How Inter-Pore Ionic Interactions Affect the Capacitive Energy Storage

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Feeling Your Neighbours Across the Walls: How InterPore Ionic Interactions Affect the Capacitive Energy Storage Svyatoslav Kondrat, Oleg A. Vasilyev, and Alexei A. Kornyshev J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b01623 • Publication Date (Web): 18 Jul 2019 Downloaded from pubs.acs.org on July 23, 2019

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The Journal of Physical Chemistry Letters

Feeling Your Neighbours Across the Walls: How Inter-Pore Ionic Interactions Affect the Capacitive Energy Storage Svyatoslav Kondrat,1 Oleg A. Vasilyev,2, 3 and Alexei A. Kornyshev4 1Department

of Complex Systems, Institute of Physical Chemistry,

PAS, Kasprzaka 44/52, 01-224 Warsaw, Poland∗ 2Max

Planck Institute for Intelligent Systems,

Heisenbergstr. 3, 70569 Stuttgart, Germany 34th

Institute for Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

4Department

of Chemistry, Imperial College London, Molecular Sciences Research Hub, White City Campus, London W12 0BZ, United Kingdom† (Dated: July 15, 2019)

The progress in low dimensional carbon materials intensified the research on supercapacitors with nanostructured/nanoporous electrodes. The theoretical and simulation work so far has focused on charging single nanopores or nanoporous networks, and the effects due to ionic interactions inside the pores, while the effect of inter-pore ion-ion correlations received less attention. Herein, we study how the interactions between the ions in the neighbouring pores across the pore walls affect the capacitive energy storage. We develop a simple lattice model for the ions in a stack of parallel-aligned nanotubes, and solve it by using the perturbation and ‘semi meanfield’ theories, and test the results by Monte Carlo simulations. We demonstrate that the inter-pore ionic interactions can have a profound effect on the charge storage, in particular, such interactions can enhance or diminish the stored energy density, depending on the sign of like-charge interactions. We also find that the charging can proceed either continuously or via a phase transition. Our results call for more detailed investigations of the properties of carbon pore walls and suggest that tuning their electrostatic response may be promising for rational design of an optimal supercapacitor.

ACS Paragon Plus Environment


2 Table of Content Graphic

Capacitance (—F/cm2 )

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3 Electrical double layer capacitors (EDLCs), also known as super or ultra capacitors, attract growing interest of the research community due to their unique properties, such as remarkable cyclability and high power densities [1]. EDLCs consist of two nanoporous electrodes (typically carbon-based) immersed in electrolyte, separated by an ion-permeable membrane, to prevent short-cut. They store energy by a potential driven separation of charge of ions between the cathode and anode in their nanopores. Much effort has been dedicated to understanding the properties of electrolytes in such narrow confinements [2–7] for maximizing the stored energy density [8–11] and speed of charging [12–15]. In prior theoretical and simulation work, porous electrodes have been modelled as perfectly conducting nanoslits [2, 3, 5, 14, 15] or cylindrical pores [16–18], or even as complex nanoporous networks [4, 13, 19]. The conducting nature of electrodes was taken into account either via analytical interaction potentials, available for some geometries [3, 20], or by modelling the pore surface by Gaussian [21, 22] or ICC∗ [23–25] charges, or by using periodic Green functions [26, 27]. Such modelling assumes that the pore surface is an ideal metal and allows no electrical field penetration, implying no electrostatic interaction between the ions from the neighbouring pores (for sufficiently thick pore walls). However, it is known that thin carbon pore walls are transparent to electrostatic and van der Waals interactions. For instance, Rafiee et al. have reported on ‘wetting transparency’ of graphene, observed for up to 6 graphene layers, and attributed it to the extreme thinness of graphene and its partial transparency to the van der Waals interactions [28]; other examples include lattice transparency [29] and even electron-transfer transparency [30]. In a recent study, MendezMorales et al. have modeled ‘graphene’ layers by Gaussian charges and observed that the ions from neighbouring pores are strongly correlated; in particular, the same-type ions on the two sides of a ‘graphene’ layer prefer to be in front of each other [7]. Similarly, the quantum density functional calculations showed that the ions of the same sign attract each other when placed inside and outside of a single-wall carbon nanotube [31]. The goal of the present work is to study how transparency of pore walls to electrostatic interactions affects the capacitive energy storage in nanoporous supercapacitors. To this end, we consider a stack of carbon nanotubes, arranged on a lattice (Fig. 1), and introduce a simple lattice model, in the spirit of Ref. [16–18], which allows the analytical approximations to be developed, providing clear physical insight. These approximate solutions are then compared with more rigorous Monte Carlo simulations of the same model.


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4

FIG. 1. Model of ions confined in a stack of nanotubes. Ions of diameter d are confined into nanotubes of radius a. The interaction energy between two ions in the same nanotube is characterized by a coupling constant J = J++ = J−− = −J+− = −J−+ > 0 (cf. Eq. (2)). The interaction between the ions from the neighbouring nanotubes is described by a coupling constant K = K++ = K−− = −K+− = −K−+ , which can be positive or negative. The right panel shows schematically the lattice model for nanotube charging and the hexagonal arrangement of nanotubes, used in Monte Carlo simulations.

The simplest model of charge storage in a (single) carbon nanotube is the one-dimensional Ising model [16], for which the Hamiltonian reads (hereafter we use systematically the Gaussian units) βH0 ({Si }) = J

X

Si Sj + u

X

hiji

Si .

(1)

i

Here Si = ±1 is a ‘spin’ at site i, which corresponds to a ‘+’ and ‘−’ ion, respectively; β = (kB T )−1 , with kB being the Boltzmann constant and T temperature, u = βev is the applied potential measured with respect to the bulk electrolyte and expressed in units of thermal voltage, vT = (βe)−1 ≈ 26mV at room temperature, and e is the elementary charge (h = ev plays here the same role as external magnetic field in magnetism). For ions inside a perfectly conducting nanotube, the coupling constant J is [20] (in units of kB T ) J = ψ(r = d; a) ≈ 3.08(λB /a)e−2.4d/a ,

(2)

where ψ(r; a) is the electrostatic interaction energy between two ions located in the center of a pore of radius a, d is the ion diameter (Fig. 1), which plays the role of a lattice


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5 constant in Eq. (1); λB = βe2 /ε is the Bjerrum length and ε the dielectric constant inside the pore. Although Eq. (2) was derived for a metallic cylinder, recent quantum density functional calculations [32, 33] suggest that it provides a reliable approximation also for carbon nanotubes; interestingly, the effective value of a fitting Eq. (2) to these calculations seems to be even smaller than the physical pore radius. The one-dimensional Ising model, Eq. (1), is exactly solvable [34]. The accumulated charge per surface area, which is proportional to the average ‘spin’ hSi0 , is given by [16, 35] Q0 = Qmax hSi0 =

Qmax sinh(u) 1/2 , sinh2 (u) + e4J

(3)

where Qmax = e/(2πad) is the maximum surface charge density. The differential capacitance is C0 (u) = −βe

dQ0 = −(CH λB /a)χ0 (u), du

(4a)

where χ0 (u) =

dhSi0 e4J cosh(u) = 3/2 du e4J + sinh2 (u)

(4b)

is the response function [35] and CH = ε/(2πd). This model and its extensions have turned out to capture surprisingly well the qualitative behaviour of the voltage-dependent capacitance obtained from simulations [17, 18]. We now extend this model to account for inter-pore ion-ion interactions. We consider a system of parallel nanotubes arranged on a lattice and write for the Hamiltonian of the system of ions inside these nanotubes βH({Sµi }) = β

X

H0 ({Sµi }) + K

XX

µ

i

Sµi Sνi ,

(5)

hµνi

where Sµi is spin at i’th site of µ’th nanotube, and the sums over µ and hµνi run over all nanotubes and all pairs of the nearest-neighbour nanotubes, respectively (we consider the interactions in rows of ions between the nearest-neighbour nanotubes). We note that this model is also applicable to the case when the ions reside between the nanotubes, or between and inside the nanotubes (see Fig. S2 for the case of the hexagonal lattice of nanotubes). In Eq. (5), K is a coupling constant characterizing the interactions between the ions from the neighboring nanotubes. As mentioned, a recent molecular dynamics study demonstrated



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6 that the ions of the same sign are positively correlated across a single ‘graphene’ layer, modeled as a layer of Gaussian charges [7]; even more recent quantum density functional calculations showed a similar attraction between the like-charge ions from the inside and outside of a single-wall carbon nanotube [31]. These results suggest that coupling constant K is negative (or ‘ferromagnetic’, using the language of magnetism). Other scenarios are also possible, however. For instance, for a two-layer pore wall, the charge-image charge can form dipoles on each side of the wall, which implies an effective attraction between the oppositely charged ions across the wall and hence positive K. However, in the absence of detailed investigations of the inter-pore ionic interactions, we shall treat K (including its sign) as a phenomenological parameter to be determined explicitly in future work. Model (5) is an anisotropic Ising model in three dimensions in external field (u), and its exact analytical solution is not known. We have therefore decided to apply to this model the perturbation analysis and semi mean-field theory, and test both approaches by Monte Carlo simulations (Supporting Information, Section S1). Assuming that the coupling constant K is small, as compared to J, we employed the standard perturbation theory, which gives for the accumulated charge (Supporting Information, Section S2) Q(u) ≈ Q0 [1 − Knχ0 (u)] ,

(6)

where n is the number of nearest-neighbour nanotubes (e.g., n = 6 for the hexagonal and n = 4 for the square lattice of nanotubes). Equation (6) has a clear physical meaning. If K is negative, i.e., if two ions of the same sign prefer to be near each other when separated by a pore wall [7, 31], then the charging is enhanced in the sense that the pores acquire a higher charge at the same potential, as compared to the case K = 0 (Fig. 2a). The reason is that for negative K the ions (of the same sign) from the neighbouring nanotubes act as an additional effective potential collectively amplifying the nanotube charging (see also the mean-field theory below). In the case of ‘anti-ferromagnetic’ coupling (K > 0), the effect is opposite and higher potentials are required to obtain the same charge as at K = 0, because in this case it is unfavourable for the ions (of the same sign) to face each other across the pore walls. Correspondingly, the peaks in the differential capacitance, C = −βe

dQ , du

(7)

are shifted towards lower and higher potentials, respectively (Fig. 2b). The response function


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a K = −0:1 K=0 K = 0:1

8 6 4 2 0

0

2

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Charge (—C/cm2 )

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b K = −0:1 K=0 K = 0:1

150 100 50 0

0

2

4 6 Voltage, v =vT

8

FIG. 2. Effect of inter-pore ionic interactions on the charge storage and capacitance. (a) Absolute value of the charge accumulated in the pore and (b) differential capacitance as a function of the applied potential from the perturbation analysis (Eq. (6)). All curves were produced for temperature T = 300K, relative dielectric constant inside pores ε = 2.2 (giving the Bjerrum length λB ≈ 25nm), pore radius a = 0.4nm and ion diameter d = 0.7nm; this gives the coupling constant J ≈ 2.9 in units of kB T (Eq. (5)). The number of nearest-neighbour nanotubes n = 6.

in Eq. (6) signifies that the magnitude of this shift depends on the charging properties along the pore, and not only on the direct ion-ion interactions across the pore walls. While the perturbation analysis (Eq. (6)) captures some physics of inter-pore interactions correctly, its quantitative predictions are limited to small values of Kn (Fig. S3). However, it has turned out that a semi mean-field approach, which we describe below, provides surprisingly reliable results in a wider range of parameters. Similarly as in the perturbation analysis above, we assume that, still, K is small as compared to J, and incorporate the effect of the inter-pore ion-ion interactions into the external field (applied potential u). This P P amounts to approximating the last term in Eq. (5) by (K/2) i hµνi sSµi , where s = hSi is an average spin (factor 1/2 is to avoid double counting). The Hamiltonian in Eq. (5) then reduces to the one-dimensional Ising model, but with the effective potential u˜(s) = u + Kns. Note that u˜ is larger (smaller) than u for positive (negative) K, and positive s, which is in line with the perturbation analysis, Eq. (6). Using the exact solution of the one-dimensional Ising model, we have obtained

u(s) = −

h i p Kns 1 − ln(1 − s2 ) + ln −sY + 1 + (Y 2 − 1)s2 2 2


(8a)


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8 and the inverse of the response function p Y − s (Y 2 − 1) / 1 + s2 (Y 2 − 1) s p , χ (s) = Kns − + 1 − s2 sY − 1 + s2 (Y 2 − 1) −1

(8b)

where Y = exp(2K). Equation (8) is a parametric equation for the differential capacitance C = CH λB χ/a, Eq. (7), where CH = ε/(2πd), as before. In addition to the analytical approximations, Eqs. (6) and (8), we have also performed Monte Carlo (MC) simulations of the same model on the three-dimensional hexagonal lattice (Fig. 1). The system size was L3 lattice sites, with L = 32 spins/ions in each direction, and periodic boundary conditions were applied in all three directions (we have checked for the final size effects and did not observe any significant deviation for L > 32) . Each MC step consisted of L3 single spin Metropolis updates [36], and the averaging was performed over 2 × 108 MC steps (Supporting Information, Section S1) Figure 3 compares the results of the mean-field theory (lines) and MC simulations (symbols). The agreement is remarkably good even for a relatively high positive values of K. For negative K, however, our simulations display the emergence of a phase transition between the uncharged (or weekly charged) and highly charged states, corresponding to low and high capacitances, respectively (Fig. 3a and Fig. S3c). The reason is that, for K < 0 and at u = 0, the ionic interactions along the nanotubes stabilize the positions of the same-sign ions from different nanotubes in front of each other (i.e., within the x, y plane, Fig. 1), leading to a strong in-plane ordering (Fig. S4). Thus, at u = 0, the system consists of strongly correlated alternating layers of cations and anions, which creates an additional energy barrier for the charging to commence. The transition occurs when the applied potential is sufficiently strong to overcome this barrier. We note that, at a negative K, the mean-field and perturbation analyses predict slightly enhanced charging at low potentials (Fig. 2 and Fig. 3a), as compared to the K = 0 case, which is opposite to the MC results (showing almost vanishing capacitance before the onset of the transition, Fig. 3a). The reason for this discrepancy is unclear. It is interesting to note, however, that at higher applied potentials, above the transition, the system starts to follow the mean-field predictions, and the theory and simulations agree remarkably well again (Fig. 3a). In stark contrast to the case of negative K, we have not observed any phase transition for K > 0. In this case, the system favours an ‘antiferromagnetic-type’ in-plane ordering of ions from the neighboring nanotubes, i.e., alternation of cations and anions within the x, y


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200

a Energy (—J/cm2 )


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


K = −0:1 K=0 K = 0:1

150 100 50 0

0

2

4 6 Voltage, v =vT

1:5

b K = −0:1 K=0 K = 0:1

1 0:5

8

0

0

2

4 6 Voltage, v =vT

8

FIG. 3. Effect of inter-pore ionic interactions on the differential capacitance and energy storage. (a) Differential capacitance and (b) stored energy per surface area for a few values of the inter-pore coupling constant K. The lines show the results of the mean-field theory (Eq. (8)), and the symbols denote the results of Monte Carlo simulations. The model parameters are the same as in Fig. 2.

plane (Fig. 1). However, the hexagonal lattice, used in our simulations, is inconsistent with such ordering, which causes the system to be frustrated in the x, y plane (Fig. S4). This makes the charging proceed continuously, as predicted by the mean-field theory. (We expect a phase transition to occur for lattices consistent with the ‘antiferromagnetic’ ordering. Our preliminary simulations on a square lattice indicated that this is indeed so; the details will be published elsewhere.) Figure 3b shows the energy stored in a pore per surface area, Z U E(U ) = uC(u)du.

(9)

0

Remarkably, negative K decreases the stored energy density at high applied potentials (close to the charge saturation), even though it provides a much higher peak in the capacitance, as compared to the case K ≥ 0. On the contrary, positive K enhances the energy storage at high voltages, but reduces it at lower potentials. This effect is similar to the effect of pore ionophobicity. Indeed, the charging is hindered or reduced for K > 0, see Eq. (6), which is analogous to making the pore more ionophobic. This shifts the charging process more towards higher potentials (red lines in Fig. 3), leading to higher stored energy densities at high potentials [11], as compared to the case K ≤ 0. The behaviour of the stored energy versus inter-pore coupling constant K is summarized in Fig. 4. It shows that non-zero K can enhance or hamper the energy storage, depend-



10

Energy, E=E(K = 0)

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2 1:5

u = 10vT u = 5vT

1 0:5 0

−0:2 0 0:2 Interpore coupling, K (kB T )

FIG. 4. Effect of inter-pore interactions on the energy storage. Energy E is shown as a function of the interpore coupling parameter K (Eq. (5)), and is expressed in terms of the stored energy at K = 0. The lines show the results of the mean-field approximation (Eq. (8)), and the symbols denote the results of Monte Carlo simulations. The applied voltage 10vT corresponds to the charge saturation (Figs. 2 and 3). The thin dotted line shows the line E = E(K = 0). The model parameters are the same as in Fig. 2. The stored energy can increase or decrease depending on the coupling constant K and applied potential.

ing on the applied potential and the sign of K. The mean-field theory provides a good approximation in the region of positive K, but it is not precise for K < 0, particularly at low potentials. As discussed, this is because for negative K the charging proceeds via a phase transition, which is not captured by the mean-field theory and perturbation analysis (but in all other respects they describe the system behaviour qualitatively well). It is also worth noting that in real systems the disorder and dispersion of pores [11, 16, 37] may smear out the transitions and hence such differences are unlikely to be manifested in experimental data. In summary, we have developed a simple lattice model to study the effect of pore-wall transparency and inter-pore ionic interactions on charging nanoporous electrodes, consisting of a hexagonal stack of nanotubes. Using perturbation analysis, mean-field theory and Monte Carlo simulations, we demonstrated that the inter-pore interactions can enhance or deteriorate charging, depending on the sign of these interactions. Our simulations predict that charging can proceed via a phase transition, but only if the inter-pore interactions between the ions of the same sign are attractive. Qualitatively this is similar to the theory


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11 of absorption on a flat surface of particles with inter-particle attraction [38]. Since there is insufficient information on the interactions between the ions from the neighbouring pores (either nanotubes or slits), we treated such interactions phenomenologically via a single inter-pore coupling constant, K. Clearly, it would be highly interesting to systematically explore the inter-pore ion-ion interaction energy, which can be done, e.g., by using quantum density functional theory [31]. Such calculations would not only shed more light on the effect of carbon pore-wall transparency, and its implications for capacitive energy storage, but may also help to optimize the pore-wall structure in order to judiciously maximize the energy storage in nanoporous supercapacitors. A.A.K. thanks Humboldt foundation for supporting his visit to Max-Planck Institute for Intelligent Systems (Stuttgart), and Professor Dietrich for hospitality and fruitful discussions.

∗

[email protected]; [email protected]

†

[email protected]

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