Symmetrizing cathode-anode response to speed up charging of nanoporous supercapacitors
Tangming Mo, Liang Zeng, Zhenxiang Wang, Svyatoslav Kondrat, Guang Feng
1 Symmetrizing cathode-anode response to speed up charging of nanoporous supercapacitors
Tangming Mo,
Liang Zeng,
Zhenxiang Wang,
Svyatoslav Kondrat, Guang Feng State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China Nano Interface Centre for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, 430074, China Department of Complex Systems, Institute of Physical Chemistry, PAS, Kasprzaka 44/52, 01-224 Warsaw, Poland * Corresponding author’s email: [email protected] Abstract
The asymmetric behaviors of capacitance and dynamics features in the cathode and anion are universal at porous supercapacitors. Understanding that is important to design optimal supercapacitors. In this work, we performed constant-potential molecular dynamics simulations to reveal asymmetric features of porous supercapacitors and their effects on capacitance and charging dynamics. Results revealed the capacitance and charging dynamics are correlated strongly with asymmetric response of the cation and anion in cathode and anode, and symmetrizing ion response could boost the charging dynamics. Counterintuitively, the charging dynamics of negatively charged narrow pore is slow, although its in-pore ions diffuse fast, which could be attributed to longer and roundabout motion paths for co-ion desorption caused by ion overfilling. Compared with the electrode with single size pore, the ion overfilling also occurs in the electrode with multiple pore sizes, while co-ion desorption is accelerated, associated with over-charging phenomenon, thus enhancing the charging dynamics.
Keywords : porous carbon; charging dynamics; charge storage mechanism; overfilling; ion motion paths; over-charging Introduction
Supercapacitors have attracted significant attention owing to their fast charging/discharging rate and long cycle life. [1-3]
However, their wide application is limited by their moderate energy density, compared with batteries. [4-6]
To address this issue, porous carbons have been used as electrode in supercapacitors, due to their high specific surface area and large specific capacitance. [7-10]
Among capacitive behaviors of porous carbon supercapacitors, an interesting feature that the cathode and anode have unequal capacitance has been reported in both experiment and modeling works. [11-16]
Electrochemical measurements on supercapacitors composed of carbide-derived carbons (CDCs) and an organic electrolyte of tetraethyl-phosphonium tetrafluoroborate in acetonitrile ([TEA][BF ]/ACN) revealed that the capacitance of the negatively polarized electrode is much larger than the positive one, [11] while supercapacitors, with CDCs and an ionic liquid (IL) of 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([EMIM][TFSI]), exhibit a higher capacitance in the positive electrode. [12] Experiments on CDCs with ILs [EMIM][TFSI] and [EMIM][BF ] disclosed that such an asymmetric behavior of the capacitance in negative and positive electrodes limits their voltage window, thereby, the energy density. [15] Recently, supercapacitors, involving the positive electrode with microporous carbons for the exclusive electrosorption of small anions and the negative electrode of mesoporous carbons accessible to both ions, have been found to provide high energy storage capacity. [17]
These studies indicate that the ion and pore size have a significant impact on the asymmetric capacitance behavior, which would affect their energy storage. Dynamic features of electrolyte ions in porous electrodes have also been revealed to exhibit the asymmetric behavior at negative and positive electrodes. [3, 18-23]
Tsai et al. utilized electrochemical quartz crystal microbalance (EQCM) to explore ion dynamics during the charging process in CDCs with [EMIM][TFSI] and found that the charge storage is dominated by the counterion adsorption under negative and highly positive polarizations, but by ion exchange under lower positive polarizations. [19]
Pean et al. performed molecular dynamics (MD) simulations of CDC electrodes in IL 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF ]), and predicted that cations are less mobile than anions during the charging process. [24] The diffusion of ions inside nanopores was found to strongly correlate with the change of cations and anions. [25, 26]
Specifically, using NMR, Forse et al. found that the much reduced diffusion of ions in negative electrodes and weak-changed diffusion of ions in positive electrodes are associated with the cation adsorption under negative polarizations and the exchange of ions under positive polarizations, respectively. [25]
Whereas, it is still unclear what the origin of asymmetric features in capacitance and dynamics should be and how these asymmetric behaviors would affect the charging process. In this work, MD simulations were performed to investigate asymmetric features of the cathode and anode response ( i.e. , the cation/anion change in positive and negative electrodes) and their relation of capacitance and charging dynamics of nanoporous supercapacitors. As presented in
Fig. 1a and
Fig. S1 , constant-potential MD simulation systems consist of an IL [EMIM][BF ] filled in two symmetric or asymmetric nanoporous electrodes with single or multiple pore size ( Methods ). Our simulations revealed that the capacitance and charging dynamics in either symmetric or asymmetric supercapacitors are strongly correlated with the ion change: symmetrizing ion change helps to speed up the charging process. Despite fast diffusion of in-pore ions, the negatively charged narrow pore would slow down the charging dynamics, which is attributed to the fact that the overfilling phenomenon occurs during charging, leading to roundabout and long motion paths for the co-ion desorption. Such co-ion desorption could be accelerated in electrodes with multiple pore sizes.
Results and discussion
The asymmetric characteristics of cathode-anode capacitance
We first concentrate on simulation systems with single pore size electrodes ( i.e. , theoretically, one size distribution of porous electrode), denoted as d s1 //d s2 to represent the system of the negatively polarized electrode with pore size of s1 and the positive one with pore size of s2 ( Fig. ). To scrutinize the asymmetric charge storage behavior of negative and positive electrodes, we calculated the integral capacitance, , defined as: , where is the surface charge density and is the potential of the electrode relative to potential of zero charge (PZC, Table S1 ). As shown in
Fig. 1b , there is a big difference of capacitance between symmetric systems d //d and d //d . Specifically, the system d //d exhibits an obviously asymmetric capacitance behavior: the capacitance of the negative electrode is larger than that of the positive electrode. For instance, the charge density is 5.54 μF/cm for the negative electrode, but 3.94 μF/cm for the positive one under a voltage between two electrodes ( i.e. , cell voltage) of 4 V ( Fig. S3 ). In sharp contrast to d //d , the capacitance of the negative electrode in d //d is almost the same with that at the positive side, and the capacitance varies slightly with electrode potentials. Different capacitance behaviors of d //d and d //d could be understood by the number density of cations and anions at different electrode potentials (
Fig. 1c ). For system d //d , the charge storage is dominated by the anion desorption under negative polarizations and by the anion absorption under positive polarizations, consistent with previous EQCM measurements which reported that the charge storage of nanoporous electrodes is controlled by the change of small anions [27] , while the charge storage of d //d is driven by the very similar cation and anion exchange under both negative and positive polarizations. To quantify these features of the cation and anion change, we adopt the charging mechanism parameter, , defined as (
Eq. 1 ): [21] (1) where and are the total numbers of ions inside pores under polarizations and at PZC, respectively, ( and for counterions, and and for co-ions). equals to +1 (-1) for pure counterion absorption (co-ion desorption) and 0 for an exact one-to-one cation-anion exchange. Based on the domination of cations or anions, the ion change is further categorized into three parts: counterion absorption domination, ion exchange domination and co-ion desorption domination. By trisecting the overall range of (i.e., from -1 to +1), counterion absorption (co-ion desorption) domination means the change of counterion (co-ion) is at least twice as large as the change of co-ion (counterion), i.e. , > 1/3 (< -1/3); the ion exchange domination represents the change of co-ion and counterion is comparable ( i.e. , -1/3 < < 1/3). As shown in Fig. 1d , for system d //d , values are almost less than -1/3 under negative polarizations and larger than 1/3 under positive polarizations, indicating that the charge storage is driven by co-ion desorption domination and by counterion absorption domination, respectively. For system d //d , it can be found that -1/3 < < 1/3, suggesting that the charge storage is dominated by ion exchange under both negative and positive polarizations. Comparison of and ( Fig. 1b and
Fig. 1d ) reveals that for d //d , the capacitance dominated by co-ion desorption under the negative polarization is larger than that governed by counterion absorption under the positive one, exhibiting an asymmetric capacitive behavior. It could be explained by the mean-field theory that the counterion must overcome entropic and electrostatic barriers to enter a pore, while those barriers are weaker or absent for co-ion desorption. [28]
For system d //d , values change little with electrode potential, resulting in the nearly constant capacitance. These results imply that the correlation of ion changes between positive and negative electrodes is important to understand the asymmetric capacitance behavior. Therefore, we calculate the net charging mechanism parameter, , to characterize such a correlation, which is defined as ( Eq. 2 ): [29] (2) where and are the charging mechanism parameters of the positive and negative electrodes, respectively. Similar to the categorization of , is accordingly divided to three parts: cation domination ( < -2/3), similar contribution between cation and anion (-2/3 < < 2/3), and anion domination ( > 2/3), by trisecting the overall range of ( i.e. , from -2 to +2). Herein, cation domination ( < -2/3) and anion domination ( > 2/3) could be identified as the asymmetric ion change, while similar contribution between cation and anion (-2/3 < < 2/3) could be considered as the symmetric ion change. For instance, and are calculated as 0.3 and -0.4, respectively, for an IL 1-methyl-1-propylpyrrolidinium bis(trifluoromethanesulfonyl)imide [Pyr ][TFSI] in porous carbon electrodes from NMR measurements, [30] and then the value is calculated to be 0.7 using Eq. 2 , meaning that its charging dynamics is driven by the asymmetric ion change (TFSI - domination). Figure 1e displays that of d //d is over 2/3, resulting in an evidently asymmetric charge-storage performance, while of d //d is close to zero, leading to the symmetric capacitance. The dissimilar responses of EMIM + and BF in d //d and d //d may originate from the different nanoconfinement effects. As shown in Fig. S4a , there is only one layer of either cation or anion, and the ethyl group of EMIM + is closer to the pore wall than BF in the 0.5 nm pore, suggesting that EMIM + has a larger nanoconfinement effect than BF , and thus BF could move more easily than EMIM + . For a larger pore ( e.g. , 0.75 nm pore in Fig. S4b ), the pore has more space to accommodate more than one layer of ions, so that the nanoconfinement effects on cations and anions become weaker, which could be further testified by diffusion of ion-pore cations [31, 32] (detail in
Method ): the cation in d //d at PZC (12.6×10 -11 m /s) is much faster than in d //d (3.0×10 -11 m /s), resulting in the more symmetric ion change in d //d . We further inspect how the asymmetry of electrodes affects the charge storage by analyzing the simulation system consisting of negative and positive electrodes with different-sized pore ( i.e. , the asymmetric electrode system). The capacitance of d //d has an obviously asymmetric behavior ( Fig. 1f ), which is associated with the far beyond zero (
Fig. S5 ). Besides, the capacitance of d //d becomes more symmetric than that of d //d (red circles in
Fig. S6 vs.
Fig. 1f ), as its is closer to zero (
Fig. S5 ). The association between capacitance and for symmetric and asymmetric electrode systems suggests that is a good indicator for evaluating the asymmetry behavior of the capacitance: the symmetric change of ions in cathode and anode leads to the symmetric capacitance behavior, no matter the cathode and anode are symmetric or not. It is worth noting that this symmetric capacitance behavior could facilitate expanding the operating potential window of supercapacitors. [15] Equivalent principle in asymmetric supercapacitor
In supercapacitor, the cathode and anode are separated by electrolyte-filled separator, [33] and therefore the capacitance of the negative electrode in the asymmetric electrode system d //d should be equal to that of the symmetric electrode system d //d under the same electrode polarization, and the positive one equals to that of system d //d . This could be illustrated in
Fig. S7 for the theoretical equivalent principle in asymmetric supercapacitor. As predicted by such equivalent principle, the capacitance of overall system d //d , directly computed from MD simulation, matches well with the theoretical prediction (
Fig. 1f ). The capacitance of system d //d also shows the same matching result (
Fig. S6 ). These results imply that the capacitance of the asymmetric supercapacitor could be directly taken from the symmetric ones. Does this approach work for the charging dynamics? To answer this question, the resistances of ionic transport in cathode and anode were computed for the system d //d (red histograms in
Fig. 1g ), using the equivalent circuit model (
Fig. S8 ), to evaluate the charging dynamics. Meanwhile, analogous with the capacitance, theoretically predicted ionic transport resistances of d //d were obtained (black histograms in
Fig. 1g ). Different from the capacitance, ionic transport resistance predicted from symmetric electrode system significantly deviates from those directly from MD simulations, which is further evidenced by d //d ( Fig. S9 ). These results imply that the ionic transport resistance in one electrode would have a strong correlation with the other electrode. Therefore, the charging dynamics of asymmetric electrode system could not be directly predicted by symmetric electrode system.
Charging dynamics of system with single pore size Figure 2a shows charging curves of systems with single pore size, quantitatively analyzed by equivalent circuit model (
Fig. S10 and Fig. 2b ). Specifically, the symmetric electrode system d //d has the fastest charging with a charging time constant of 0.28 ns, while the charging of the system d //d is the slowest (3.46 ns). The charging dynamics of asymmetric electrode systems d //d and d //d are comprised between those of d //d and d //d . Specifically, the charging of d //d is several times faster than that of d //d (0.51 ns vs. , [34] where is the MD-obtained charging time constant, is thickness of carbon electrode in a real cell, and is the length of the electrode in MD simulation. Therefore, for 200-µm-thick porous carbon electrodes of d //d and d //d will be 1.71 and 0.32 s, respectively, which are compatible with previous experiment showing ~1 s for the charging time of an individual nanopore. [35] The charging performance of the supercapacitor system could be further evaluated by the maximum power density ( ), which is calculated by , [36, 37] where is the applied cell voltage and is the cell resistance estimated by . As presented in Fig. 2b , systems d //d and d //d have the power density of 0.34 W/m and 0.19 W/m , respectively, much higher than that of d //d (0.03 W/m ) and d //d (0.04 W/m ). These results indicate that the pore size and the asymmetry of electrodes have a remarkable influence on the power density. The time-evolution of ion number inside pores was analyzed to disclose the charging mechanism. The charging dynamics of d //d is ruled by co-ion desorption domination in the negative electrode and by counterion absorption domination in the positive electrode, and requires a long time (>20 ns) to reach equilibrium ( Fig. 2c ), while for system d //d , the charging dynamics is dominated by ion exchange, arriving at equilibrium very quickly (<5 ns,
Fig. 2d ). For the asymmetric electrode system d //d , the charging of the positively charged 0.75 nm pore is slowed down by the negative 0.50 nm pore (
Fig. 2e ), while for d //d , the positive 0.50 nm pore gets charged faster as combined with a negative 0.75 nm pore (
Fig. 2f ). To demonstrate why the positive 0.50 nm pore has much faster charging than the negative one (0.51 ns vs.
Fig. 2g reveals that during the charging of the negative 0.50 nm pore, the total number density of ions increases first, achieving a maximum, and then decreases, eventually reaching equilibrium. Herein, the increasing process is called ‘overfilling’, and the decreasing one is de-filling. [38]
Previous simulation work revealed that the phenomenon of the ion overfilling followed by de-filling would slow down the charging dynamics, [38-42] which could account for the slow charging process by the negative 0.50 nm pore in both symmetric and asymmetric electrode systems ( i.e. , d //d and d //d , see
Fig. S11 ). Accordingly, the absence of overfilling in the positive 0.50 nm pore of d //d renders a fast-charging process (
Fig. 2g ). It is worthy of notice that the positive 0.50 nm pore has larger ion density than the negative one (
Fig. 2g ). Both modeling and experiment have suggested that the higher ion density could result in slower ion diffusion. [25, 26, 43]
We then calculated the diffusion of in-pore ions and found the same trend (
Method ). As shown in
Fig. 2h , the diffusion of cations in the negative 0.50 nm pore of d //d (~51.5×10 -11 m /s), comparable with the bulk diffusion (53.8×10 -11 m /s), is much larger than that at PZC (3.0×10 -11 m /s), while in the positive 0.50 nm pore of d //d , the cation diffusion (0.4×10 -11 m /s) is much smaller than at PZC. The same trend is observed for the anion diffusion ( Fig. 2h ). Many studies have demonstrated that the faster-diffused ions help to accelerate charging dynamics in nanoporous electrode. [38, 44, 45]
Surprisingly, our results demonstrate an opposite relationship that the charging is much faster in the 0.50 nm pore of d //d with quite slower diffusion of ions, compared with that of d //d . To understand this abnormal relationship, we investigated the time-evolution of the densities of in-pore cations and anions along the direction of pore length (
Fig. 3 and
Fig. S12 ). For the negatively charged 0.5 nm pore in d //d , cations at the pore entrance (ranges of -4~-2 nm & 2~4 nm) are more crowded than at center (range of -2~2 nm,
Fig. 3b and
Fig. S12a ), while anions at entrance are much fewer (
Fig. 3c and
Fig. S12b ). Based on the change of ion density, the charging dynamics of this pore could be divided into two steps: (i) during 0~2 ns, anions at the pore entrance get out of the pore, meanwhile, more cations get into the pore, mainly accumulating at the entrance; (ii) after ~2 ns, anions in the center move to the entrance and then get out of the pore, while very few cations further move into the pore. In the first step, since cations getting into the pore are more than anions getting out, the overfilling occurs (
Fig. S11c ). For the second step ( i.e. , the de-filling process), anions moving to the pore entrance need to overcome the block of cations accumulated at the entrance, thus, experiencing longer and more roundabout motion paths (
Fig. 3a and
Movie 1 with the description in
Part 4 of SI), which results in a slower charging process. For the positive 0.5 nm pore of d //d , its charging dynamics is driven mainly by the anion absorption, and the cation desorption plays a minor role (
Fig. 1d and
Fig. 2f ). Cations at the pore entrance get out of the pore (
Fig. 3e and
Fig. S12c ), meanwhile, anions enter the pore (
Fig. 3f and
Fig. S12d ). All in-pore anions, including newly-entered ones, are moving towards the pore center (
Fig. S13 and
Movie 2 with the description in
Part 4 of SI), in a manner that each anion only needs to successively occupy its neighboring site where another anion leaves, moving closer to the center. Therefore, anions’ motion paths in the positive 0.5 nm pore of d //d are shorter and straighter, compared with those in the negative 0.5 nm pore of system d //d . This process is similar to multi-ion concerted migrations in ionic conductors, which could reduce energy barriers for ion motion, leading to higher ionic conductivity. [46-48]
While the longer anion motion path in d //d is similar to single-ion migrations, resulting in a larger energy barrier for ion motion. These explain why the negative 0.50 nm pore in d //d , despite its much faster ion diffusion, is charged more slowly than the positive 0.50 nm pore in d //d . Occurrence of over-charging and accelerated de-filling in system with multiple pore sizes
In practice, porous electrodes are composed of pores in different sizes. [49]
Therefore, it is crucial to understand the capacitance and charging dynamics of electrodes with multiple pore sizes (
Fig. 4a ) and their relationship with those of the single pore size system. We obtained prediction values of the capacitance and ionic resistance of each pore in the system d d //d d ( Fig. S14 ) from symmetric electrode systems with single pore size ( i.e. , d //d and d //d ), with the same approach used for the asymmetric electrode system (
Fig. 1f-g ). Results reveal that the capacitance directly obtained by MD simulation matches well with that from the prediction, in accordance with results for electrodes with single pore size (
Fig. 1f and
Fig. S6 ). This implies that the capacitance of systems with multiple pore sizes could be taken from systems with single pore size, which agrees with the previous combined modeling and experiment study that the capacitance of the porous electrode could be estimated through averaging the capacitance of single pores by the weight of pore size distribution. [50] However, for charging dynamics, the in-pore ionic transport resistances in system d d //d d directly from MD simulation mostly deviate from the theoretical prediction, similar to asymmetric electrode systems ( Fig. 1g and
Fig. S9 ). Thereby, we focus on the charging dynamics of electrodes with multiple pore sizes. As shown in
Fig. 4b , the charging of system d d //d d (1.39 ns) is not only between that of symmetric electrode systems with single pore size, but also between asymmetric ones ( Fig. 2b ). We then dig into the charging dynamics of each pore in d d //d d . As seen in Fig. 4c , both negative and positive 0.50 nm pores are charged faster than the system with single pore size (d //d ), suggesting that the very slow charging dynamics of 0.50 nm pore could be enhanced in system with multiple pore sizes. Besides, the charging of positive 0.50 nm pore is faster than the negative one, in agreement with the results of asymmetric electrode systems (d //d vs. d //d , see Fig. 2a ). For the negative 0.50 nm pore, the total number of ions (
Fig. 4d ) shows that the overfilling occurs in the negative 0.5 nm pore of both d d //d d and d //d , and the maximum occupation of ions inside those pores are comparable. While the de-filling of the negative 0.50 nm pore of d d //d d is faster than that of d //d , leading to a faster charging process. The absence of overfilling inside the positive 0.50 nm pore contributes to its faster charging dynamics than that of the negative one. For the positive 0.75 nm pore in d d //d d , the charging is slower than that of d //d ; unexpectedly, an over-charging phenomenon occurs in the charging dynamics of negative 0.75 nm pore: the charge, , increases quickly, and becomes larger than the equilibrium. Then decreases to that at equilibrium ( i.e ., de-charging) ( Fig. 4e ). The over-charging and de-charging phenomena occur in system with multiple pore sizes, but are absent in systems with single pore size (
Fig. 2 ). The over-charging phenomenon could be explained by the time-dependent electrode potential (
Fig. 4f ): the negative electrode of d d //d d is more polarized at the beginning of the charging process than that at equilibrium. Therefore, the electrostatic interaction between electrode and ions in the negative 0.75 nm pore is stronger at the beginning than that at equilibrium. So, more cations and anions could exchange before the electrode potential arrives at equilibrium, resulting in more charge accumulated inside pore. While for the negative 0.50 nm pore, the ion change is much slower than that of 0.75 nm pore, so that the accumulated charge could not reach the maximum before the electrode potential arrives at equilibrium, and thus the over-charging is not able to happen. The over-charging could increase the intermediate surface charge of the electrode, thus enhancing the charging dynamics. As seen in Fig. 4g , the overfilling occurs during the charging dynamics inside negative 0.75 nm pore in d d //d d , and the following de-filling arrives at equilibrium soon (<10 ns), which is different from the situation inside negative 0.50 nm pore where it takes long time to reach equilibrium (>20 ns) ( Fig. 4d ). For the 0.75 nm pore, absorbed cations could move into pore quickly and then would not accumulate at the entrance, so that the anions would not be blocked by cations, leading to their straighter motion paths compared with those in 0.50 nm pore (
Fig. S15 ). On the contrary, roundabout motion paths produce slow charging dynamics of negative 0.50 nm pore, which slow down the overall charging process of system d d //d d . Moreover, during the over-charging process, the number of cations is larger than that at equilibrium ( i.e. , ion overfilling), and the number of anions is smaller than that at equilibrium ( i.e. , ion over-de-filling). It is worth noting that the ion overfilling and over-de-filling are also absent in electrodes with single pore size ( Fig. 2 ). We further found that the ion overfilling occurs in negative 0.50 nm pore of system with multiple pore sizes d d //d d ( Fig. S17 ), and its de-filling is faster than that of d //d . Meanwhile, the over-charging and de-charging phenomena occur in negative 0.45 nm pore of d d //d d . However, those phenomena are absent in system d d //d d that is combined by two single pore size with the symmetric ion change behavior in negative and positive electrodes ( i.e. , d //d and d //d ) . In particular, system d //d has perfect-symmetric ion change ( = 0), since the 0.45 nm pore is not wetted by ions at PZC ( i.e. , ionophobic [38] ), and consequently, only counterions could enter the charged pore when polarized ( Fig. S17d ). As seen in
Fig. S18 , except capacitance, the charging of system d d //d d is very similar to the systems with single pore size, that is, the charging dynamics of 0.45 nm and 0.75 nm pores in d d //d d are almost equivalent to that in d //d and d //d . As could be a good indicator of capacitance and charging dynamics of systems with single pore size ( Figs. 1-2 ), for multiple pore sizes, it is found that the charging of system d d //d d (0.26 ns) is faster than d d //d d (1.39 ns) and d d //d d (0.96 ns), which could be attributed to closer to zero for d d //d d . To conclude the asymmetric behavior of ion change effect on charging dynamics, we explore the correlation between and the charging time constant. As shown in Fig. S19 , closer to zero, representing the more symmetric ion change between positive and negative electrode, would generate faster charging dynamics. On the contrary, far from zero, demonstrating the asymmetric ion change, leads to slow charging dynamics. Conclusion
In summary, we investigated asymmetric features of porous supercapacitors, including the asymmetry of the negative/positive electrode and ion change, and their effect on capacitance and charging dynamics. We found that the capacitance and charging dynamics are correlated strongly with asymmetric ion change, no matter negative and positive electrodes are symmetric or not. Specifically, the net charging mechanism parameter ( ) close to zero could not only lead to the symmetric capacitance behavior but also help to speed up the charging dynamics of the porous electrodes.
For the asymmetric electrode system, simulation revealed that the direction of applied voltage on cathode and anode would dramatically affect their charging dynamics and power density. Counterintuitively, we found that negatively charged narrow pores ( e.g. , 0.50 nm) have a slow charging process, despite its fast ion diffusion. This is attributed to that the overfilling phenomenon occurs during their charging dynamics, leading to longer and roundabout motion paths for co-ion desorption. In contrast, for the positive one, the overfilling is absent and the multi-ion connected migrations help to form shorter and straighter ion motion paths, resulting in faster charging dynamics. Furthermore, we found the overfilling also takes place in electrodes with multiple pore sizes, while the following de-filling is faster than that in electrodes with single pore size, thus enhancing the charging dynamics. Besides, the over-charging occurs during the charging dynamics, which could increase the intermediate surface charge of the electrode, further boosting the charging dynamics. We uncovered the relationship of capacitance and charging dynamics between symmetric electrode systems with single pore size and the systems with asymmetric electrodes or with multiple pore sizes. Our results proved that the capacitance of those systems could be directly obtained by superposing the capacitances of the symmetric electrode system with singe pore size. However, for the charging dynamics, the in-pore ionic transport resistance in one electrode would have strong correlation with that in the other, so that they could not be taken from the system with single pore size, unless the multiple pore system is composed of single pore size systems with the symmetric ion change behavior. Our work builds up a bridge to understand the charge storage and charging dynamics between single-sized pores and porous electrodes, which is vital to design supercapacitors with both high energy and power densities. Methods
Molecular dynamics simulation
In MD simulations, the electrode is modeled by identical slit pores. In detail, the symmetric electrode system with single pore size consists of positive and negative electrodes with the same pore size, while for the asymmetric one, the pore size of the positive electrode is different from the negative one (
Fig. S1a-b ). The pore length was 8 nm and the access widths of the pore could be 0.45, 0.50 and 0.75 nm. The system was placed in a box with a size of 3×3.53×32 nm . The system with multiple pore sizes is modeled by slit pores of two sizes ( Fig. S1c ). The system was placed in a box with a size of 3×7.06×32 nm . The periodic boundary conditions were applied in all three directions and two ILs reservoirs were separating pores. The four-site coarse-grained model of [EMIM][BF ] [51] with larger cation and smaller quasi-spherical anion and the Lennard-Jones model of carbon atoms [52] was employed. All simulations were performed by using a customized MD code based on the software GROMACS. [53] Constant potential method
In our simulations, the applied electrical potential between two electrodes is maintained constant by the constant potential method (CPM), as it allows fluctuations of charges on electrode atoms during simulations, which is significant for charging dynamics. [54, 55]
The CPM is implemented in a methodology proposed by Siepmann et al. [56] and refined by Reed et al. [57] . Our CPM directly applies a potential on electrode atoms instead of the electrode surface and accords well with other implementations of CPM. [58, 59]
The electrolyte temperature was maintained at 400 K using the v-rescale thermostat in the NVT ensemble. Electrostatic interactions were computed using the PME method, and the FFT grid spacing was 0.1 nm. A cutoff distance of 1.2 nm was used in the calculation of electrostatic interactions in the real space. To obtain capacitance behaviors ( e.g. , Fig. 1b ), simulations were performed under the cell voltage ranging from 0 to 5 V. To get the charging dynamics ( e.g. , Fig. 2a ), the system was first taken 20 ns for equilibrated with no applied potential and then was taken 5 ns with a cell voltage of 0 V using constant potential between two electrodes. Then the cell voltage between two electrodes was initially set to 4 V, and the system could equilibrate enough to unsure that the change arrived equilibrium (> 40 ns). Each charging case was carried out more than three times independently. In those approaches, the charge on each electrode was updated at each time step. The diffusion coefficients of ions is calculated by: [60] (3) where is the position of the ion at time , and is the position of the ion at . Here, the diffusion of ions is calculated along the direction of pore length in equilibrium state. Acknowledgments
Authors acknowledge the funding support from the National Natural Science Foundation of China (51876072) and the Hubei Provincial Natural Science Foundation of China (2019CFA002, 2020CFA093). The authors also thank Beijng PARATERA Tech CO., Ltd. for providing HPC resources to accomplish simulations in this work.
This work is also supported by Program for HUST Academic Frontier Youth Team.
Supplementary Information
Details of MD simulation, additional results of ion structure inside pores, capacitance and charging dynamics, movies of charging process. This material is available free of charge via the Internet at http://...
Conflict of interest
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Snapshot of the system with single pore size. The system was named by d s1 //d s2 , representing the system of the negative electrode with pore size of s1 and the positive electrode with pore size of s2.(unit: nm) b, Integrate capacitance, , of symmetric electrode systems versus the electrical potential. c, The number densities of cation and anion, , as a function of the electrode potential. d, The charging mechanism parameter, , as a function of the electrode potential. e, The net charging mechanism parameter, , as a function of cell voltage. f, The integral capacitance, , of asymmetric electrode system d //d obtained by MD simulations and theoretical prediction. g, The resistance, , of ion transport in positive and negative electrodes of system d //d obtained by MD simulation and theoretical prediction. Figure 2. Charging dynamics of systems with single pore size. a,
Comparison of charging process of different systems with a cell voltage of 4 V. represents the electrode surface charge density at equilibrium. b, The charging time constant ( in left y-axis) and power density (right y-axis) of different systems with a cell voltage 4 V. c-f,
Evolution of the number densities of cations and anions, , of systems d //d ( c ), d //d ( d ), d //d ( e ) and d //d ( f ) under negative ( upper plane ) and positive ( lower plane ) polarization. Red lines represent the number density of cation and blue lines represent the number density of anion. g , Evolution of the total number of ion (sum cation and anion), , of negatively charged 0.50 nm pore of d //d (red line) and positively charged 0.50 nm pore of d //d (blue line). h , The diffusion of cation and anion inside negatively charged 0.50 nm pore of d //d , 0.50 nm pore at PZC, and positively charged 0.50 nm pore of d //d . Figure 3. Ion motion paths inside 0.5 nm pores. a,
Top-view snapshot of negatively charged 0.50 nm of system d //d . b-c, Time-evolution of the cations ( b ) and anions ( c ) inside negatively charged 0.50 nm pore of the system d //d along the direction of the pore length, . d, Top-view snapshot of ions inside positively charged 0.50 nm pore of the system d //d . Black arrows represent the anion motion paths. e-f,
Time-evolution of the cations ( e ) and anions ( f ) inside the positively charged 0.50 nm pore of the system d //d , respectively. Red spheres and blue spheres represent the cations and anions, respectively. Write hexagonal spheres are carbon atoms. Black arrows represent the anion motion paths. = 0 is the center of the pore alone the direction of pore length. Unit of colorbar: . Figure 4. Charging dynamics of system with multiple pore sizes d d //d d . a, Snapshot of the system with multiple pore sizes. The system was named by d s1 d s2 //d s3 d s4 , which specifies as the system is composed of the negatively charged pores with sizes of s1 and s2, and the positively charged pores with sizes of s3 and s4. b, Comparison of charging process of system with multiple pore sizes and systems with single pore size. c, Comparison of charging process inside 0.50 nm pore of system d d //d d and d //d . d, Comparison of the ion number density, , inside 0.50 nm pore of system d d //d d and negatively charged 0.50 nm pore of d //d . e, Comparison of charging process inside 0.75 nm pore of system d d //d d and d //d . f , The electrode potential, , versus time in negative electrode in d d //d d . The red solid line represents the time-evolution electrode potential and the black dotted line represents the electrode potential at equilibrium. g , The ion number density, , inside negatively charged 0.75 nm pore of system d d //d d . Supporting Information
Symmetrizing cathode-anode response to speed up charging of nanoporous supercapacitors
Tangming Mo,
Liang Zeng,
Zhenxiang Wang,
Svyatoslav Kondrat, Guang Feng State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China Nano Interface Centre for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, 430074, China Department of Complex Systems, Institute of Physical Chemistry, PAS, Kasprzaka 44/52, 01-224 Warsaw, Poland * Corresponding author’s email: [email protected]
Contents
Part 1. Ion sturcture inside pore ........................................................................ 3
Part 2. The rationale of asymmetric electrode supercapacitor ....................... 5
Part 3. Charging dynamics of system with single pore size ............................ 9
Part 4. Description of movies ........................................................................... 11
Part 5. Systems with multiple pore sizes ......................................................... 12
Part 6. The correlation between charging dynamics and ion change .......... 16 Part 1. Ion sturcture inside pore
As shown in
Fig. S1 , in MD simulation, the electrode is modeled by identical slit pores. The symmetric electrode system with single pore size consists of positive and negative electrodes with the same pore size (
Fig. S1a ), while in asymmetric one, the pore size of positive and negative electrode is different (
Fig. S1b ). Specifically, the symmetric electrode system d //d is composed of two identical pores with a size of 0.50 nm. The asymmetric electrode system d //d is composed of the negative pore with a size of 0.50 nm and the positive pore with a size of 0.75 nm. The system with multiple pore sizes is modeled by slit pores of two sizes (
Fig. S1c ). For instance, the system with multiple pore sizes d d //d d is composed of 0.50 nm and 0.75 nm pores as the negative electrode and 0.50 nm and 0.75 nm pores as the positive electrode. Three typical sub-nanometer pores with sizes of 0.45 nm, 0.50 nm and 0.75 nm were employed. As shown in Fig. S2 , there is no ion inside 0.45 nm pore at PZC ( i.e. , ionophobic [1] ), and only counterions are driven into the pore and accumulate at the central plane of the pore by applied cell voltage. For the 0.50 nm pore, the size of the pore is comparable with the ion size [2] and there is only a single layer of cations and anions accumulating at the central plane of the pore, with or without applied working potential. As for larger pore with a size of 0.75 nm, the EDL structure transits from one layer to two layers with mixing cations and anions.
Figure S1. Schematics of molecular dynamics simulation system setup. a , A schematic of the system with symmetric electrode system with single pore size d //d . b, A schematic of the asymmetric electrode system d //d . c , A schematic of system with multiple pore sizes d d //d d . For all systems, the left electrode is negative and the right one is positive. Table S1:
The potential of zero charge (PZC) at different systems
MD system d //d d //d d //d d //d PZC (V) 0.0024 0.1103 0.0698 0.0698
Figure S2. Number density profiles, , of ions across slit pores with various pore sizes. a-b,
The number density inside pores under the negative polarization ( a ), and the positive polarization ( b ). Red solid lines represent the cation and blue dotted lines represent the anion. The positions of ions are based on their mass center. = 0 represents the central plane between two pore walls. Note: There is no ion inside 0.45 nm pore at PZC, and only counterions are driven into the pore by applied cell voltage. Part 2. The rationale of asymmetric electrode supercapacitor
Figure S3. The surface charge density as a function of electrode potential.
Surface charge density, , versus the electrical potential of symmetric electrode systems d //d and d //d , and asymmetric electrode systems d //d and d //d . Figure S4. Atom number density inside pores at PZC. a-b,
Atom number density inside pore with a size of 0.50 nm (a) and 0.75 nm (b) at PZC. A microstructure sketch of the cation is shown at the upper left corner. ET, IM, ME represent the ethyl, imidazole ring and methyl, respectively. Y =0 is the center plane between two pore walls. Figure S5. The ion change in asymmetric electrode system. a, The charging mechanism parameter, , of asymmetric electrode systems as a function of electrode potential. b, The net charging mechanism parameter, , of asymmetric electrode systems as a function of cell voltage. is divided to two parts: asymmetric ion change ( < -2/3 or > 2/3) and symmetric ion change (-2/3 ≤ ≤ 2/3 ) . Figure S6. The capacitance of asymmetric electrode system d // d . The integral capacitance, , of asymmetric electrode system d //d by simulation and theoretical prediction.
Figure S7. The scheme of equivalent principle in asymmetric supercapacitor . Based on the RC equivalent circuit (
Fig. S8a ), the capacitance ( ) and resistance ( ) of the negative electrode in the asymmetric electrode system d //d should be equal to that of the symmetric electrode system d //d under the same electrode polarization, and the positive one ( and ) equals to that of d //d The charge evolution of RC equivalent circuit model (
Fig. S8a ) is (
Eq. S1 , Ref [3] ): (S1) where is the charging time constant, is the evolution of net charge accumulated on the pore with charging time , and denotes the electrode charge at equilibrium. Specifically, , where and are the total resistance and capacitance, respectively. The total resistance could be obtained by fitting the charging curve. However, the resistance on each side, positive or negative electrode, cannot be obtained in that way. To calculate the resistance of positive or negative electrode, we can calculate the electric current firstly by ( Eq. S2 ): (S2) Then the electrode potential of the positive electrode, , could be calculated from the formula ( Eq. S3 ): (S3) where the and are resistance and capacitance of the positive electrode, respectively. So, the is given by ( Eq. S4 ): (S4) , and can be obtained by the constant potential method based molecular dynamics simulations. In the same way, the resistance of the negative electrode is given by ( Eq. S5 ): (S5) where the and are resistance and capacitance of the negative electrode, respectively. 8 Figure S8. The resistance inside pore. a, The model of RC equivalent circuit. b-c , The potential vs. electric current during charging process inside negative ( b ) and positive ( c ) electrode. Red solid lines are the MD-obtained data and green dotted lines are the fitted results. Figure S9. The resistance of asymmetric electrode system d // d . The resistance, , of asymmetric electrode system d //d obtained by MD simulation and theoretical prediction. is showed by , where is the sectional area of the pore and is the length of the pore. Part 3. Charging dynamics of system with single pore size
Figure S10. Total accumulated charge of the electrodes as a function of time for different systems. a-d , Time evolution of charge density per unit surface area of the pore, , in the positive electrode of system d //d ( a ), d //d ( b ), d //d ( c ) and d //d ( d ) with a cell voltage of 4 V. In each panel, the blue solid line represents the data from molecular dynamics simulations, and the green dashed line represents curves by fitting molecular dynamics data to the equivalent circuit model . Figure S11. Evolution of the total ion number density of difference systems. a-d , Evolution of the total number density of ions inside system d //d ( a ), d //d ( b ), d //d ( c ) and d //d ( d ) under negative ( upper plane ) and positive polarization ( lower plane ). Figure S12. Evolution of the number density of ion in 0.50 nm pore of asymmetric electrode system. a-b , Evolution of the number density of cation ( a ) and anion ( b ) inside negative 0.50 nm pore of system d //d . c-d , Evolution of the number density of cation ( c ) and anion ( d ) inside positive 0.50 nm pore of system d //d . = 0 is the center of the pore alone the direction of pore length. Figure S13. Ion motion paths inside 0.50 nm pores. a,
A snapshot inside positively charged 0.50 nm pore of the system d //d at PZC (Time = 0 ns). b, A snapshot inside positively charged 0.50 nm pore of the system d //d (Time = 10 ns). White hexagonal spheres are electrode atoms and red spheres represent cations. Gray spheres represent anions at ILs reservoirs at Time=0 ns and spheres of other colors represent anions inside pore at Time = 0 ns. Part 4. Description of movies
Two movies showing the top view of ions’ trajectory are made to clarify the charging process in system d //d ( Movie 1 ) and system d //d ( Movie 2 ). In both movies, the gray anions accumulate at ILs reservoirs; the purple anions accumulate at the pore entrance; the yellow anions accumulate between the entrance and the center; and the blue anions locate at the center.
Movie 1 reflects the 10 ns charging dynamics of the negatively charged 0.5 nm pore in system d //d . At the beginning of charging dynamics, the purple anions previously at the entrance get out of the pore gradually. Meanwhile, more red cations get into the pore and accumulate at entrance. Then, more anions, including the blue ones previously at center, move through the entrance to the outside of the pore.
Movie 2 reflects the 10 ns charging dynamics of positively charged 0.5 nm pore of d //d . At the beginning of charging dynamics, it can be found that red cations previously accumulating at the entrance get out of the pore. Meanwhile, gray anions, previously out of the pore, get into the pore. Notably, all anions move toward the center in a successive way and each anion only need to occupy the site of its neighboring anion nearer to the center. Part 5. Systems with multiple pore sizes
Figure S14. The capacitance and ionic resistance system with multiple pore sizes d d //d d . a-b , The integral capacitance, , inside 0.50 nm pore ( a ) and 0.75 nm pore ( b ) of system d d //d d by MD simulation and theoretical prediction from symmetric electrode system with single pore size. c-d , The ionic resistance, , inside 0.50 nm pore ( c ) and 0.75 nm pore ( d ) of system d d //d d by MD simulation and theoretical prediction from symmetric electrode system with single pore size. The theory results are obtained by the single pore size system d //d and d //d under the same electrode potential. is showed by , where is the sectional area of the pore and is the length of the pore. Figure S15. Ion motion paths inside 0.75 nm pore. a-b,
Time-evolution of the cations ( a ) and anions ( b ) inside negatively charged 0.75 nm pore of the system d d //d d along the direction of the pore length, . = 0 is the center of the pore alone the direction of pore length. Unit of colorbar: . For system d d //d d , the capacitance of each pore in the system d d //d d ( Fig. S16 ) is match well with that of prediction values from symmetric electrode systems with single pore size d //d and d //d . The charging of system d d //d d is between that of systems with single pore size d //d and d //d ( Fig. S17a ). For the positive 0.45 nm pore in d d //d d , the charging is slower than that of d //d ( Fig. S17b ). An over-charging phenomenon occurs in the charging dynamics of negative 0.45 nm pore, which is similar with negative 0.75 nm pore in d d //d d . Besides, both negative and positive 0.50 nm pores are charged faster than the system with single pore size (d //d ), suggesting that the very slow charging dynamics of 0.50 nm pore could be enhanced in system with multiple pore sizes ( Fig. S17c ). For the negative 0.50 nm pore, the total number of ions shows that the overfilling occurs in both d d //d d and d //d , and the maximum occupation of ions inside those pores are comparable. While the de-filling of d d //d d is faster than d //d ( Fig. S17e ), so the charging of negative 0.50 nm pore in d d //d d is faster than that of d //d . Figure S16. The capacitance of system with multiple pore sizes d d //d d . a-b , The integral capacitance, , inside 0.45 nm pore ( a ) and 0.50 nm pore ( b ) of system d d //d d by MD simulation and theoretical prediction from symmetric electrode system with single pore size. The theory results are obtained by the single pore size systems d //d and d //d . Figure S17. Charging dynamics of system with multiple pore sizes d d //d d . a, Comparison of charging process of system with multiple pore sizes and systems with single pore size. represents the electrode surface charge density at equilibrium. b, Comparison of charging process inside 0.45 nm pore of system d d //d d and d //d . c, Comparison of charging process inside 0.50 nm pore of system d d //d d and d //d . d, The ion number density, , inside negatively charged 0.45 nm pore of system d //d . e , Comparison of the ion number density, , inside negatively charged 0.50 nm pore of system of system d d //d d and d //d . Figure S18. The capacitance and ionic resistance system with multiple pore sizes d d //d d . a-b , The integral capacitance, , inside 0.45 nm pore ( a ) and 0.75 nm pore ( b ) of system d d //d d by MD simulation and theoretical prediction from symmetric electrode system with single pore size. c, The net charging mechanism parameter, , as a function of cell voltage. d, Comparison of charging process of system with multiple pore sizes and systems with single pore size. represents the electrode surface charge density at equilibrium. e-f , Comparison of charging process inside 0.45 nm ( e ) and 0.75 nm ( f ) of system with multiple pore sizes and systems with single pore size. Part 6. The correlation between charging dynamics and ion change
To conclude the asymmetric behavior of ion change effect on charging dynamics, we explore the correlation between and the charging time constant. As shown in
Fig. S19 , the red markers represent the symmetric electrode system with single pore size ( i.e. , d //d , d //d and d //d ) and results show that d //d and d //d , which are closer to zero, are charging faster. The blue ones are asymmetric electrode system with single pore size ( i.e. , d //d and d //d ) and d //d ( ) is charging faster than d //d ( ). For multiple pore sizes (the black markers), the charging of system d d //d d is faster than d d //d d and d d //d d , which could be attributed to closer to zero for d d //d d . Additionally, of system with multiple pore size d d //d d (0.67) is between systems with single pore size d //d and d //d (1.03 and -0.06), that why the charging of d d //d d is faster than d //d and d //d Figure S19. The correlation between charging time constant and ion change.
The correlation between charging time constant, , and net charging mechanism parameter, , with the cell voltage of 4 V.
Red circles and blue squares represent the data from symmetric ( i.e. , d //d , d //d and d //d ) and asymmetric electrode systems with single pore size ( i.e. , d //d and d //d ), respectively, and black triangles represent the data from system with multiple pore sizes ( i.e. , d d //d d , d d //d d and d d //d d ). References [1] Kondrat, S.; Wu, P.; Qiao, R.; Kornyshev, A. A. Accelerating charging dynamics in subnanometre pores.
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