Understanding charge storage in low-dimensional electrodes is crucial for developing novel ecologically friendly devices for capacitive energy storage and conversion and water desalination. Exactly solvable models allow in-depth analyses and essential physical insights into the charging mechanisms. So far, however, such analytical approaches have been mainly limited to lattice models. Herein, we develop a versatile, exactly solvable, one-dimensional off-lattice model for charging single-file pores. Unlike the lattice model, this model shows an excellent quantitative agreement with three-dimensional Monte Carlo simulations. With analytical calculations and simulations, we show that the differential capacitance can be bell-shaped (one peak), camel-shaped (two peaks), or have four peaks. Transformations between these capacitance shapes can be induced by changing pore ionophilicity, by changing cation–anion size asymmetry, or by adding solvent. We find that the camel-shaped capacitance, characteristic of dilute electrolytes, appears for strongly ionophilic pores with high ion densities, which we relate to charging mechanisms specific to narrow pores. We also derive a large-voltage asymptotic expression for the capacitance, showing that the capacitance decays to zero as the inverse square of the voltage, C ∼ u−2. This dependence follows from hard-core interactions and is not captured by the lattice model.
REFERENCES
1.
J.
Vatamanu
, M.
Vatamanu
, and D.
Bedrov
, “Non-Faradic energy storage by room temperature ionic liquids in nanoporous electrodes
,” ACS Nano
9
, 5999
(2015
).2.
M.
Salanne
, B.
Rotenberg
, K.
Naoi
, K.
Kaneko
, P.-L.
Taberna
, C. P.
Grey
, B.
Dunn
, and P.
Simon
, “Efficient storage mechanisms for building better supercapacitors
,” Nat. Energy
1
, 16070
(2016
).3.
Z.
Li
, T.
Mendez-Morales
, and M.
Salanne
, “Computer simulation studies of nanoporous carbon-based electrochemical capacitors
,” Curr. Opin. Electrochem.
9
, 81
(2018
).4.
K.
Xu
, H.
Shao
, Z.
Lin
, C.
Merlet
, G.
Feng
, J.
Zhu
, and P.
Simon
, “Computational insights into charge storage mechanisms of supercapacitors
,” Energy Environ. Mater.
3
, 235
(2020
).5.
P.
Simon
and Y.
Gogotsi
, “Perspectives for electrochemical capacitors and related devices
,” Nat. Mater.
19
, 1151
(2020
).6.
C.
Cruz
and A.
Ciach
, “Phase transitions and electrochemical properties of ionic liquids and ionic liquid—Solvent mixtures
,” Molecules
26
, 3668
(2021
).7.
J. R.
Miller
and P.
Simon
, “Materials science—Electrochemical capacitors for energy management
,” Science
321
, 651
(2008
).8.
P.
Simon
and Y.
Gogotsi
, “Materials for electrochemical capacitors
,” Nat. Mater.
7
, 845
(2008
).9.
F.
Béguin
, V.
Presser
, A.
Balducci
, and E.
Frackowiak
, “Carbons and electrolytes for advanced supercapacitors
,” Adv. Mater.
26
, 2219
(2014
).10.
A.
González
, E.
Goikolea
, J. A.
Barrena
, and R.
Mysyk
, “Review on supercapacitors: Technologies and materials
,” Renewable Sustainable Energy Rev.
58
, 1189
(2016
).11.
D.
Brogioli
, “Extracting renewable energy from a salinity difference using a capacitor
,” Phys. Rev. Lett.
103
, 058501
(2009
).12.
A.
Härtel
, M.
Janssen
, D.
Weingarth
, V.
Presser
, and R.
van Roij
, “Heat-to-current conversion of low-grade heat from a thermocapacitive cycle by supercapacitors
,” Energy Environ. Sci.
8
, 2396
(2015
).13.
M.
Janssen
and R.
van Roij
, “Reversible heating in electric double layer capacitors
,” Phys. Rev. Lett.
118
, 096001
(2017
).14.
S.
Porada
, R.
Zhao
, A.
van der Wal
, V.
Presser
, and P. M.
Biesheuvel
, “Review on the science and technology of water desalination by capacitive deionization
,” Prog. Mater. Sci.
58
, 1388
(2013
).15.
M. E.
Suss
and V.
Presser
, “Water desalination with energy storage electrode materials
,” Joule
2
, 10
(2018
).16.
Y.
Zhang
, P.
Srimuk
, M.
Aslan
, M.
Gallei
, and V.
Presser
, “Polymer ion-exchange membranes for capacitive deionization of aqueous media with low and high salt concentration
,” Desalination
479
, 114331
(2020
).17.
J.
Chmiola
, G.
Yushin
, Y.
Gogotsi
, C.
Portet
, P.
Simon
, and P. L.
Taberna
, “Anomalous increase in carbon capacitance at pore sizes less than 1 nanometer
,” Science
313
, 1760
(2006
).18.
E.
Raymundo-Piñero
, K.
Kierzek
, J.
Machnikowski
, and F.
Béguin
, “Relationship between the nanoporous texture of activated carbons and their capacitance properties in different electrolytes
,” Carbon
44
, 2498
(2006
).19.
C.
Largeot
, C.
Portet
, J.
Chmiola
, P.-L.
Taberna
, Y.
Gogotsi
, and P.
Simon
, “Relation between the ion size and pore size for an electric double-layer capacitor
,” J. Am. Chem. Soc.
130
, 2730
(2008
).20.
C. C.
Rochester
, A. A.
Lee
, G.
Pruessner
, and A. A.
Kornyshev
, “Interionic interactions in electronically conducting confinement
,” ChemPhysChem
16
, 4121
(2013
).21.
A.
Goduljan
, F.
Juarez
, L.
Mohammadzadeh
, P.
Quaino
, E.
Santos
, and W.
Schmickler
, “Screening of ions in carbon and gold nanotubes—A theoretical study
,” Electrochem. Commun.
45
, 48
–51
(2014
).22.
L.
Mohammadzadeh
, A.
Goduljan
, F.
Juarez
, P.
Quaino
, E.
Santos
, and W.
Schmickler
, “Nanotubes for charge storage–towards an atomistic model
,” Electrochim. Acta
162
, 11
(2015
).23.
L.
Mohammadzadeh
, P.
Quaino
, and W.
Schmickler
, “Interactions of anions and cations in carbon nanotubes
,” Faraday Discuss.
193
, 415
(2016
).24.
C.
Merlet
, C.
Péan
, B.
Rotenberg
, P. A.
Madden
, B.
Daffos
, P.-L.
Taberna
, P.
Simon
, and M.
Salanne
, “Highly confined ions store charge more efficiently in supercapacitors
,” Nat. Commun.
4
, 2701
(2013
).25.
S.
Kondrat
and A.
Kornyshev
, “Superionic state in double-layer capacitors with nanoporous electrodes
,” J. Phys.: Condens. Matter
23
, 022201
(2011
).26.
R.
Futamura
, T.
Iiyama
, Y.
Takasaki
, Y.
Gogotsi
, M. J.
Biggs
, M.
Salanne
, J.
Ségalini
, P.
Simon
, and K.
Kaneko
, “Partial breaking of the coulombic ordering of ionic liquids confined in carbon nanopores
,” Nat. Mater.
16
, 1225
(2017
).27.
A. A.
Kornyshev
, “The simplest model of charge storage in single file metallic nanopores
,” Faraday Discuss.
164
, 117
(2014
).28.
H.
Pan
, J.
Li
, and Y. P.
Feng
, “Carbon nanotubes for supercapacitor
,” Nanoscale Res. Lett.
5
, 654
(2010
).29.
A.
Muralidharan
, L. R.
Pratt
, G. G.
Hoffman
, M. I.
Chaudhari
, and S. B.
Rempe
, “Molecular simulation results on charged carbon nanotube forest-based supercapacitors
,” ChemSusChem
11
, 1927
(2018
).30.
C.
Cao
, Y.
Zhou
, S.
Ubnoske
, J.
Zang
, Y.
Cao
, P.
Henry
, C. B.
Parker
, and J. T.
Glass
, “Highly stretchable supercapacitors via crumpled vertically aligned carbon nanotube forests
,” Adv. Energy Mater.
9
, 1900618
(2019
).31.
A. A.
Lee
, S.
Kondrat
, and A. A.
Kornyshev
, “Charge storage in conducting cylindrical nanopores
,” Phys. Rev. Lett.
113
, 048701
(2014
).32.
C. C.
Rochester
, S.
Kondrat
, G.
Pruessner
, and A. A.
Kornyshev
, “Charging ultra-nanoporous electrodes with size-asymmetric ions assisted by apolar solvent
,” J. Phys. Chem. C
120
, 16042
(2016
).33.
A. O.
Zaboronsky
and A. A.
Kornyshev
, “Ising models of charge storage in multifile metallic nanopores
,” J. Phys.: Condens. Matter
32
, 275201
(2020
).34.
M.
Dudka
, S.
Kondrat
, A.
Kornyshev
, and G.
Oshanin
, “Phase behaviour and structure of a superionic liquid in nonpolarized nanoconfinement
,” J. Phys.: Condens. Matter
28
, 464007
(2016
).35.
M.
Dudka
, S.
Kondrat
, O.
Bénichou
, A. A.
Kornyshev
, and G.
Oshanin
, “Superionic liquids in conducting nanoslits: A variety of phase transitions and ensuing charging behavior
,” J. Chem. Phys.
151
, 184105
(2019
).36.
Y.
Groda
, M.
Dudka
, A. A.
Kornyshev
, G.
Oshanin
, and S.
Kondrat
, “Superionic liquids in conducting nanoslits: Insights from theory and simulations
,” J. Phys. Chem. C
125
, 4968
(2021
).37.
A. A.
Lee
, D.
Vella
, A.
Goriely
, and S.
Kondrat
, “Capacitance-power-hysteresis trilemma in nanoporous supercapacitors
,” Phys. Rev. X
6
, 021034
(2016
).38.
A. A.
Lee
, S.
Kondrat
, G.
Oshanin
, and A. A.
Kornyshev
, “Charging dynamics of supercapacitors with narrow cylindrical nanopores
,” Nanotechnology
25
, 315401
(2014
).39.
W.
Schmickler
, “A simple model for charge storage in a nanotube
,” Electochim. Acta
173
, 91
(2015
).40.
A. J.
Pak
and G. S.
Hwang
, “Charging rate dependence of ion migration and stagnation in ionic-liquid-filled carbon nanopores
,” J. Phys. Chem. C
120
, 24560
(2016
).41.
W.
Schmickler
and D.
Henderson
, “On the capacitance of narrow nanotubes
,” Phys. Chem. Chem. Phys.
19
, 20393
(2017
).42.
W.
Schmickler
and D.
Henderson
, “Charge storage in nanotubes: The case of a 2-1 electrolyte
,” Condens. Matter Phys.
20
, 33004
(2017
).43.
Y.
Qiao
, C.
Lian
, B.
Lu
, and J.
Wu
, “Modeling selective ion adsorption into cylindrical nanopores
,” Chem. Phys. Lett.
709
, 116
(2018
).44.
S.
Kondrat
, O. A.
Vasilyev
, and A. A.
Kornyshev
, “Feeling your neighbors across the walls: How interpore ionic interactions affect capacitive energy storage
,” J. Chem. Phys. Lett.
10
, 4523
(2019
).45.
D.
Frydel
, “One-dimensional Coulomb system in a sticky wall confinement: Exact results
,” Phys. Rev. E
100
, 042113
(2019
).46.
V.
Démery
, D. S.
Dean
, T. C.
Hammant
, R. R.
Horgan
, and R.
Podgornik
, “The one-dimensional Coulomb lattice fluid capacitor
,” J. Chem. Phys.
137
, 064901
(2012
).47.
V.
Démery
, D. S.
Dean
, T. C.
Hammant
, R. R.
Horgan
, and R.
Podgornik
, “Overscreening in a 1D lattice Coulomb gas model of ionic liquids
,” Europhys. Lett.
97
, 28004
(2012
).48.
V.
Démery
, R.
Monsarrat
, D. S.
Dean
, and R.
Podgornik
, “Phase diagram of a bulk 1d lattice Coulomb gas
,” Europhys. Lett.
113
, 18008
(2016
); arXiv:1511.07170 [cond-mat.stat-mech].49.
D.
Frydel
and Y.
Levin
, “Soft-particle lattice gas in one dimension: One- and two-component cases
,” Phys. Rev. E
98
, 062123
(2018
).50.
H. C.
Longuet-Higgins
, “One-dimensional multicomponent mixtures
,” Mol. Phys.
1
, 83
(1958
).51.
M.
Heying
and D. S.
Corti
, “The one-dimensional fully non-additive binary hard rod mixture: Exact thermophysical properties
,” Fluid Phase Equilib.
220
, 85
(2004
).52.
A.
Santos
, “Exact bulk correlation functions in one-dimensional nonadditive hard-core mixtures
,” Phys. Rev. E
76
, 062201
(2007
).53.
A.
Ben-Naim
and A.
Santos
, “Local and global properties of mixtures in one-dimensional systems. II. Exact results for the Kirkwood–Buff integrals
,” J. Chem. Phys.
131
, 164512
(2009
).54.
A.
Santos
, “One-dimensional systems: Exact solution for nearest-neighbor interactions
,” in A Concise Course on the Theory of Classical Liquids: Basics and Selected Topics
(Springer International Publishing
, Cham
, 2016
), pp. 125
–156
.55.
B.
Widom
, “Some topics in the theory of fluids
,” J. Chem. Phys.
39
, 2808
(1963
).56.
S.
Kondrat
, N.
Georgi
, M. V.
Fedorov
, and A. A.
Kornyshev
, “A superionic state in nano-porous double-layer capacitors: Insights from Monte Carlo simulations
,” Phys. Chem. Chem. Phys.
13
, 11359
(2011
).57.
L.
Tonks
, “The complete equation of state of one, two and three-dimensional gases of hard elastic spheres
,” Phys. Rev.
50
, 955
(1936
).58.
H.
Takahasi
, “Eine einfache methode zur behandlung der statistischen mechanik eindimensionaler substanzen
,” Proc. Phys.-Math. Soc. Jpn.
24
, 60
(1942
).59.
R.
Kikuchi
, “Theory of one-dimensional fluid binary mixtures
,” J. Chem. Phys.
23
, 2327
(1955
).60.
M.
Blume
, “Theory of the first-order magnetic phase change in UO2
,” Phys. Rev.
141
, 517
(1966
).61.
H. W.
Capel
, “On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting
,” Physica
32
, 966
(1966
).62.
D. T.
Limmer
, C.
Merlet
, M.
Salanne
, D.
Chandler
, P. A.
Madden
, R.
van Roij
, and B.
Rotenberg
, “Charge fluctuations in nanoscale capacitors
,” Phys. Rev. Lett.
111
, 106102
(2013
).63.
M. V.
Fedorov
and A. A.
Kornyshev
, “Ionic liquids at electrified interfaces
,” Chem. Rev.
114
, 2978
(2014
).64.
D.
di Caprio
, Z.
Borkowska
, and J.
Stafiej
, “Simple extension of the Gouy-Chapman theory including hard sphere effects. Diffuse layer contribution to the differential capacity curves for the electrode–electrolyte interface
,” J. Electroanal. Chem.
540
, 17
(2003
).65.
D.
di Caprio
, Z.
Borkowska
, and J.
Stafiej
, “Specific ionic interactions within a simple extension of the Gouy-Chapman theory including hard sphere effects
,” J. Electroanal. Chem.
572
, 51
(2004
).66.
A. A.
Kornyshev
, “Double-layer in ionic liquids: Paradigm change?
,” J. Phys. Chem. B
111
, 5545
(2007
).67.
Y.
Krynytskyi
and A.
Rovenchak
, “Asymptotic estimation for eigenvalues in the exponential potential and for zeros of Kiν(z) with respect to order
,” Symmetry, Integrability, Geom.: Methods Appl.
17
, 057
(2021
); arXiv:2103.01732 [math-ph].68.
S.
Kondrat
and A. A.
Kornyshev
, “Pressing a spring: What does it take to maximize the energy storage in nanoporous supercapacitors?
,” Nanoscale Horiz.
1
, 45
(2016
).69.
J. P.
Hallett
and T.
Welton
, “Room-temperature ionic liquids: Solvents for synthesis and catalysis. 2
,” Chem. Rev.
111
, 3508
(2011
).70.
A. C.
Forse
, C.
Merlet
, J. M.
Griffin
, and C. P.
Grey
, “New perspectives on the charging mechanisms of supercapacitors
,” J. Am. Chem. Soc.
138
, 5731
(2016
).71.
K.
Breitsprecher
, M.
Abele
, S.
Kondrat
, and C.
Holm
, “The effect of finite pore length on ion structure and charging
,” J. Chem. Phys.
147
, 104708
(2017
).72.
R.
Fantoni
and A.
Santos
, “One-dimensional fluids with second nearest–neighbor interactions
,” J. Stat. Phys.
169
, 1171
(2017
).73.
M. T.
Alam
, M. M.
Islam
, T.
Okajima
, and T.
Ohsaka
, “Capacitance measurements in a series of room-temperature ionic liquids at glassy carbon and gold electrode interfaces
,” J. Phys. Chem. C
112
, 16600
(2008
).74.
C.
Cruz
, A.
Ciach
, E.
Lomba
, and S.
Kondrat
, “Electrical double layers close to ionic liquid–solvent demixing
,” J. Chem. Phys. C
123
, 1596
(2018
).75.
C.
Cruz
, S.
Kondrat
, E.
Lomba
, and A.
Ciach
, “Capillary ionization and jumps of capacitive energy stored in mesopores
,” J. Phys. Chem. C
125
, 10243
(2021
).76.
C.
Cruz
, E.
Lomba
, and A.
Ciach
, “Capacitance response and concentration fluctuations close to ionic liquid-solvent demixing
,” J. Mol. Liq.
22
, 117078
(2021
).77.
M. T.
Alam
, M. M.
Islam
, T.
Okajima
, and T.
Ohsaka
, “Measurements of differential capacitance in room temperature ionic liquid at mercury, glassy carbon and gold electrode
,” Electrochem. Commun.
9
, 2370
(2007
).78.
J.
Klos
and S.
Lamperski
, “Electrical double layer properties in diameter asymmetric molten salt investigated by grand canonical Monte Carlo method
,” J. Phys. Chem. C
114
, 13329
(2010
).79.
D.
Henderson
, S.
Lamperski
, L. B.
Bhuiyan
, and J.
Wu
, “The tail effect on the shape of an electrical double layer differential capacitance curve
,” J. Chem. Phys.
138
, 144704
(2013
).80.
S.
Kondrat
, C. R.
Pérez
, V.
Presser
, Y.
Gogotsi
, and A. A.
Kornyshev
, “Effect of pore size and its dispersity on the energy storage in nanoporous supercapacitors
,” Energy Environ. Sci.
5
, 6474
(2012
).81.
C.
Lian
, H.
Liu
, D.
Henderson
, and J.
Wu
, “Can ionophobic nanopores enhance the energy storage capacity of electric-double-layer capacitors containing nonaqueous electrolytes?
,” J. Phys.: Condens. Matter
28
, 414005
(2016
).82.
L.
He
, L.
Yang
, M.
Dincă
, R.
Zhang
, and J.
Li
, “Observation of ion electrosorption in metal–organic framework micropores with in operando small-angle neutron scattering
,” Angew. Chem.
132
, 9860
(2020
).83.
Z.
Gan
, Y.
Wang
, M.
Wang
, E.
Gao
, F.
Huo
, W.
Ding
, H.
He
, and S.
Zhang
, “Ionophobic nanopores enhancing the capacitance and charging dynamics in supercapacitors with ionic liquids
,” J. Mater. Chem.A
9
, 15985
(2021
).84.
X.
Yin
, J.
Zhang
, L.
Yang
, W.
Xiao
, L.
Zhou
, Y.
Tang
, and W.
Yang
, “Carbon electrodes with ionophobic characteristics in organic electrolyte for high-performance electric double-layer capacitors
,” Sci. China Mater.
(published online) (2021
).85.
E.
Helfand
, H. L.
Frisch
, and J. L.
Lebowitz
, “Theory of the two- and one-dimensional rigid sphere fluids
,” J. Chem. Phys.
34
, 1037
(1961
).86.
M.
Holovko
, M.
Shmotolokha
, and W.
Dong
, “Analytical theory of one- and two-dimensional hard sphere fluids in random porous media
,” Condens. Matter Phys.
13
, 23607
(2010
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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