Optimal Electrode Mass Ratio in Nanoporous Carbon Electrochemical

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Optimal Electrode Mass Ratio in Nanoporous Carbon Electrochemical Supercapacitors Srinivasa Rao Varanasi, and Suresh K Bhatia J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07759 • Publication Date (Web): 18 Nov 2016 Downloaded from http://pubs.acs.org on November 19, 2016

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The Journal of Physical Chemistry

Optimal Electrode Mass Ratio in Nanoporous Carbon Electrochemical Supercapacitors

Srinivasa Rao Varanasi and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

Abstract Electrode mass ratio is one of the important parameters that influence capacitance, cycle life and operating voltage of an electrochemical supercapacitor, however, molecular level understanding of the consequences of inappropriate electrode mass ratio on the overall performance of a supercapacitor is still lacking. Here, we performed constant voltage Gibbs ensemble based grand canonical Monte Carlo simulations on different combinations of microporous carbon electrodes of known atomic structure, and room temperatures ionic liquid as electrolyte. Our results indicate that the optimum mass ratio depends not only on the symmetry of electrodes, but also on the size symmetry of the respective counter-ions. There is an enhancement of 5% in capacitance from the usual mass ratio of 1.0 to the optimum of ~0.7-0.8, however, the imbalance in the masses of electrodes causes overloading of the lighter electrode, causing excluded volume interactions to dominate. We anticipate that highly repulsive interactions and large magnitude of forces on electrode atoms can induce irreversible strain in the electrode structure leading to mechanical instability upon charging and discharging for several cycles, and are possibly responsible for the diminished life time of the supercapacitor device, as is experimentally observed. However, such impact cannot be directly captured in our simulation study since the electrodes were treated frozen. *To whom correspondence may be addressed. Tel.: +61 7 3365 4263. Email: [email protected].

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1. Introduction With ever increasing energy demand in the world, energy storage and delivery stands crucial besides its harvesting. Carbon based supercapacitors are promising alternatives to batteries in future technologies where efficient energy storage and faster delivery is highly desirable. These systems possess superior cycle life, high power densities and moderate energy densities as the storage mechanism is the reversible formation of electric double layers (EDLs)1, involving only physical adsorption of electrolyte ions on the surface of the electrode, with no redox process that leads to corrosion of the electrode in the long run. Supercapacitors based on nanoporous carbon electrodes and room temperature ionic liquid electrolytes have been demonstrated to be an appropriate choice, since they possess high internal surface area2 and pores commensurate with the size of ionic liquid ions3 with the latter enabling higher operating voltages4 and environmentally friendly. Though the power densities, charge-discharge times and life times are far better than their battery counter-parts, efforts have been made to increase their energy densities, and asymmetric capacitors is one possibility being considered, in which one of the electrodes (usually inorganic oxide material), is battery-like where faradaic reactions take place and the other being purely nonfaradaic (EDL based). Such systems deliver superior electrochemical performances. In many theoretical, simulation and experimental studies on supercapacitors as well as capacitive deionization (CDI), the electrodes are taken to be of equal proportions for any electrolyte, and the influence of mass ratio overlooked. However, it has been experimentally found that in these systems, besides other factors, electrode mass ratio is a crucial factor that decides device capacitance, maximum operating voltage and cycle life5-9 and desalination performance of CDI electrodes10. 2


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Duffy et al5 have investigated nickel-carbon asymmetric supercapacitors at various electrode mass ratios and found that the utilization of the nickel hydroxide electrode is dependent on the mass ratio. Apart from asymmetric systems, the effect of electrode mass ratio has also been investigated in carbon-carbon symmetric supercapacitors11-13, for instance, the experimental investigation by He et al12 shows that positive to negative activated carbon (AC) electrode weight ratio of 80:120 delivers best electrochemical performance in terms of specific capacitance (39.6 F/g) and charge-discharge efficiency (97.6%) with 1M spyro (1,1’) bipyrrolidinium tetraflouroborate in propylene carbonate (PC).

However, from the

experimental investigations of Staiti and Lufrano14, the optimum anode/cathode mass ratio in a symmetric AC-AC supercapacitor is between 1.2-1.3 with polymer based electrolyte, having a mobile cation (proton) in hydrated form and an anion (R-SO3 - ) anchored through a polymer chain. The difference in the optimum mass ratios in these two studies can be attributed the asymmetry in the electrolyte ions; i.e. the cation is larger than the anion in the former case, whereas it is opposite in the latter. Comparison between these two independence experimental investigations highlights the role of the size symmetry of the electrolyte ions for a given set of electrodes with similar pore characteristics. However, from the experimental investigations, the microscopic mechanism behind the optimum performance at a particular electrode mass ratio for a given set of electrode-electrolyte combination is not immediately known, and theoretical and simulation tools are necessary to investigate supercapacitor performance at the atomic level. Monte Carlo (MC) and molecular dynamics (MD) simulation techniques have emerged as promising tools to obtain molecular level insights of the electrochemical performance of supercapacitor materials15-22. There are numerous MD and MC simulation studies on various 3



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interesting aspects associated with supercapacitors such as capacitance scaling with pore size23 and its dispersity24, various regimes of voltage dependence on capacitance25, dielectric permittivity26, electrode curvature effect15, 27. However, there is no simulation reported till date on the intriguing influence of electrode mass ratio on supercapacitor performance.

Here, we report results from the Gibbs ensemble based grand canonical Monte Carlo simulations performed on symmetric and asymmetric electrode combinations comprising activated carbon (ACF-15), SiC-DC and slits made up of pristine graphene sheets with a size asymmetric RTIL, 1-ethyl-3methylimidazolium boron tetrafluoride (EMI-BF4) and a size symmetric mono-site RTIL as electrolytes. Analysis of our simulation data indicates that the electrode mass ratio is one of the factors that influence the performance of a supercapacitor, and the optimum electrode mass ratio depends upon symmetry in the pore structure between the electrodes and size symmetry between the electrolyte ions.

2. Computational Details: Electrode models: We have considered three different microporous carbon structures, namely, activated carbon fiber (ACF-15), SiC-DC and pristine graphene. Atomistic models of ACF-15 and SiC-DC have been developed using hybrid reverse Monte Carlo (HRMC) simulation as reported elsewhere28-29. We have taken symmetric ACF-15 (cathode) – ACF-15 (anode), graphene (cathode) – graphene (anode) and asymmetric ACF-15 (cathode)- SiC-DC (anode) electrode combinations. Graphene electrodes have been modelled as slits in which the distance between two adjacent graphene planes is 0.75 nm. In most of the simulations, the mass of the anode has been fixed and that of the cathode has been varied in order to investigate the electrode mass ratio effect. In the simulations involving ACF-15-ACF-15 combination, the anode 4


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comprises of three unit cells whereas the cathode is varied between 1 and 6 unit cells, giving rise to anode/cathode mass ratio between 0.5-3.0.

In most cases of ACF-15-SiC-DC

combination, the anode comprises one unit cell and the ACF-15 cathode has been varied between 1 and 5 unit cells, yielding anode/cathode mass ratio between 0.51 and 2.58. In some cases, we have taken two SiC-DC units cells as anode. In case of graphene-graphene combination, we have taken 5 sheets of graphene separated by a distance of 0.75 nm (slit width) for the anode, and the cathode has been varied between 2 to 10 sheets with same slit width. An external electrolyte layer of 4 nm has been considered in all the electrodes in each case in order to account for the contribution of external double layers, which is important for the convergence of the results, as found in our earlier study30. The distance between electrodes is 10 nm along the z-axis and periodic boundary conditions are imposed along x and y axes. Electrolyte model and intermolecular potential: The size asymmetric (asymm) ionic liquid has been modelled using a coarse grained four site model proposed by Roy et al31, in which each imidazolium cation is represented by three connected sites whereas BF4 anions are represented as single sites (Figure 1(a)). Each site in the electrolyte ions is treated as a Lennard-Jones (LJ) sphere with a point charge placed on it. The potential parameters have been taken from the work of Merlet et al32. In the size symmetric (symm) electrolyte, both cation and anion are modelled as single sites with the same LJ parameters (σii = 0.506 nm, ϵii = 4.71 kJ/mol) with equal and opposite charges (+/0.78e) (Figure 1(b)). Following our earlier work33 and that of Vatamanu et al.34, the charge on each carbon atom of the electrodes is modelled as a Gaussian distribution centered on the atom, originally proposed by Siepmann and Sprik35, and applied by Reed et al36 for the first

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time in the electrochemical context. The total interaction energy of the system, consisting of two carbon electrodes and ionic liquid electrolyte is given by34:

Φ to ta l = Φ L J + Φ E (1) where Φ LJ

ΦE =

∑

i , j ,i ≠ j

 σ ij = ∑ 4 ε ij      rij i , j ,i ≠ j 

qi q j 4 πε r ε 0 rij

−

1 2

∑

  

i , j ,i ≠ j

    q i q j erfc ( γ rij )

12

 σ ij −  rij 

  

6

4 πε r ε 0 rij

(2)

+

∑ i

q i2 γδ ( i , G )

π

(3)

Here rij is distance between two atoms (or ions) i, j = 1, 2, qi and qj are partial charges on atoms (or ion sites) i and j respectively. σij, εij are Lennard-Jones (LJ) parameters for the i-j interaction, εo, εr and 1/γ are permittivity of vacuum, relative permittivity and width of the Gaussian distribution respectively. The first term in eqn. (1) represents the contribution from all the LJ interactions (eqn. (2)), except those between electrode atoms, while the second term (eqn. (3)) represents that from electrostatic interactions. The second and third terms in eqn. (3) are related to the Gaussian nature of the electrode charges. The delta function in eqn. (3) is defined to be zero for a point charge and unity for Gaussian charge distribution. The Modified Wolf method37-40 has been used to calculate the first term in eqn. (3). The convergence parameter in the Wolf method is set to be 1.5 nm-1, ensuring convergence of electrostatic energies to those obtained by the Ewald summation approach. The cut-off length for all the interactions is set to be 14.75 Å in all the simulations, equivalent to half of the smallest dimension of the simulation cell. The specific width of the Gaussians (1/γ) is chosen to be 0.2 Å to ensure that the short range approximation to the second term of eqn. (3) is valid within the cut-off.34 The intramolecular degrees of freedom of the porous electrode structures are assumed to be frozen, and hence the LJ interactions between the atoms of the electrodes are not taken into account. This is justified by electrochemical dilatometry studies41 6


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indicating only limited dilatation due to pumping/injecting in and out the electrodes during the charging-discharging cycles. The cross interaction parameters between dissimilar atoms are calculated using standard Lorentz-Berthelot mixing rules.

All the simulations are

performed using ϵr = 1.

Figure 1: Schematic of the structural models of the (a) size asymmetric electrolyte in which the cationic sites S1, S2 and S3 (yellow spheres) are the coarse-grained representations of imidazolium ring, methyl and ethyl groups respectively, whereas the anion (BF4) is modelled as mono-atomic site (pink sphere), and (b) The size symmetric electrolyte in which the cation and anion are modelled as mono-atomics sites and are represented by blue and pink spheres respectively. Simulation procedure Constant voltage Gibbs ensemble based GCMC simulations: Gibbs ensemble based constant voltage grand canonical Monte Carlo simulations have been carried out on porous carbon electrodes, following the algorithm of Punnathanam42, as

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implemented in our previous works30, 33. The grand canonical partition function at constant voltage is given by: ∞

∞

A =−∞

N =0

Ξ(µ , Φ,V , T ) =

eβ µ ∑ ∑ σ

[ N +Φ θ t ]

Q ( N ,θt ,V , T )

(4)

where

Q( N ,θt ,V , T ) =

N

∑

N1 A = 0

N

∑Q

AC

( N1 A , N1C , N , θ t , V , T )

(5)

N1C = 0

where

QAC ( N1A , N1C , N,θt ,V , T ) =

(qV )2 N ..... e−βU ds2 N N1A ! N1C ! N2 A ! N2C ! ∫ ∫

(6)

and the probability density distribution is given by:

f (s2N , N1A ,N1C ,N2 A ,N2C ,θt ) ∝

(qV)2N βµN βΦθt −βU e e e N1A!N1C!N2 A!N2C!

(7)

Here, µ, Φ, V and T are chemical potential, applied voltage, volume and temperature respectively. θt is the total charge on the electrode, β is (kBT)-1 and q is eβµ. The number of cations and anions in each of the electrodes are N1C, N2C and N1A, N2A respectively, where 1, 2 are the indices of the electrodes. The following condition has been imposed to ensure charge neutrality in the overall system:

N 1 A + N 2 A = N 1C + N 2 C = N

(8)

Seven different Monte Carlo moves (formulated using Eqn. 6) are performed during the simulation, namely, translation, insertion, deletion, charge plus ion transfer (CPIT), ion transfer (IT), charge transfer (CT) between the electrodes and charge transfer within (CTW) the electrodes. In a translation move, a random ion is chosen from a randomly selected electrode and displaced by a small distance (determined by the maximum allowed acceptance, 35%), and rotated (in case of EMI+) by a small angle with respect to an axis 8


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passing through the center of mass of the selected ion, which is selected randomly. In an insertion (or deletion) move, an ion is created in a randomly chosen electrode at a random position and its counter-ion is chosen according to distance-bias scheme, proposed by Panagiotopoulos and coworkers43-44, using a bias potential, proposed by Rane and Errington45. Similarly in a deletion move, an ion is chosen randomly and its counter-ion is chosen according to the distance-bias scheme. In a charge plus ion transfer move (CPIT), a randomly selected ion (cation or anion) from a randomly selected electrode is transferred to the other electrode, and a charge equivalent to the ion’s charge is added/subtracted to/from the origin/destination electrode respectively in order to preserve charge neutrality of both the electrodes. In an ion transfer (IT) move, a randomly selected ion (cation or anion) from a randomly chosen electrode is transferred to the other electrode, and no charge compensation on the respective electrodes is performed, so that the charge neutrality of the individual electrodes is not preserved. In a charge transfer move (CT), a small amount of charge (maximum of +/- 1.0 e) is transferred between electrodes. The transferred charge is shared equally only between randomly selected 500 atoms in the destination electrode (non-uniform charges among the electrode atoms) and in a similar manner; the same amount is subtracted from the atoms in the origin electrode. Similarly, in a charge transfer within the electrode (CTW) move, a small amount of charge is transferred between two different parts of the randomly chosen electrode. The Metropolis scheme has been followed to accept or reject all the moves, and the acceptance probability for any move is:

 f new  acc PMC = min 1,  move  f old  9


(9)


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where fnew and fold

represent probability density distributions (see Eqn. (6)) for the

microstates after and before the moves respectively. All Monte Carlo moves performed here are given equal priority. Further details of the Monte Carlo moves and their acceptance criterion are reported elsewhere30, 33, 42. The simulations are initiated starting with dry and neutral electrodes (a zero charge on each carbon atom). The voltage has been set at 2V. The system is equilibrated until no new molecules are inserted into the system and the number of molecules only fluctuates around a constant average value, which typically requires 10,000 Monte Carlo (MC) cycles, comprising about 5 million MC moves. In each MC cycle, we have attempted as many MC moves as the number of molecules (ions) in the system but with a minimum of 100 MC moves. Thus, if the number of molecules is less than 100 then there will be 100 MC attempts in each MC cycle. A production run is carried out for about 25,000 MC cycles. The coordinates of the ions and charges on the electrode atoms are stored at the end of each MC cycle during the production run for subsequent analysis. The acceptance ratio for displacement and charge transfer moves is around 35% and it is 1-2% for the remaining moves. The temperature and the bulk activity are set to be 300 K and 0.0001 nm-3 respectively in all the simulations.

3. Results and Discussion We have conducted Gibbs ensemble based grand canonical ensemble simulations on four different electrode-electrolyte combinations comprising ACF-15, SiC-DC and graphene slits electrodes with EMI-BF4 ( size asymmetric) and size symmetric (cation and anion are equal size) electrolytes. Our simulations with electrodes of unequal masses indicate that the extent 10


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and nature of wetting (whether it is counter-ion or co-ion) of the heavier electrode is influenced by the lighter electrode when the masses of the electrodes are significantly different from each other. Typical configurations of ions electro-sorbed inside cathode and anode for mass ratios 3.0 and 0.75 have been illustrated in Figures 2(a) and 2(b) respectively. They show that the cathode is largely wetted by cations (counter-ions) at mass ratios 3.0 and 0.75, while at the mass ratio 3.0, the anode is predominantly wetted by cations (co-ions) rather than anions (counter-ions) and at mass ratio of 0.75, the wetting by anions (counterions) appears to dominate that of cations (co-ions). While the masses of these electrodes are distinctly different, the influence of lighter electrode on the wetting of heavier electrode will be reflected in the effective capacitance of the supercapacitor device.

Figure 2: Snapshots of ion-electrode configurations comprising ACF-15-ACF15 electrodes with size asymmetric electrolyte with anode/cathode mass ratio of (a) 3.0 and (b) 0.;75 . Yellow and pink spheres represent EMI+ (three sites) cation and BF4 (one site) anions respectively. The electrodes are shown in cyan colored bonded representation.

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The integral capacitance of the system has been calculated from:

Cint = Q / ∆V

(10)

where Q is the total charge accumulated on the given electrode, with ‘< >’ representing the ensemble average. The denominator, ∆V is the applied potential difference between the two electrodes. The effective gravimetric specific capacitance of the simulated supercapacitor is given by:

C=

Cint (mc + ma )

(11)

where ‘C’ is the effective specific capacitance of the simulated supercapacitor device, ,mc and ma are the dry masses of cathode and the anode respectively. The effective capacitance is plotted against anode/cathode mass ratio for ACF-15-ACF-15 electrodes in combination with size symmetric and asymmetric electrolytes, ACF-15-SiC-DC and graphene-graphene electrodes with size asymmetric electrolyte in Figure 3. The effective capacitance values are much smaller than the cell capacitances reported by Cericola et al11 with activated carbon electrodes with a solvent based electrolyte, whereas in the present study an RTIL has been used. Figure 3 demonstrates that the capacitance exhibits a maximum at a particular mass ratio of value less than or equal to 1.0, depending on the electrode-electrolyte the combination. The optimum mass ratio for graphene-graphene (slit), ACF-15-SiC-DC electrodes where the capacitance exhibits a maximum is 0.814 with the size asymmetric electrolyte. The capacitance maximum occurs at 0.75 for the ACF-15-ACF-15 electrodes with the size asymmetric electrolyte, whereas it occurs at 1.0 for the same electrodes but with size symmetric electrolyte, indicating that size symmetric electrodes need size symmetric electrolyte for better performance, as expected. Although, the gain in capacitance from the 12


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usual mass ratio of 1.0 to the ratio where it exhibits a maximum is only about 5-10% (for instance, ACF-15-ACF-15 with asymmetric electrolyte) in these cases, it is required to understand the intriguing mechanism behind the loss in capacitance due to mass imbalance between the electrodes.

32 graphene-graphene (asym) ACF15-ACF15 (asym) ACF15-SiC-DC (asym) ACF15-ACF15 (sym)

30

capacitance (F/g)

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28

26

24

22

20 0.5

1.0

1.5

2.0

2.5

3.0

anode/cathode mass ratio

Figure 3: Effective specific capacitance as a function of anode/cathode mass ratio for ACF15-ACF-15 electrodes with size asymmetric (red squares) and size symmetric (pink rhombuses), ACF-15-SiC-DC electrodes (blue down triangles) and graphene-graphene (black circles) with size asymmetric electrolyte. The error bars have been estimated as the standard deviations from the average value calculated from the 5 equal parts of the simulated trajectory. The solid lines are drawn to guide the eye and the dashed line at mass ratio 1.0 is drawn to distinguish cathode dominating (left) and anode dominating (right) regions. We note here that the changes that occur in the fixed mass electrode (while the other electrode’s mass being varied) are crucial in understanding the capacitance variation with 13



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mass ratio. In order to get deeper insights into the mass imbalance impact on the device capacitance, we have first analyzed the residual porosity of the electrodes at all the mass ratios studied. Residual porosity is a measure of the volume unoccupied by the electrolyte by the end of the charging process and is defined by:

γ res = γ empty −

ions VVDW Vsolid

(12)

Here γres and γempty are the residual (unoccupied) and empty electrode porosities respectively. The porosity of the empty electrode has been estimated to be 0.615, using helium probe. VVDW ions

is the van der Waals (vdW) volume of adsorbed electrolyte ions and Vsolid is the volume of

porous solid. Figure 4 illustrates the variation of the residual porosity of the electrodes of ACF15-ACF15 (asymmetric electrolyte) system with mass ratio. It is seen that the residual porosity of the cathode decreases significantly between mass ratios of 0.5 and 3.0, whereas the anode’s residual porosities varied only within 10% in the entire range of mass ratios. Since the anode mass is fixed in our simulations, it is apparent that the residual porosity of the cathode, the lighter electrode, decreases quite significantly when anode to cathode mass ratio is greater than 1.0. This indicates that electrolyte is being forced to occupy more porous volume because of the larger electrolyte capacity of the counter-electrode and the global charge neutrality condition. The porosity changes in the anode, the lighter electrode, when the mass ratio is below 1.0, are comparatively small, which can be attributed to the size asymmetry between the cation and anion. The drastic decrease in residual porosity of the lighter electrode for mass ratios above 1.0 can be a result of overcrowding; this is readily shown based on interaction energies between ions and electrode atoms, and the forces acting on the electrode atoms, and is subsequently discussed in this section. 14


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Although residual porosities calculated from vdW volumes of the occupied electrolyte ions give reasonable estimates, it is controversial to define ionic volume in presence of attractive interactions such as dispersive, electrostatic interactions and for the molecular ions which are non-spherical.

0.28

Cathode Anode (fixed)

0.26 ACF15-ACF15 (asymmetric) 0.24

residual porosity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.22

0.20

0.18

0.16

0.14 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

mass ratio

Figure 4: Variation of residual porosity of cathode and anode as a function of mass ratio in ACF15-ACF15 and asymmetric electrolyte combination. The residual porosity is the difference between the porosity of empty electrode (0.615 with helium probe) and fraction of the volume (vdW) occupied by the electrolyte. Besides analyzing the unoccupied volume in the electrodes, to track the changes in the wetting behavior of the anode (fixed mass) with the electrolyte ions, following our earlier work33, we have defined a wetting length, a parametric distance between the ion and carbon atom, within which the number of carbon atoms around a given ion type (cation or anion) are enumerated. The fraction of carbon atoms within the wetting length from the electrolyte ions 15



electro-sorbed inside the fixed mass electrode (anode) have been evaluated at various wetting lengths between 0.3 and 0.5 nm. The variation of fraction of carbon atoms with wetting length, by anion and cation inside the fixed mass ACF-15 anode is illustrated in Figures 5 and 6 respectively. At the mass ratio of 3.0, the amount of carbons wetted by anions (counterions) is about 28% for 0.5 nm wetting length and this fraction increases with decrease in mass ratio or increasing cathode mass (see Figure 5). The fraction of carbons atoms covered within 0.5 nm from the anion (counter-ion) is about 45% for 0.75 and did not increase greatly until the lowest mass ratio (0.5) studied here. As illustrated in Figure 6, the amount of carbons covered by cations (co-ions) is about 32% at a wetting length of 0.5 nm and mass ratio of 0.75, and this fraction of carbon atoms decreases with decrease in mass ratio (increasing cathode mass). The trends of cations coverage at 0.5 nm wetting length between the mass ratios 0.75 and 0.5 is exactly opposite to that of anions, indicating there is little change in the number of ions adsorbed into the fixed mass ACF-15 electrode (anode) despite the increase in the mass of the cathode. 0.6

anode wetted by anion anode/cathode mass ratio 0.5

fraction of carbons

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.5 0.6 0.75 1.0 1.5 3.0

0.4

cathode mass increasing

0.3

0.2

0.1

0.0 3.0

3.5

4.0

4.5

5.0

wetting parameter (Å)

Figure 5: Fraction of carbon atoms wetted by the anions (counter-ions) in the ACF-15 anode of ACF15-ACF15 electrode pair with asymmetric electrolyte, as a function of wetting length. The ‘arrow’ indicates increasing cathode mass for the curves from bottom to top. 16


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0.5 anode/cathode mass ratio 0.5 0.6 0.75 1.0 1.5 3.0

0.4

fraction of carbons

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.3

anode wetted by cation

cathode mass increasing 0.2

0.1

0.0 3.0

3.5

4.0

4.5

5.0

wetting parameter (Å)

Figure 6: Fraction of carbon atoms wetted by the cations (co-ions) in the ACF-15 anode of ACF15-ACF15 electrode pair with asymmetric electrolyte, as a function of wetting length. The ‘arrow’ indicates the decreasing cathode mass for the curves from top to bottom. Having analyzed the wetting behavior of the fixed mass electrode (anode), we further note that the counter-ion-co-ion association inside the electrode is crucial for its charging behavior. From our earlier work33 and work from other groups15, 46, it has been demonstrated that the coordination environment around a counter-ion is the key for efficient charging despite the electrode being highly accessible to the electrolyte. Figure 7 illustrates the distribution of co-ion (cation) coordination around the counter-ion (anion) in the ACF-15 (fixed mass) anode at various anode/cathode mass ratios (varying cathode mass). It shows that the distribution is maximum for coordination numbers 3 and 4 for mass ratio of 3 whereas the maximum shifts towards the value of 2 with decreasing mass ratio which is in accordance with our earlier observation on wetting behavior (c. f. Figures 5 and 6). The average coordination number in the anode (fixed electrode) at mass ratio 3.0 is 3.45 and it decreases to 2.48 when the mass ratio decreased to 0.75. Also, the coordination number distribution does not change significantly for mass ratios between 0.75 and 0.5 (with 17



average coordination number of 2.515), except that the peak value of the distribution function at the coordination number is higher at 0.75 than at 0.6 and 0.5, which supports our earlier observation of maximum capacitance at the mass ratio 0.75 (Figure 2). Also, we note here that the average coordination number does not give sufficient insights into the charging mechanism since these distributions are broad and skewed. Although, the device capacitance is a combined result of the two electrodes, the lighter electrode (at mass ratio 0.75, the anode is lighter) determines the overall capacitance. Upon decreasing the mass ratio from 0.75 to 0.6 and 0.5, the coordination number distribution shifts towards higher coordination numbers, and as a result the induced charge on the electrode is reduced, leading to diminished capacitance of the electrode.

0.35


inside ACF15 anode 0.30

3.0 1.5 1.0 0.75 0.6 0.5

0.25

fraction of anions

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0.20 0.15 0.10 0.05 0.00

0

2

4

6

8

10

anion coordination number

Figure 7: Distribution of cation (co-ion) coordination around an anion (counter-ion) inside ACF-15 anode (fixed mass) at varying anode/cathode mass ratios (varying cathode mass). The coordination numbers are calculated using the first minimum distance of the radial distribution function between the anions and cations. 18


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As demonstrated earlier33, the de-coordinated ions (ions with low or no coordination) induce maximum charge on the surrounding electrode atoms and the induced charge distributions reflect the nature of the pore structure around the counter-ion, for instance the graphitic domains will attain a nearly symmetric distribution around the peak value, whereas nongraphitic domains show skewed distributions15.

Figure 8 illustrates the distribution of

induced partial charges on the individual atoms of the ACF-15 anode (fixed mass) at different anode/cathode mass ratios (varying cathode mass). It shows that the distributions are nearly symmetric around the peak value, indicating that ACF15 contains graphitic domains which are also evident from the snapshots shown in Figure 2. The peak height of the distribution decreases with decrease in the mass ratio from 3.0 to 0.75 and the decrease in the peak height is compensated by the extension of both positive and negative wings which is a result of the entry of additional ions (cations and anions). The distribution does not change much upon decreasing the mass ratio from 0.75 to 0.5 (increasing the cathode mass) which indicates that the number density of ions electrode-sorbed into anode did not increase upon increasing cathode mass. The results shown in Figures 5, 6, 7 and 8 for other electrode-electrolyte combinations are similar and therefore we focus on the ACF15-ACF15 system to avoid repetition of discussion.

19



12

ACF15 - anode (fixed)


10

distribution function

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3.00 1.50 1.00 0.75 0.60 0.50

8

6

4

2

0 -0.2

-0.1

0.0

0.1

0.2

charge on carbon atom (e)

Figure 8: Distribution of individual partial charges of the electrode atoms in the fixed mass ACF15 anode at various anode/cathode mass ratios (varying mass of the cathode). The y-axis (distribution function) represents the fraction of atoms divided by the bin width (0.001e), where ‘e’ is the electronic charge. While the above analysis suggests that an inappropriate electrode mass ratio brings adversary changes in the capacitance behavior of the simulated supercapacitor device, we further note that the study of changes in the energetics in the lighter electrode (low mass) will help elucidating the mechanisms behind the relative loss of capacitance after several chargedischarge cycles or lower cycle life associated with devices of inappropriate mass ratio compared to those with appropriate mass ratio11. Figure 9 depicts the variation of ensemble averaged Lennard-Jones (LJ) energy between ions and electrode atoms per ion electro-sorbed into the lighter electrode for each mass ratio studied. It is to be noted that for mass ratios greater than or equal to 1.0, the lighter electrode is the cathode, and it is the anode otherwise. 20


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It shows that the ion-electrode LJ energy per ion is at a minimum at the mass ratios where the capacitance shows a maximum (Figure 3). The ion-electrode energy inside the lighter electrode increases sharply when the mass ratio is increased/decreased from the optimum value. Although the electrodes in the present study are treated as frozen (no intramolecular degree of freedom for carbon atoms), it should be noted that the relatively higher repulsive ion-electrode LJ energy when the mass ratio is varied beyond the optimum value, is an indication of increased repulsion between ions and electrode atoms due to overcrowding. However, the consequences of such increased repulsion can only be seen through the forces on the electrode atoms due to ions electro-sorbed into the electrode, particularly the lighter one. 0 Lighter electrode -5

Eion-electrodeLJ (kJ/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-10

-15

-20

ACF15-ACF15 (asym) ACF15-ACF15 (symm) ACF15-SiC-DC (asym) graphene-graphene (asym)

-25

-30 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0


Figure 9: Average ion-electrode Lennard-Jones energy per ion in the lighter electrode of the simulated supercapacitor device. In present work, the cathode is the lighter electrode when the anode/cathode mass ratio is greater than or equal to 1.0 whereas it is the anode otherwise. The legends ‘asymm’ and ‘symm’ stand for asymmetric and symmetric electrolytes respectively.

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Figure 10 (a) shows the variation of magnitude of the average force on the electrode atoms due ion-electrode LJ interactions, for those in the lighter electrodes as a function of anode/cathode mass ratio. The values of the forces are within the range of forces measured on carbon electrode-electrolyte interface using atomic force microscopy (AFM)47. The average magnitude of force acting on the atoms of the lighter electrode increases significantly between the mass ratios 1.0 and 3.0 in all the systems we have studied, which supports the observation based on the results presented in Figure 3 and Figure 9, that the increased repulsive short-range interactions and magnitude of the corresponding forces on the lighter electrode at mass ratios away from where maximum capacitance occurs are due to overcrowding whereas the magnitude of average force on the atoms in heavier electrode decreases at mass ratios away from the optimal mass ratio (see Figure 10(b)) since there is more unoccupied volume than in the lighter electrode (see Figure 4). The repulsive force on the lighter electrode atoms increases significantly further when the mass ratio is further away from the optimal value, despite a significant decrease in the force on the heavier electrode atoms. Such an increase in the magnitude of force on the atoms of the lighter electrode can have adverse effects on its mechanical stability by inducing more strain than at optimal mass ratio, on each charge-discharge cycle, and may lead to mechanical collapse of the structure of the electrode in fewer cycles than for electrodes at the optimal mass ratio. However, such mechanical effects cannot be captured in the present study since the electrodes are considered to be frozen. A detailed analysis of mechanical impacts on the electrode upon charging with a flexible potential for the electrodes will be considered in future studies.

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3.5

magnitude of average force (pN)

(a) ACF15-ACF15 (Asymm) ACF15-ACF15 (Symm) Lighter electrode ACF15-SiC-DC (Asymm) Graphene-Graphene (Asymm)

3.0

2.5 Lighter Electrode 2.0

1.5

1.0

0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

3.0

3.5


2.2

(b)

2.0

magnitude of average force (pN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ACF15 - ACF15 (Asymm)

cathode anode

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.0

0.5

1.0

1.5

2.0

2.5


Figure 10: Variation of magnitude of average force on the electrode atom due to ions adsorbed (a) for the lighter electrode in all the systems studied, and (b) for both the electrodes (ACF15-ACF15 with size asymmetric electrolyte) with the mass ratio. Cathode is the lighter electrode when mass ratio is greater than 1.0 and it is the anode otherwise. The legends ‘asymm’ and ‘symm’ stand for size asymmetric and size symmetric electrolytes respectively.

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4. Conclusions Electrode mass ratio is one of the key parameters for the electrochemical performance of the carbon supercapacitors. We have conducted Gibbs ensemble based GCMC simulations on three different sets of electrodes, namely, ACF15-ACF15, ACF15-SiC-DC and graphenegraphene in combination with size symmetric and asymmetric electrolytes. As the above results suggest, the filling of one electrode is largely limited by the mass of its counterelectrode especially when the mass ratio is far below or above its optimum value. For instance, the residual porosity of the lighter electrode decreases significantly when the mass ratio is away from where maximum capacitance is seen, due to overcrowding of ions, which is a consequence of global charge neutrality (the total charge of the system is zero). The optimum mass ratio is dependent on the size symmetry between the ions of the electrolyte. In the case of a size symmetric electrolyte in combination with symmetric electrodes (same material for both electrodes), the optimum mass ratio is found to be 1.0 as expected, and the case in which either the electrolyte ions or the electrodes are asymmetric, the optimum mass ratio is not 1.0. When the electrode mass ratio is quite different from the optimum mass ratio, the ion-electrode LJ energy per ion in the lighter electrode becomes much more repulsive and magnitude of the force on the electrode atom is greater than that at the optimum mass ratio. It is possible that the excess repulsive energy and high magnitude of the corresponding force on the electrode atoms is responsible for the gradual irreversible structural collapse of the lighter electrode, ultimately leading to diminished performance after several charge-discharge cycles, as observed in experimental investigations11. However, due to intra-molecular degrees of freedom of electrodes being frozen in our systems, such adverse impacts on electrodes cannot be directly realized in our simulation study. A detailed study with consideration intra-

24


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molecular degrees of freedom of electrodes is required to investigate the mechanical impacts on the electrode structure upon charging/discharging.

5. Acknowledgements This research has been supported by a grant (DP150101824) from the Australian Research Council under the Discovery Scheme.

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TABLE OF CONTENTS GRAPHIC

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Optimal Electrode Mass Ratio in Nanoporous Carbon Electrochemical

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