Properties of solid-liquid interfaces are of immense importance for electrocatalytic and electrochemical systems, but modeling such interfaces at the atomic level presents a serious challenge and approaches beyond standard methodologies are needed. An atomistic computational scheme needs to treat at least part of the system quantum mechanically to describe adsorption and reactions, while the entire system is in thermal equilibrium. The experimentally relevant macroscopic control variables are temperature, electrode potential, and the choice of the solvent and ions, and these need to be explicitly included in the computational model as well; this calls for a thermodynamic ensemble with fixed ion and electrode potentials. In this work, a general framework within density functional theory (DFT) with fixed electron and ion chemical potentials in the grand canonical (GC) ensemble is established for modeling electrocatalytic and electrochemical interfaces. Starting from a fully quantum mechanical description of multi-component GC-DFT for nuclei and electrons, a systematic coarse-graining is employed to establish various computational schemes including (i) the combination of classical and electronic DFTs within the GC ensemble and (ii) on the simplest level a chemically and physically sound way to obtain various (modified) Poisson-Boltzmann (mPB) implicit solvent models. The detailed and rigorous derivation clearly establishes which approximations are needed for coarse-graining as well as highlights which details and interactions are omitted in vein of computational feasibility. The transparent approximations also allow removing some of the constraints and coarse-graining if needed. We implement various mPB models within a linear dielectric continuum in the GPAW code and test their capabilities to model capacitance of electrochemical interfaces as well as study different approaches for modeling partly periodic charged systems. Our rigorous and well-defined DFT coarse-graining scheme to continuum electrolytes highlights the inadequacy of current linear dielectric models for treating properties of the electrochemical interface.

1.
A. J.
Bard
and
L. R.
Faulker
,
Electrochemical Methods: Fundamentals and Applications
, 2nd ed. (
John Wiley and Sons, Inc.
,
2001
).
2.
R.
Jinnouchi
,
K.
Kodama
, and
Y.
Morimoto
, “
Electronic structure calculations on electrolyte–electrode interfaces: Successes and limitations
,”
Curr. Opin. Electrochem.
8
,
103
(
2018
).
3.
M.
Tuckerman
,
Statistical Mechanics: Theory and Molecular Simulations
(
Oxford University Press
,
2010
).
4.
J.
Cheng
,
X.
Liu
,
J.
VandeVondele
,
M.
Sulpizi
, and
M.
Sprik
, “
Redox potentials and acidity constants from density functional theory based molecular dynamics
,”
Acc. Chem. Res.
47
(
12
),
3522
3529
(
2014
).
5.
T.
Ikeshoji
and
M.
Otani
, “
Toward full simulation of the electrochemical oxygen reduction reaction on pt using first-principles and kinetic calculations
,”
Phys. Chem. Chem. Phys.
19
,
4447
4453
(
2017
).
6.
N.
Holmberg
and
K.
Laasonen
, “
Ab initio electrochemistry: Exploring the hydrogen evolution reaction on carbon nanotubes
,”
J. Phys. Chem. C
119
(
28
),
16166
16178
(
2015
).
7.
J.-P.
Hansen
and
J.-P.
Hansen
,
Theory of Simple Liquids
, 3rd ed. (
Academic Press
,
2006
).
8.
J.
Forsman
,
C. E.
Woodward
, and
R.
Szparaga
,
Computational Electrostatics for Biological Applications
(
Springer International Publishing
,
2015
).
9.
T. J.
Sluckin
, “
Applications of the density-functional theory of charged fluids
,”
J. Chem. Soc., Faraday Trans. 2
77
,
575
586
(
1981
).
10.
N.
David Mermin
, “
Thermal properties of the inhomogeneous electron gas
,”
Phys. Rev.
137
,
A1441
A1443
(
1965
).
11.
A.
Pribram-Jones
,
S.
Pittalis
,
E. K. U.
Gross
, and
K.
Burke
, “
Thermal density functional theory in context
,” in
Frontiers and Challenges in Warm Dense Matter
, edited by
F.
Graziani
,
M. P.
Desjarlais
,
R.
Redmer
, and
S. B.
Trickey
(
Springer International Publishing
,
2014
), pp.
25
60
.
12.
S. A.
Petrosyan
,
A. A.
Rigos
, and
T. A.
Arias
, “
Joint density-functional theory: Ab initio study of Cr2O3 surface chemistry in solution
,”
J. Phys. Chem. B
109
(
32
),
15436
15444
(
2005
).
13.
A.
Kovalenko
and
F.
Hirata
, “
Self-consistent, Kohn-Sham DFT and three-dimensional RISM description of a metal-molecular liquid interface
,”
J. Mol. Liq.
90
(
1
),
215
224
(
2001
).
14.
A.
Kovalenko
and
S.
Gusarov
, “
Multiscale methods framework: Self-consistent coupling of molecular theory of solvation with quantum chemistry, molecular simulations, and dissipative particle dynamics
,”
Phys. Chem. Chem. Phys.
20
,
2947
2969
(
2018
).
15.
S.
Nishihara
and
M.
Otani
, “
Hybrid solvation models for bulk, interface, and membrane: Reference interaction site methods coupled with density functional theory
,”
Phys. Rev. B
96
,
115429
(
2017
).
16.
M.
Otani
and
O.
Sugino
, “
First-principles calculations of charged surfaces and interfaces: A plane-wave nonrepeated slab approach
,”
Phys. Rev. B
73
,
115407
(
2006
).
17.
N.
Bonnet
,
T.
Morishita
,
O.
Sugino
, and
M.
Otani
, “
First-principles molecular dynamics at a constant electrode potential
,”
Phys. Rev. Lett.
109
,
266101
(
2012
).
18.
R.
Sundararaman
,
W. A.
Goddard
 III
,
A.
Tomas
, and
Arias
, “
Grand canonical electronic density-functional theory: Algorithms and applications to electrochemistry
,”
J. Chem. Phys.
146
(
11
),
114104
(
2017
).
19.
T.
Kreibich
,
R.
van Leeuwen
, and
E. K. U.
Gross
, “
Multicomponent density-functional theory for electrons and nuclei
,”
Phys. Rev. A
78
,
022501
(
2008
).
20.
S.
Smidstrup
,
D.
Stradi
,
J.
Wellendorff
,
P. A.
Khomyakov
,
U. G.
Vej-Hansen
,
M.-E.
Lee
, and
T.
Ghosh
,
E.
Jónsson
,
H.
Jónsson
, and
K.
Stokbro
, “
First-principles Green’s-function method for surface calculations: A pseudopotential localized basis set approach
,”
Phys. Rev. B
96
,
195309
(
2017
).
21.
C. D.
Taylor
,
S. A.
Wasileski
,
J.-S.
Filhol
, and
M.
Neurock
, “
First principles reaction modeling of the electrochemical interface: Consideration and calculation of a tunable surface potential from atomic and electronic structure
,”
Phys. Rev. B
73
,
165402
(
2006
).
22.
J. D.
Goodpaster
,
A. T.
Bell
, and
M.
Head-Gordon
, “
Identification of possible pathways for C–C bond formation during electrochemical reduction of CO2: New theoretical insights from an improved electrochemical model
,”
J. Phys. Chem. Lett.
7
(
8
),
1471
1477
(
2016
).
23.
R.
Jinnouchi
and
A. B.
Anderson
, “
Electronic structure calculations of liquid-solid interfaces: Combination of density functional theory and modified Poisson-Boltzmann theory
,”
Phys. Rev. B
77
,
245417
(
2008
).
24.
E.
Skúlason
,
V.
Tripkovic
,
M. E.
Bjørketun
,
S.
Gudmundsdóttir
,
G.
Karlberg
,
J.
Rossmeisl
,
T.
Bligaard
,
H.
Jónsson
, and
J. K.
Nørskov
, “
Modeling the electrochemical hydrogen oxidation and evolution reactions on the basis of density functional theory calculations
,”
J. Phys. Chem. C
114
(
42
),
18182
18197
(
2010
).
25.
K.
Letchworth-Weaver
and
T. A.
Arias
, “
Joint density functional theory of the electrode-electrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge
,”
Phys. Rev. B
86
,
075140
(
2012
).
26.
Y.-H.
Fang
and
Z.-P.
Liu
, “
Mechanism and Tafel lines of electro-oxidation of water to oxygen on RuO2(110)
,”
J. Am. Chem. Soc.
132
(
51
),
18214
18222
(
2010
).
27.
I.
Dabo
,
B.
Kozinsky
,
N. E.
Singh-Miller
, and
N.
Marzari
, “
Electrostatics in periodic boundary conditions and real-space corrections
,”
Phys. Rev. B
77
,
115139
(
2008
).
28.
O.
Andreussi
and
N.
Marzari
, “
Electrostatics of solvated systems in periodic boundary conditions
,”
Phys. Rev. B
90
,
245101
(
2014
).
29.
S.
Ringe
,
H.
Oberhofer
, and
K.
Reuter
, “
Transferable ionic parameters for first-principles Poisson-Boltzmann solvation calculations: Neutral solutes in aqueous monovalent salt solutions
,”
J. Chem. Phys.
146
(
13
),
134103
(
2017
).
30.
G.
Fisicaro
,
L.
Genovese
,
O.
Andreussi
,
N.
Marzari
, and
S.
Goedecker
, “
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
,”
J. Chem. Phys.
144
(
1
),
014103
(
2016
).
31.
K.
Mathew
and
R. G.
Hennig
, “
Implicit self-consistent description of electrolyte in plane-wave density-functional theory
,” e-print arXiv:1601.03346 (
2016
).
32.
D.
Gunceler
,
K.
Letchworth-Weaver
,
R.
Sundararaman
,
K. A.
Schwarz
, and
T. A.
Arias
, “
The importance of nonlinear fluid response in joint density-functional theory studies of battery systems
,”
Modell. Simul. Mater. Sci. Eng.
21
(
7
),
074005
(
2013
).
33.
G.
Kastlunger
,
P.
Lindgren
, and
A. A.
Peterson
, “
Controlled-potential simulation of elementary electrochemical reactions: Proton discharge on metal surfaces
,”
J. Phys. Chem. C
122
(
24
),
12771
12781
(
2018
).
34.
G.
Fisicaro
,
L.
Genovese
,
O.
Andreussi
,
S.
Mandal
,
N. N.
Nair
,
N.
Marzari
, and
S.
Goedecker
, “
Soft-sphere continuum solvation in electronic-structure calculations
,”
J. Chem. Theory Comput.
13
(
8
),
3829
3845
(
2017
).
35.
O.
Andreussi
,
I.
Dabo
, and
N.
Marzari
, “
Revised self-consistent continuum solvation in electronic-structure calculations
,”
J. Chem. Phys.
136
(
6
),
064102
(
2012
).
36.
R.
Sundararaman
and
K.
Schwarz
, “
Evaluating continuum solvation models for the electrode-electrolyte interface: Challenges and strategies for improvement
,”
J. Chem. Phys.
146
(
8
),
084111
(
2017
).
37.
R.
Sundararaman
,
K.
Letchworth-Weaver
, and
K. A.
Schwarz
, “
Improving accuracy of electrochemical capacitance and solvation energetics in first-principles calculations
,”
J. Chem. Phys.
148
(
14
),
144105
(
2018
).
38.
J. F.
Capitani
,
R. F.
Nalewajski
, and
R. G.
Parr
, “
Non–Born–Oppenheimer density functional theory of molecular systems
,”
J. Chem. Phys.
76
(
1
),
568
573
(
1982
).
39.
K.
Chan
and
J. K.
Nørskov
, “
Electrochemical barriers made simple
,”
J. Phys. Chem. Lett.
6
(
14
),
2663
2668
(
2015
).
40.
E. A.
Guggenheim
,
Thermodynamics: An Advanced Treatment for Chemists and Physicists
(
North Holland Physics Publishing
,
1967
).
41.
D.
Petz
, “
Entropy, von Neumann and the von Neumann entropy
,” e-print arXiv:math-ph/0102013v1 (
2001
).
42.
M.
Levy
, “
Electron densities in search of Hamiltonians
,”
Phys. Rev. A
26
,
1200
1208
(
1982
).
43.
A.
Migliore
,
N. F.
Polizzi
,
M. J.
Therien
, and
D. N.
Beratan
, “
Biochemistry and theory of proton-coupled electron transfer
,”
Chem. Rev.
114
(
7
),
3381
3465
(
2014
).
44.
A.
Nitzan
,
Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems
(
Oxford University Press
,
2006
).
45.
R.
Crespo-Otero
and
M.
Barbatti
, “
Recent advances and perspectives on nonadiabatic mixed quantum–classical dynamics
,”
Chem. Rev.
118
(
15
),
7026
7068
(
2018
).
46.
M. V.
Pak
,
A.
Chakraborty
, and
S.
Hammes-Schiffer
, “
Density functional theory treatment of electron correlation in the nuclear electronic orbital approach
,”
J. Phys. Chem. A
111
(
20
),
4522
4526
(
2007
).
47.
J.
Wu
,
Classical Density Functional Theory for Molecular Systems
(
Springer Singapore
,
Singapore
,
2017
), pp.
65
99
.
48.
W.
Schmickler
and
R.
Guidelli
, “
The partial charge transfer
,”
Electrochim. Acta
127
,
489
505
(
2014
).
49.
L. D.
Chen
,
M.
Bajdich
,
J. M. P.
Martirez
,
C. M.
Krauter
,
J. A.
Gauthier
,
E. A.
Carter
,
A. C.
Luntz
,
K.
Chan
, and
J. K.
Nørskov
, “
Understanding the apparent fractional charge of protons in the aqueous electrochemical double layer
,”
Nat. Commun.
9
(
1
),
3202
(
2018
).
50.
R.
de Levie
, “
The electrosorption valency and partial charge transfer
,”
J. Electroanal. Chem.
562
(
2
),
273
276
(
2004
).
51.
M.
Franco-Perez
,
P. W.
Ayers
,
J. L.
Gazquez
, and
A.
Vela
, “
Local chemical potential, local hardness, and dual descriptors in temperature dependent chemical reactivity theory
,”
Phys. Chem. Chem. Phys.
19
,
13687
13695
(
2017
).
52.
R.
Haase
,
Thermodynamics of Irreversible Processes
(
Addison-Wesley
,
1969
).
53.
O.
Anatole von Lilienfeld
and
M. E.
Tuckerman
, “
Molecular grand-canonical ensemble density functional theory and exploration of chemical space
,”
J. Chem. Phys.
125
(
15
),
154104
(
2006
).
54.
C. R.
Jacob
and
J.
Neugebauer
, “
Subsystem density functional theory
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
4
(
4
),
325
362
(
2014
).
55.
A.
Patrykiejew
,
S.
Sokolowski
, and
O.
Pizio
, , edited by
K.
Wandelt
(
Wiley-Blackwell
,
2016
), Chap. 46, pp.
883
1253
.
56.
L.
Mier-y-Teran
,
S. H.
Suh
,
H. S.
White
, and
H. T.
Davis
, “
A nonlocal free–energy density–functional approximation for the electrical double layer
,”
J. Chem. Phys.
92
(
8
),
5087
5098
(
1990
).
57.
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
1998
).
58.
D. J.
Bonthuis
and
R. R.
Netz
, “
Beyond the continuum: How molecular solvent structure affects electrostatics and hydrodynamics at solid–electrolyte interfaces
,”
J. Phys. Chem. B
117
(
39
),
11397
11413
(
2013
).
59.
A.
Held
and
M.
Walter
, “
Simplified continuum solvent model with a smooth cavity based on volumetric data
,”
J. Chem. Phys.
141
(
17
),
174108
(
2014
).
60.
Y.
Nakayama
and
D.
Andelman
, “
Differential capacitance of the electric double layer: The interplay between ion finite size and dielectric decrement
,”
J. Chem. Phys.
142
(
4
),
044706
(
2015
).
61.
D.
Ben-Yaakov
,
D.
Andelman
, and
R.
Podgornik
, “
Dielectric decrement as a source of ion-specific effects
,”
J. Chem. Phys.
134
(
7
),
074705
(
2011
).
62.
N.
Gavish
and
K.
Promislow
, “
Dependence of the dielectric constant of electrolyte solutions on ionic concentration: A microfield approach
,”
Phys. Rev. E
94
,
012611
(
2016
).
63.
M. Z.
Bazant
,
M.
Sabri Kilic
,
B. D.
Storey
, and
A.
Ajdari
, “
Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions
,”
Adv. Colloid Interface Sci.
152
(
1
),
48
88
(
2009
).
64.
I.
Borukhov
,
D.
Andelman
, and
H.
Orland
, “
Steric effects in electrolytes: A modified Poisson-Boltzmann equation
,”
Phys. Rev. Lett.
79
,
435
438
(
1997
).
65.
E.
Gongadze
,
A.
Velikonja
,
S.
Perutkova
,
P.
Kramar
,
A.
Macek-Lebar
,
V.
Kralj-Iglic
, and
A.
Iglic
, “
Ions and water molecules in an electrolyte solution in contact with charged and dipolar surfaces
,”
Electrochim. Acta
126
,
42
60
(
2014
).
66.
F.
Booth
, “
The dielectric constant of water and the saturation effect
,”
J. Chem. Phys.
19
(
4
),
391
394
(
1951
).
67.
S.
Trasatti
, “
The absolute electrode potential: An explanatory note (recommendations 1986)
,”
J. Electroanal. Chem. Interfacial Electrochem.
209
(
2
),
417
428
(
1986
).
68.
S.
Trasatti
, “
The ‘absolute’ electrode potential—The end of the story
,”
Electrochim. Acta
35
(
1
),
269
271
(
1990
).
69.
K.
Leung
, “
Surface potential at the air–water interface computed using density functional theory
,”
J. Phys. Chem. Lett.
1
(
2
),
496
499
(
2010
).
70.
M.
Otani
,
I.
Hamada
,
O.
Sugino
,
Y.
Morikawa
,
Y.
Okamoto
, and
T.
Ikeshoji
, “
Electrode dynamics from first principles
,”
J. Phys. Soc. Jpn.
77
(
2
),
024802
(
2008
).
71.
M. N.
Tamashiro
and
H.
Schiessel
, “
Where the linearized Poisson–Boltzmann cell model fails: Spurious phase separation in charged colloidal suspensions
,”
J. Chem. Phys.
119
(
3
),
1855
1865
(
2003
).
72.
B.
Zoetekouw
and
R.
van Roij
, “
Volume terms for charged colloids: A grand-canonical treatment
,”
Phys. Rev. E
73
,
021403
(
2006
).
73.
F. G.
Donnan
, “
The theory of membrane equilibria
,”
Chem. Rev.
1
(
1
),
73
90
(
1924
).
74.
L.
Wan
,
S.
Xu
,
M.
Liao
,
C.
Liu
, and
P.
Sheng
, “
Self-consistent approach to global charge neutrality in electrokinetics: A surface potential trap model
,”
Phys. Rev. X
4
,
011042
(
2014
).
75.
A.
Baskinz
and
D.
Prendergast
, “
Improving continuum models to define practical limits for molecular models of electrified interfaces
,”
J. Electrochem. Soc.
164
,
E3438
E3447
(
2017
).
76.
A. R.
Denton
, “
Charge renormalization, effective interactions, and thermodynamics of deionized colloidal suspensions
,”
J. Phys.: Condens. Matter
20
(
49
),
494230
(
2008
).
77.
J. S.
Hub
,
B. L.
de Groot
,
H.
Grubmüller
, and
G.
Groenhof
, “
Quantifying artifacts in Ewald simulations of inhomogeneous systems with a net charge
,”
J. Chem. Theory Comput.
10
(
1
),
381
390
(
2014
).
78.
J. J.
Mortensen
,
L. B.
Hansen
, and
K. W.
Jacobsen
, “
Real-space grid implementation of the projector augmented wave method
,”
Phys. Rev. B
71
,
035109
(
2005
).
79.
J.
Enkovaara
,
C.
Rostgaard
,
J. J.
Mortensen
,
J.
Chen
,
M.
Dułak
,
L.
Ferrighi
,
J.
Gavnholt
,
C.
Glinsvad
,
V.
Haikola
,
H. A.
Hansen
,
H. H.
Kristoffersen
,
M.
Kuisma
,
A. H.
Larsen
,
L.
Lehtovaara
,
M.
Ljungberg
,
O.
Lopez-Acevedo
,
P. G.
Moses
,
J.
Ojanen
,
T.
Olsen
,
V.
Petzold
,
N. A.
Romero
,
J.
Stausholm-Møller
,
M.
Strange
,
G. A.
Tritsaris
,
M.
Vanin
,
M.
Walter
,
B.
Hammer
,
H.
Häkkinen
,
G. K. H.
Madsen
,
R. M.
Nieminen
,
J. K.
Nørskov
,
M.
Puska
,
T. T.
Rantala
,
J.
Schiøtz
,
K. S.
Thygesen
, and
K. W.
Jacobsen
, “
Electronic structure calculations with GPAW: A real-space implementation of the projector augmented-wave method
,”
J. Phys.: Condens. Matter
22
(
25
),
253202
(
2010
).
80.
L.
Bengtsson
, “
Dipole correction for surface supercell calculations
,”
Phys. Rev. B
59
,
12301
12304
(
1999
).
81.
P. E.
Blöchl
, “
Projector augmented-wave method
,”
Phys. Rev. B
50
,
17953
17979
(
1994
).
82.
A. H.
Larsen
,
J. J.
Mortensen
,
J.
Blomqvist
,
I. E.
Castelli
,
R.
Christensen
,
M.
Dułak
,
J.
Friis
,
M. N.
Groves
,
B.
Hammer
,
C.
Hargus
,
E. D.
Hermes
,
P. C.
Jennings
,
P.
Bjerre Jensen
,
J.
Kermode
,
J. R.
Kitchin
,
E. L.
Kolsbjerg
,
J.
Kubal
,
K.
Kaasbjerg
,
S.
Lysgaard
,
J.
Bergmann Maronsson
,
T.
Maxson
,
T.
Olsen
,
L.
Pastewka
,
A.
Peterson
,
C.
Rostgaard
,
J.
Schiøtz
,
O.
Schütt
,
M.
Strange
,
K. S.
Thygesen
,
T.
Vegge
,
L.
Vilhelmsen
,
M.
Walter
,
Z.
Zeng
, and
K. W.
Jacobsen
, “
The atomic simulation environment—A Python library for working with atoms
,”
J. Phys.: Condens. Matter
29
(
27
),
273002
(
2017
).
83.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
84.
A.
Bondi
, “
van der Waals volumes and radii
,”
J. Phys. Chem.
68
(
3
),
441
451
(
1964
).
85.
S. S.
Batsanov
, “
van der Waals radii of elements
,”
Inorg. Mater.
37
(
9
),
871
885
(
2001
).
86.
A.
Gross
,
Theory of Solid/Electrolyte Interfaces
, Psi-k Highlight No. 125 (
Wiley
,
2014
).
87.
R. I.
Slavchov
,
J. K.
Novev
,
T. V.
Peshkova
, and
N. A.
Grozev
, “
Surface tension and surface Δχ-potential of concentrated Z+:Z electrolyte solutions
,”
J. Colloid Interface Sci.
403
,
113
126
(
2013
).
88.
A.
Hamelin
,
Z.
Borkowska
, and
J.
Stafiej
, “
A double layer study of the (210) and (111) faces of gold in aqueous nabf4 solutions
,”
J. Electroanal. Chem. Interfacial Electrochem.
189
(
1
),
85
97
(
1985
).
89.
J.
Rossmeisl
,
E.
Skúlason
,
M. E.
Bjørketun
,
V.
Tripkovic
, and
J. K.
Nørskov
, “
Modeling the electrified solid–liquid interface
,”
Chem. Phys. Lett.
466
(
1-3
),
68
71
(
2008
).
90.
A.
Hamelin
, “
Study of the (210) face of gold in aqueous solutions
,”
J. Electroanal. Chem. Interfacial Electrochem.
138
(
2
),
395
400
(
1982
).
91.
S.
Trasatti
, “
Surface science and electrochemistry: Concepts and problems
,” in
Proceedings of the IUVSTA Workshop on Surface Science and Electrochemistry

Supplementary Material

You do not currently have access to this content.