Capacitance-Power-Hysteresis Trilemma in Nanoporous Supercapacitors

Nanoporous supercapacitors are an important player in the field of energy storage that fill the gap between dielectric capacitors and batteries. The key challenge in the development of supercapacitors is the perceived trade-off between capacitance and power delivery. Current efforts to boost the capacitance of nanoporous supercapacitors focus on reducing the pore size so that they can only accommodate a single layer of ions. However, this tight packing compromises the charging dynamics and hence power density. We show via an analytical theory and Monte Carlo simulations that charging is sensitively dependent on the affinity of ions to the pores, and that high capacitances can be obtained for ionophobic pores of widths significantly larger than the ion diameter. Our theory also predicts that charging can be hysteretic with a significant energy loss per cycle for intermediate ionophilicities. We use these observations to explore the parameter regimes in which a capacitance-power-hysteresis trilemma may be avoided.

Popular Summary

The global energy crisis necessitates the development of devices that can store energy efficiently. Nanoporous supercapacitors, which store energy by accumulating ions in nanosized pores, have become increasingly popular. However, a major stumbling block hindering the implementation of such supercapacitors is the trade-off between capacitance and power delivery: Capacitance is maximized only when the pore size approaches the ion size, and the rate of charging is slow for such close-fitting pores. In addition, theory and simulations reveal that charging can proceed via an abrupt adsorption or desorption transition, which may cause hysteresis and therefore increase the energy dissipation during each charge or discharge cycle. This hysteresis adds yet another hurdle to the capacitance-power dilemma already noted. Here, we study how the capacitance-power-hysteresis trilemma can be avoided.

Using analytical calculations together with Monte Carlo simulations, we develop a theoretical model for nanoconfined ions. Our model reveals that the key control parameter for supercapacitors is the ionophobicity or ionophilicity of pores: It tells us how favorable the pore is to ions and determines the pore occupation at no applied voltage. We find that the capacitance of ionophobic pores can be optimized at pore widths significantly larger than the ion diameter, which avoids the issue of slow charging. Moreover, we show that charging is hysteretic only for intermediate ionophilicities. Therefore, by tuning the ionophobicity, nanoporous supercapacitors can be simultaneously optimized to provide high power and energy densities without hysteretic losses.

We hope that our findings will provide a framework to assess and direct future efforts to experimentally realize an optimal supercapacitor and, in particular, to control the pore ionophobicity.

Figure 1

Schematic drawing of the model porous electrode under consideration: Ions of diameter $d$ are confined between two metallic surfaces separated by a distance $L$. A potential $V$ (relative to the bath) is applied to the surfaces.

Figure 2

An ion’s resolvation energy, $\delta E$, determines the ion density (per surface area) inside nanopores at no applied voltage. At large $\delta E$, ions prefer to stay outside the pores, and the pores are (nearly) free of ions; we term such pores ionophobic. In the opposite case of small or negative $\delta E$, the pores are occupied by ions, and we call them ionophilic. The crossover between ionophilic and ionophobic occurs when $\delta E_{\text{crossover}}=l_B\ln (2)/L$, where $l_B$ is the Bjerrum length and $L$ the pore width (see text); $\delta E_{\text{crossover}}$ for two pore sizes ($L=0.6\mathrm{nm}$ and $L=0.7\mathrm{nm}$) are shown by thin vertical lines. Pores with $\delta E$ close to $\delta E_{\text{crossover}}$ have moderate ion densities at zero applied voltages, and we term them weakly ionophilic/ionophobic. The ion diameter is $d=0.5\mathrm{nm}$, and the Bjerrum length is $l_B=25\mathrm{nm}$. This plot has been obtained by using the mean-field theory (see Appendix pp1).

Figure 3

(a) Differential capacitance at zero voltage, $C_D(0)$, as a function of pore width and (b) the stored energy as a function of voltage (for $L=0.75\mathrm{nm}$) from the mean-field theory, plotted for weakly ionophobic ($\delta E=25k_{B}T$) and strongly ionophilic ($\delta E=10k_{B}T$) pores, as indicated by the line styles on the plots. The ion diameter is 0.5 nm. The inset in (a) shows how the total ion density at zero voltage depends on pore width. The inset in (b) highlights the stored energy at low voltages.

Figure 4

The calculated dependence of the differential capacitance, $C_D(0)$, as the pore width varies. The results of Monte Carlo simulations are shown for a strongly ionophilic pore ($\delta E=-2.5k_{B}T$) and for weakly ionophobic pores ($\delta E=38.5k_{B}T$ for $d=0.5\mathrm{nm}$ and $\delta E=33.2k_{B}T$ for $d=0.6\mathrm{nm}$). The insets show the 3D packing fraction ${\eta }_{3\mathrm{D}}=(\pi /6)\rho d^3$ as a function of pore width.

Figure 5

(a) Capacitance at zero voltage, $C_D(0)$, as a function of ion diameter ($d=d_{\pm }$) for weakly ionophobic ($\delta E=25k_{B}T$) and strongly ionophilic ($\delta E=10k_{B}T$) pores calculated using mean-field theory (MFT). The ratio between the pore width and the ion diameter is fixed to $L/d=1.2$. The inset shows $C_D(0)$ for small ions with $d\lesssim 0.5\mathrm{nm}$ and $L/d=1.1$. For such ultra narrow pores, $C_D(0)$ is practically independent of ionophobicity (for the values of $\delta E$ used in this plot). This is because the image forces [see Eq. (a9) in Appendix pp1-s3] overcompensate $\delta E$ for slit widths $L\lesssim 0.55$, making the pores completely filled with ions. (b,c) Capacitance map for strongly ionophilic and weakly ionophobic nanopores and room-temperature ionic liquids (with $d\gtrsim 0.5\mathrm{nm}$) in the plane of ion diameter ($d$) and the pore width ($L$) calculated using MFT. This figure suggests a “two-step optimization strategy,” in which an ionophobicity-dependent optimal pair ($d$, $L$) exists that maximizes the differential capacitance.

Figure 6

Charging proceeds via a discontinuous phase transition for some parameter regimes. In all plots, ion diameter $d=0.5\mathrm{nm}$, and the mean-field theory is used to compute the thermodynamic properties. (a) Capacitance as a function of voltage in the continuous charging regime. The ionophilic and ionophobic pores have resolvation energies $\delta E=10k_{B}T$ and $\delta E=32k_{B}T$, respectively, and width $L=0.8\mathrm{nm}$. (b) Capacitance as a function of voltage in the discontinuous charging regime. The solid curve indicates the minimum free-energy path, and the dotted curves denote the hysteresis loop with the arrows showing charging or discharging routes. The curve is plotted for $\delta E=18k_{B}T$ and $L=0.8\mathrm{nm}$. (c) Discontinuous charging is connected with an abrupt expulsion of coions. We identify here coion-deficient and coion-rich phases with vanishing and appreciable amounts of coions in the pore, respectively. The arrows show charging and discharging paths, which follow the metastable branches denoted by dotted lines, with the solid curve giving the minimum energy path, as in panel (b). The vertical dashed line marks a voltage $V_{\mathrm{coex}}\approx 0.49$ volts at which the two phases have the same free energy and coexist. (d) The free-energy landscape for applied voltage $V_{\mathrm{coex}}$ corresponding to the dashed line in panel (c). White lines are a contour plot of constant free energy, and the arrows point to the ion densities that simultaneously minimize the free energy at coexistence. The contour plot has been obtained by evaluating the free energy for given densities of cations and anions using Eq. (3). (e) The phase diagram plotted for different slit widths. (f) The percentage of energy that is lost because of hysteresis during discharging of a nanopore charged at 1.3 volts (see Appendix pp1-s4).