A for the PolyIL films studied at 370 K.
Figure 2A shows that thickness of the lowdielectric layer (λ_{s}) had a nonmonotonic dependence on the applied voltage exhibiting a finite nonzero value at zero bias, followed by an initial decrease before increasing and leveling off with increase in DC voltage. Furthermore, in the experiments with different polarities of applied DC bias, the positive and negative wings had almost the same values at the minima (corresponding to λ_{s} = 2.47 and 2.28 nm for 0.7 and −1.3 V, respectively), suggesting almost symmetric capacitancevoltage relations shown in Fig. 2B. In contrast, the length scale of mutual diffusion (L_{m}) was unchanged with the applied voltage, suggesting that we are in the linear response regime. The length scale of mutual diffusion for PolyILs was found to be ∼0.60 nm, which is reasonable and in excellent agreement with values reported in (37) using a different analysis of the BDS data. We should point out that the value of L_{m} estimated here is
$\mathrm{\sim}\sqrt{6}$times the value reported in our previous work (31), which means that the mutual diffusion coefficient D_{m} is ∼6 times the selfdiffusion constant of the counterions (estimated using PFGNMR measurements), in contrast to strict enforcement of D_{m} being equal to the selfdiffusion constant of counterions used in (31) for estimating L_{m}. The fit parameters were plugged into Eq. 1 to construct capacitance of the electrodePolyIL interfaces, and these results are shown in Fig. 2B. Derivation of Eq. 1 from the electrode polarization model, analyzed in the steady state, is presented in Appendix B in the Supplementary Materials. Briefly, capacitance can be shown to be
$${C}_{\text{interface}}=\frac{{\mathrm{\u03f5}}_{o}{\mathrm{\u03f5}}_{r}}{{L}_{\mathrm{m}}{\mathrm{\lambda}}_{\mathrm{s}}}\frac{[\text{cosh}(d/{L}_{\mathrm{m}})\mathrm{}1]}{[\frac{\text{cosh}(d/{L}_{\mathrm{m}})}{{L}_{\mathrm{m}}}+\frac{\text{sinh}(d/{L}_{\mathrm{m}})}{{\mathrm{\lambda}}_{\mathrm{s}}}]}$$(1)where ϵ_{o}, ϵ_{r}, L_{m}, λ_{s}, and 2(d + λ_{s}) are the permittivity of vacuum, static dielectric constant of the film, length scale of mutual diffusion, apparent thickness of lowdielectric layer on each electrode, and apparent total distance between the electrodes, respectively. Also, λ_{s} = l_{s}ϵ_{r}/ϵ_{s}, where l_{s} and ϵ_{s} are the real thickness and static dielectric constant of the lowdielectric layer, respectively. Except for d, which was set by experimental conditions used to measure BDS spectra, all of the parameters were readily obtained by fitting the BDS spectra. Unfortunately, l_{s} and ϵ_{s} are not directly accessible from the BDS spectra. This is because the impedance measured using BDS shows dependence on l_{s} and ϵ_{s} only via λ_{s}. We should point out that in the limit, where d/L_{m} → ∞, 1/ cosh (d/L_{m}) → 0, and tanh (d/L_{m}) → 1, Eq. 1 can be simplified as
${C}_{\text{interface}}={C}_{\text{interface}}^{\mathrm{\infty}}$, where
$${C}_{\text{interface}}^{\infty}={[\frac{{L}_{\mathrm{m}}}{{\mathrm{\u03f5}}_{\mathrm{o}}{\mathrm{\u03f5}}_{\mathrm{r}}}+\frac{{l}_{\mathrm{s}}}{{\mathrm{\u03f5}}_{\mathrm{o}}{\mathrm{\u03f5}}_{\mathrm{s}}}]}^{1}$$(2)
Equation 2 simply shows that the lowdielectric layer and another layer characterized by L_{m} and relative permittivity ϵ_{o}ϵ_{r} act as two capacitors in series leading to sequential voltage drop across each of them. Also, in the limit of d/L_{m} → ∞, the capacitance is independent of the parameter d.
The camel shape of capacitancevoltage curves shown in Fig. 2B is qualitatively similar to the one predicted in atomistic MD simulations of ILs on rough surfaces (21). The two maxima have roughly the same positions. Unexpectedly, we did not detect any substantial asymmetry in the shape of the individual peaks as would be expected based on studies with ILs because of the different sizes of the ions on the chains and their counterions [e.g., van der Waals radii of the mobile counterion (38) and the monomer (39) used in this study are estimated to be 0.33 and 0.57 nm, respectively]. As the shape of the capacitancevoltage curve is dictated by the lowdielectric layer, such a lack of asymmetry in the curve implies that reorganization of the lowdielectric layer in applied voltages is almost symmetric as shown in Fig. 2A. However, because of the lack of information about molecular mechanisms underlying reorganization of the lowdielectric layer, reasons for the symmetric λ_{s}voltage curve (cf. Fig. 2A) are not clear. Note that the apparent thickness of the lowdielectric layer at 0 V is not zero and has a value of ∼2.7 nm (cf. Fig. 2A). The observed nonzero initial thicknesses can originate from the physically adsorbed layer of PolyILs and/or can be due to the rough layer of native oxides found on most of the electrodes. This statement is in line with results of reflectivity measurements (cf. Fig. 3B), which revealed the presence of an interfacial layer without any applied bias with a thickness of 1.4 ± 0.3 nm while the scattering length density (SLD) of the interfacial layer is
$1.50\mathrm{\times}1{0}^{\mathrm{}6}\stackrel{\mathrm{\u030a}}{\mathrm{A}}{}^{\mathrm{}2}$, which is lower than a bulk polymer film value of
$2.55\mathrm{\times}1{0}^{\mathrm{}6}\stackrel{\mathrm{\u030a}}{\mathrm{A}}{}^{\mathrm{}2}$. Note that the SLD value for bulk polymer is in good agreement with the theoretical value of
$2.63\mathrm{\times}1{0}^{\mathrm{}6}{\stackrel{\mathrm{\u030a}}{\mathrm{A}}}^{\mathrm{}2}$calculated based on the pure polymer density. The disparity in the SLD values for bulk and interface can be attributed to a decrease in counterion density in the interfacial region, which lowers its SLD value closer to the value expected of the polymer without a mobile counterion. Unfortunately, direct comparisons between neutron and BDS studies are not possible due to the different types of substrates used in these measurements. Nevertheless, it should be clear that both the silicon and metallic electrode surfaces had preadsorbed layers, whose voltage dependence dictated the capacitancevoltage relations. Transformation of this preadsorbed layer with changes in applied voltage determined the capacitancevoltage relation, highlighting that the quality and chemistry of preadsorbed layer are important in designing efficient energy storage devices. Similar adsorbed layers have been observed in atomistic MD simulations of polymerized ILs on neutral and charged substrates (40).
In constructing capacitancevoltage relations from BDS measurements, we have used a minimal model with simplifying assumptions such as symmetry of lowdielectric layers on the two electrodes and ignored explicit nonlinear dependence of the surface charge on applied voltage and asymmetry in molar volume of monomers with respect to counterions. To go beyond these simplifying assumptions and study their effects on capacitancevoltage curves, we have numerically solved (41, 42) underlying eqs. A1 to A4 of the electrode polarization model in a steady state. Representative results obtained from the numerical solutions are shown in Fig. 4. In the absence of any molecular model, voltage dependence of the λ_{s} was ignored while solving the equations. For experimentally studied systems for which L_{m}/d → 0, the right panel in Fig. 4 shows that capacitance in a steady state is mainly dictated by the lowdielectric layer, which yields nonzero capacitance at zero applied voltage. However, if L_{m}/d ≫ 1, then there is an additional contribution to the capacitance, which is voltage dependent and decreases with an increase in the applied voltage, V. It can be readily shown that the capacitance must decrease as V^{−1/2} in the absence of the lowdielectric layer (see Appendix C for details), and this is a direct consequence of the crowding enforced by the local incompressibility condition in the model. Specifically, taking λ_{s} = 0 for an applied voltage V, capacitance can be derived to be
$$C=\frac{\sqrt{8\mathrm{\pi}{l}_{\mathrm{B}}{\mathrm{\rho}}_{\text{co}}}\mid {z}_{\mathrm{p}}{v}_{\mathrm{r}}{\scriptscriptstyle \frac{\text{exp}(({z}_{\mathrm{c}}{z}_{\mathrm{p}}{v}_{\mathrm{r}})\overline{V})}{1+{\scriptscriptstyle \frac{{v}_{\mathrm{r}}}{{N}_{\mathrm{p}}}}{\overline{\mathrm{\varphi}}}_{\mathrm{c}}\text{exp}(({z}_{\mathrm{c}}{z}_{\mathrm{p}}{v}_{\mathrm{r}})\overline{V})}}\mid}{2\sqrt{\mid {z}_{\mathrm{p}}{v}_{\mathrm{r}}\overline{V}+{\scriptscriptstyle \frac{{N}_{\mathrm{p}}}{{v}_{\mathrm{r}}{\overline{\mathrm{\varphi}}}_{\mathrm{c}}}}\text{ln}\left({\scriptscriptstyle \frac{1+{\scriptscriptstyle \frac{{v}_{\mathrm{r}}}{{N}_{\mathrm{p}}}}{\overline{\mathrm{\varphi}}}_{\mathrm{c}}\text{exp}(({z}_{\mathrm{c}}{z}_{\mathrm{p}}{v}_{\mathrm{r}})\overline{V})}{1+{\scriptscriptstyle \frac{{v}_{\mathrm{r}}}{{N}_{\mathrm{p}}}}{\overline{\mathrm{\varphi}}}_{\mathrm{c}}}}\right)\mid}}$$(3)where
$\overline{V}=\mathrm{eV}/{k}_{\mathrm{B}}T$is the dimensionless voltage so that e is the charge of an electron; k_{B} is the Boltzmann constant; T is the temperature; l_{B} is the Bjerrum length; z_{c} and 1/ρ_{co} are the valency and the molar volume of the counterions, respectively; v_{r} is the ratio of molar volume of the counterions to the monomers; z_{p} is the valency of a monomer; and N_{p} is the number of Kuhn segments along a polymer chain. The only unknown in this equation is
${\overline{\mathrm{\varphi}}}_{\mathrm{c}}$, which is an integration constant. Equation 3 shows that the capacitancevoltage relations are weakly dependent on N_{p}, which is confirmed by numerical solutions in Appendix D. Comparing Figs. 2B and 4, we conclude that if the dependence of λ_{s} on applied voltage was to be ignored, then the capacitance should either be independent or decrease with an increase in applied voltage similar to the curves shown in Fig. 4. Nevertheless, letting λ_{s} vary with applied voltage led to the camelshaped capacitancevoltage curve, indicating the strong dependence of the capacitance on the lowdielectric layer. Note that our treatment of the lowdielectric layer is conceptually similar to the Stern layer in a classical treatment of electric double layers in dilute solutions, as presented by Grahame in 1954 (2). In addition, temperature can be varied to alter magnitude of the steadystate capacitance at 0 V due to changes in apparent thickness of the lowdielectric layer (31). However, qualitative features of capacitancevoltage relations are expected to be similar to Fig. 2.
CONCLUSIONS
We have studied electrodePolyIL interfaces using BDS, neutron reflectometry, and modeling based on the Rayleighian approach. It was shown that a preadsorbed layer exists at the electrode at zero applied voltage, which dictates measured impedance and capacitance of the electrodePolyIL interfaces. The preadsorbed layer, which was treated as a lowdielectric layer in the electrode polarization model studied here, is expected to be present in most of the other films containing similarly charged PolyILs. Such an expectation results from the hypothesis that adsorption of the PolyILs is driven by their attraction toward negatively charged surfaces and metallic electrodes. Not only did we estimate the apparent thickness of the lowdielectric layer from the BDS spectrum, but we also found the thickness to exhibit a nonmonotonic dependence on the applied voltages. The nonmonotonic dependence manifests in a similar dependence of the capacitancevoltage curves for the films. We found that the preadsorbed layer at 0 V plays an important role in determining changes in the interfacial layer thickness in an applied voltage up to 1 V, as we see clear monotonic decrease in its thickness with applied electric field. These changes can be associated with rearrangement of the already adsorbed layer similar to those observed in atomistic simulations (40). Capacitancevoltage curves are found to be camelshaped exhibiting almost no substantial asymmetry with the reversal of voltage. Our numerical investigations revealed that size asymmetry between monomers and counterions does not affect the capacitancevoltage relations significantly. This suggests that only the preadsorbed layer near the electrode contributes substantially to the capacitance. Last, the capacitance is, on average, found to be ∼22 mF/cm^{2} for the PolyIL studied in this work, and the shape of capacitancevoltage curve is qualitatively similar to the corresponding IL. As an outlook, preadsorbed layers are expected to be present in most of the films containing similarly charged polymers, and this study should be of interest to researchers engaged in research related to energy storage and harvesting devices.
MATERIALS AND METHODS
BDS measurements
The chemical structure of the PolyIL used in this study is presented in Fig. 1A. Synthesis of poly(Nvinyl ethyl imidazolium) is detailed elsewhere (31, 43). Direct measurements of the molecular weight of the PolyIL are challenging and were not performed here as it required a combination of certain solvents to neutralize the charge of the polymer, which are not known. However, some information about molecular weight and dispersity of the PolyIL is known. Specifically, the PolyIL was synthesized from poly(Nvinyl imidazole). The latter was synthesized using conditions reported in (44), and the reaction conditions would give M_{n} = 42,000 g/mol with a dispersity of 1.84. The BDS measurements were performed in a frequency range 10^{−2} to 10^{7} Hz using an AlphaA analyzer from Novocontrol. DC voltages of different polarities and amplitudes were applied to the sample. For BDS measurements, the sample of PolyILs was spincoated from an acetone solution between two stainless steel plates separated by silica fibers with a thickness of 50 μm. The spectra were accumulated at 97°C, which is 30°C above the glass transition temperature (T_{g}) of this polymer. Before the measurements, the temperature was stabilized within 0.2 K by a Quattro temperature controller, also provided by Novocontrol. In the experiments with DC voltage, the AC amplitude was kept at 0.01 V, while DC bias was changed gradually from 0.1 to 20 V. Spectra obtained after application of different DC voltages are presented in Fig. 1A. In the experiments with DC voltage, first positive bias was applied, followed by zero bias and negative bias at each frequency. A schematic of the experimental protocol is presented in Fig. 1B. Note that spectra were analyzed in the range of −3 to 3 V in the absence of Faradaic current, where changes in the spectra were reversible.
Rayleighian approach–based model for electrode polarization and capacitance
Fitting BDS spectra using the model. We used the Rayleighian approach (29–31) for understanding capacitancevoltage relations in films of PolyILs. Recently, we have used the approach to develop a model for electrode polarization phenomenon (31), which can be studied using BDS measurements. A brief description of the model is presented in Appendix A in the Supplementary Materials. In this work, the same model for electrode polarization was used to fit the BDS spectra, and the fit parameters were used to construct capacitance of the films using an analytical expression derived in the current work. An example showing typical fits of the BDS spectra is presented in Fig. 1D. Briefly, fittings of the BDS spectra were performed only for the peak region in the ratio of imaginary and real parts of the frequencydependent dielectric response [known as tan δ or the loss tangent in the literature on the BDS (45)]. The main advantage of fitting the peak is that the fit results are independent of the values of the static permittivity of the medium, leading to a reduction in the number of fit parameters. Also, the location of the peak provides an estimate of the characteristic time involved in the electrode polarization, which provides reasonable initial estimates of the fit parameters. Specifically, the fitting was performed through constrained multivariable optimization of the ratio of the imaginary and the real part of eq. A5 in the Supplementary Materials to the corresponding BDS data using the software package Mathematica. Three parameters were optimized: the characteristic time scale of the mutual diffusion (τ_{m}), the apparent thickness of a lowdielectric layer with respect to the separation between the electrodes (λ_{s}/d), and the length scale of mutual diffusion (L_{m}). The apparent thickness of the lowdielectric layer (λ_{s}) was assumed to be the same on each electrode, which were taken to be separated by distance 2d so that the total distance between the electrodes was taken to be 2(d + λ_{s}) (see Fig. 1C for a schematic). Numerical results obtained by solving equations (eqs. A1 to A4) for the electrode polarization model showed that asymmetry in sizes of the lowdielectric layers on two sides of the films had negligible effect on capacitancevoltage relations. For the fitting of the data, d was taken to be 25 μm. Lower limits of the three parameters were constrained to zero, and the upper limits were chosen as follows. The upper limit of λ_{s} was set to an arbitrarily high value of 100 nm as it did not have any significant impact on the fitting. However, the choice of the upper limit of τ_{m} had a significant impact on the fit results. If the choice was too far off from the correct value (∼1/ω_{peak}, where ω_{peak} is the circular frequency for the peak in tan δ), then we could not fit the experimental results. Thus, the upper limit of τ_{m} was chosen close to ∼1/ω_{peak}. The upper limit of L_{m} was chosen on the basis of the complimentary PFGNMR measurements of the diffusion constants presented in our previous work (31). In particular, we used the relation
${D}_{\mathrm{m}}={L}_{\mathrm{m}}^{2}/{\mathrm{\tau}}_{\mathrm{m}}$to estimate the mutual diffusion constant (D_{m}). For different choices of the upper limit of L_{m}, the peak in tan δ can be fitted using eq. A5, but the extracted D_{m} may be orders of magnitude off from the PFGNMR results. This results from the fact that there are multiple parameter sets, which can fit the same experimental data [see (31) for the details]. For the results presented in Fig. 2A, the upper limit of L_{m} was set at 1.5 nm for all the PolyIL samples. The static permittivity of the medium (ϵ_{0}ϵ_{r}) was determined by fitting the real and the imaginary parts of the dielectric response at the lowest voltage. In particular, fitting with one additional multiplicative factor was performed for ϵ′ and ϵ′′ by using eq. A5 with the extracted τ_{m}, λ_{s}/d, and L_{m}. However, we kept ϵ_{0}ϵ_{r} fixed as a function of DC voltage as we did not expect any dielectric saturation effect for small applied voltages. After estimating the four parameters from the BDS spectra, capacitance at the steady state was determined by using Eq. 1 (see Appendix B in the Supplementary Materials for its derivation). However, Eq. 1 has no explicit dependence on the applied DC voltage. To understand the capacitancevoltage relations, we assumed that voltage dependence appears implicitly via the thickness of the lowdielectric layer (λ_{s}) and the length scale of mutual diffusion (L_{m}). Thus, we let them vary while fitting the experimental BDS data at different voltages. In other words, we determined voltage dependencies of λ_{s} and L_{m} from the BDS data.
Numerical results obtained from the model. To understand explicit dependence of the capacitance on applied voltage, we solved the equations appearing in the electrode polarization model and studied the effects of various parameters on capacitancevoltage curves (see Appendices C and D in the Supplementary Materials). Key representative results are presented in Fig. 4 and figs. S2 to S4, and details about numerical methods are presented in the Supplementary Materials.
Neutron reflectometry
To probe structure in the thin films, we performed neutron reflectometry measurements (32–35). The PolyIL film was spincoated on a silicon wafer and placed in a heating cell for the measurements. The measurements were performed at 40° and 100°C. Apart from the oscillations originating from the film interference, the reflectivity decayed with the expected Q^{4} (Q being the wave vector) dependence up to
$\mathrm{\sim}0.13\stackrel{\mathrm{\u030a}}{\mathrm{A}}{\mathrm{\u030a}}^{\mathrm{}1}$. For higher values of Q, the reflected intensity became indistinguishable from the substrate background scattering (i.e., reflectivity, R < 10^{−6}); data beyond this Q range were not included in the fitting. Models to the reflectivity data were simulated by generating SLD profiles. Each layer was associated with individual thickness, SLD (ρ), and interfacial width (σ). The theoretical SLDs, estimated using the compositions and densities of the polymer with counterion (
$2.63\mathrm{\times}1{0}^{\mathrm{}6}\stackrel{\mathrm{\u030a}}{\mathrm{A}}{\mathrm{\u030a}}^{\mathrm{}2}$), polymer without counterion (
$1.18\mathrm{\times}1{0}^{\mathrm{}6}\stackrel{\mathrm{\u030a}}{\mathrm{A}}{\mathrm{\u030a}}^{\mathrm{}2}$), and the counterion alone (
$2.76\mathrm{\times}1{0}^{\mathrm{}6}\stackrel{\mathrm{\u030a}}{\mathrm{A}}{\mathrm{\u030a}}^{\mathrm{}2}$), were used as initial fitting parameters and limits of the fitting. The fitting was performed using three layers, which consisted of (i) the native oxide layer, (ii) an interface between the silicon wafer and the polymer, and (iii) the bulk polymer film. We also fitted the reflectivity curves including an additional airpolymer interface layer. However, the resulting fits were similar to the ones without the added layer, suggesting that the threelayer model was adequate for describing this interface. The latter statement is also supported by literature data obtained for air–ionic liquid interfaces (15). Furthermore, in our case, fitting with only a single polymer layer without the depleted zone proved to be inadequate. Between both fits, values of ρ for the silicon (Si), oxide layer (SiO_{2}), and air in addition to σ for the substrate/(1) and (1)/(2) and (2)/(3) interfaces were held constant, since these values were found to not depend significantly on the temperature. The thickness of the native oxide was fixed to be 1.1 nm. The thickness of the bulk polymer was found to be 104 to 105 nm, in agreement with our ellipsometry measurements. The SLD and thickness of the interfacial layer were found to be highly coupled. To eliminate this codependence, the SLD was restricted to the lower limit of the pure polymer (without counterions), while the thickness was free to vary. For the bulk polymer layer, both thickness and ρ were free to assume any values. The spectra for 0 V overlaid for 40° and 100°C together with the fittings are presented in Fig. 3A. The SLD profiles generated from the fittings are presented in Fig. 3B.
This is an openaccess article distributed under the terms of the Creative Commons AttributionNonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
 ↵
 ↵

E. Verwey, J. Overbeek, K. van Nes, Theory of the Stability of Lyophobic Colloids: The Interaction of Sol Particles Having an Electric Double Layer (Elsevier Publishing Company, 1948).

S. L. Carnie, G. M. Torrie, in The Statistical Mechanics of the Electrical Double Layer (John Wiley & Sons Ltd., 2007), pp. 141–253.
 ↵
J. Israelachvili, Intermolecular and Surface Forces, Intermolecular and Surface Forces (Elsevier Science, 2015).
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
M. B. Armand, P. G. Bruce, M. Forsyth, B. Scrosati, W. Wieczorek, in Polymer Electrolytes (John Wiley & Sons Ltd., 2011), chap. 1, pp. 1–31.
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
T. A. Driscoll, N. Hale, L. N. Trefethen, Chebfun Guide (Pafnuty Publications, 2014).
 ↵
 ↵
 ↵
 ↵
F. Kremer, Broadband Dielectric Spectroscopy (SpringerVerlag, 2003).
Acknowledgments: We gratefully acknowledge contributions from G. Smith related to neutron reflectivity experiments in the early stages of the works presented in this manuscript. Funding: This research was sponsored by the Laboratory Directed Research and Development (LDRD) Program of Oak Ridge National Laboratory (ORNL), managed by UTBattelle, LLC, for the U.S. Department of Energy. The research was conducted at the Center for Nanophase Materials Sciences, which is a U.S. Department of Energy Office of Science User Facility. K.S.S. was supported by the Department of Energy Computational Science Graduate Fellowship program under grant no. DEFG0297ER25308. A.P.S. and B.G.S. acknowledge support from the Division of Materials Sciences and Engineering, DOE Office of Basic Energy Sciences. Author contributions: R.K., J.P.M., K.S.S., and B.G.S. contributed to the computational aspects in this work. Z.W., V.B., and A.P.S. contributed to the materials and BDS measurements. A.E. and J.F.A. contributed to the neutron reflectivity–based analysis presented in the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
 Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BYNC).