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Weierstrass Institute forApplied Analysis and Stochastics
The Lippmann equation for liquid metal electrodes
Wolfgang Dreyer, Clemens Guhlke, Manuel Landstorfer, Rüdiger Müller
Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de
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Lippmann equation
doi:10.1351/goldbook.L03577
IUPAC Compendium of Chemical Terminology Copyright © 2014 IUPAC
Lippman's equationAn equation which gives the electric charge per unit area of an interface (electrode):
where is the interfacial tension, is the potential of a cell in which the referenceelectrode has an interfacial equilibrium with one of the ionic components of A, isthe charge on unit area of the interface, is the chemical potential of the combinationof species whose net charge is zero, is the thermodynamic temperature and isthe external pressure. Since more than one type of reference electrode may be chosen,more than one quantity may be obtained. Consequently cannot be considered asequivalent to the physical charge on a particular region of the interphase. It is in factan alternative way of expressing a surface excess or combination of surface excess ofcharged species.
Source:PAC, 1974, 37, 499 (Electrochemical nomenclature) on page 508PAC, 1986, 58, 437 (Interphases in systems of conducting phases (Recommendations1985)) on page 445
doi:10.1351/goldbook.L03577
IUPAC Compendium of Chemical Terminology Copyright © 2014 IUPAC
Lippman's equationAn equation which gives the electric charge per unit area of an interface (electrode):
where is the interfacial tension, is the potential of a cell in which the referenceelectrode has an interfacial equilibrium with one of the ionic components of A, isthe charge on unit area of the interface, is the chemical potential of the combinationof species whose net charge is zero, is the thermodynamic temperature and isthe external pressure. Since more than one type of reference electrode may be chosen,more than one quantity may be obtained. Consequently cannot be considered asequivalent to the physical charge on a particular region of the interphase. It is in factan alternative way of expressing a surface excess or combination of surface excess ofcharged species.
Source:PAC, 1974, 37, 499 (Electrochemical nomenclature) on page 508PAC, 1986, 58, 437 (Interphases in systems of conducting phases (Recommendations1985)) on page 445
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Lippmann equation
ΩS
Ω−
+
γS
qS
LippmanndγdE = −Q
E well defined ?
Q = Q(qs, q+, q−)
γ = γ(γs, γ+, γ−)
differential capacity
C := dQdE = − d2γ
dE2
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Lippmann equation
ΩS
Ω−
+
γS
qS
LippmanndγdE = −Q
E well defined ?
Q = Q(qs, q+, q−)
γ = γ(γs, γ+, γ−)
differential capacity
C := dQdE = − d2γ
dE2
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Lippmann equation
ΩS
Ω−
+γ~+
q+~
γS
E ~−qqS
~γ−
LippmanndγdE = −Q
E well defined ?
Q = Q(qs, q+, q−)
γ = γ(γs, γ+, γ−)
differential capacity
C := dQdE = − d2γ
dE2
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Lippmann equation
ΩS
Ω−
+γ~+
q+~
γS
E ~−qqS
~γ−
LippmanndγdE = −Q
E well defined ?
Q = Q(qs, q+, q−)
γ = γ(γs, γ+, γ−)
differential capacity
C := dQdE = − d2γ
dE2
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Outline
1 Ion transport in electrolyte bulkShortcomings of Nernst-Planck model
General continuum model for the bulk
2 Electrochemical double layerSurface balances and adsorption
Free energy models for metals
Double-layer capacity
3 Lippmann equation for liquid mercury electrodeMatched asymptotic analysis
Numerical results
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Shortcomings of Nernst-Planck model
diluted solution of anions A, cations C in solvet S
∂t(mαnα) + div(mαnαv + Jα) = 0 for α = A,C
−ε∆ϕ = zAnA + zCnC
Nernst-Plack (1890):
Jα = −M(∇nα + zαnα∇ϕ) for α = A,C
+
+
+
+
+
− +
+
+
−
−
−
−
−
−
−
−
−−electrolyte
ϕE
ϕs
metal
equilibrium→ Poisson-Boltzmann
0 = ∇nα + zαnα∇ϕ for α = A,C
−ε∆ϕ = zAnA exp(−zAϕ) + zC nC exp(−zCϕ)
0 10
0
0.5
1
1.5
ξL
y
y
A
yC
yS
−10 0ξ
R
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Shortcomings of Nernst-Planck model
diluted solution of anions A, cations C in solvet S
∂t(mαnα) + div(mαnαv + Jα) = 0 for α = A,C
−ε∆ϕ = zAnA + zCnC
Nernst-Plack (1890):
Jα = −M(∇nα + zαnα∇ϕ) for α = A,C
+
+
+
+
+
− +
+
+
−
−
−
−
−
−
−
−
−−electrolyte
ϕE
ϕs
metal
equilibrium→ Poisson-Boltzmann
0 = ∇nα + zαnα∇ϕ for α = A,C
−ε∆ϕ = zAnA exp(−zAϕ) + zC nC exp(−zCϕ)
0 10
0
0.5
1
1.5
ξL
y
y
A
yC
yS
−10 0ξ
R
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“Extended” Nernst-Planck models
postulated layer structure[Stern:1924]
finite size of ions, volume exclusion
[Bikerman:1942], [Eigen,Wicke:1952], [Carnahan,Starling:1969],
[Borukhov,Andelman,Orlando:1997], [Ajdari,Bazant,Kilic:2007]
... molecular dynamics, statistical mechanics,
Langevin, Monte Carlo, DFT ...
→ Poisson-Fermi
+
+
+
+
+
− +
+
+
−
−
−
−
−
−
−
−
−−electrolyte
ϕE
ϕs
metal
layers not diluted
pressure dependence,
momentum balance
[Freise:1952], [Sanfeld:1968],
[DGM13] 0 100
100
200
300
400
ξL
p [M
Pa]
−10 0ξ
R
0 10
0
0.5
1
1.5
ξL
y
y
A
yC
yS
−10 0ξ
R
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“Extended” Nernst-Planck models
postulated layer structure[Stern:1924]
finite size of ions, volume exclusion
[Bikerman:1942], [Eigen,Wicke:1952], [Carnahan,Starling:1969],
[Borukhov,Andelman,Orlando:1997], [Ajdari,Bazant,Kilic:2007]
... molecular dynamics, statistical mechanics,
Langevin, Monte Carlo, DFT ...
→ Poisson-Fermi
+
+
+
+
+
− +
+
+
−
−
−
−
−
−
−
−
−−electrolyte
ϕE
ϕs
metal
layers not diluted
pressure dependence,
momentum balance
[Freise:1952], [Sanfeld:1968],
[DGM13] 0 100
100
200
300
400
ξL
p [M
Pa]
−10 0ξ
R
0 10
0
0.5
1
1.5
ξL
y
y
A
yC
yS
−10 0ξ
R
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“Extended” Nernst-Planck models
postulated layer structure[Stern:1924]
finite size of ions, volume exclusion
[Bikerman:1942], [Eigen,Wicke:1952], [Carnahan,Starling:1969],
[Borukhov,Andelman,Orlando:1997], [Ajdari,Bazant,Kilic:2007]
... molecular dynamics, statistical mechanics,
Langevin, Monte Carlo, DFT ...
→ Poisson-Fermi
+
+
+
+
+
− +
+
+
−
−
−
−
−
−
−
−
−−electrolyte
ϕE
ϕs
metal
layers not diluted
pressure dependence,
momentum balance
[Freise:1952], [Sanfeld:1968],
[DGM13] 0 100
100
200
300
400
ξL
p [M
Pa]
−10 0ξ
R
0 10
0
0.5
1
1.5
ξL
y
y
A
yC
yS
−10 0ξ
R
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General Framework
non-equilibrium thermodynamics distinguishes
general balance laws
material dependent constitutive equations
constitutive equations restricted by
2nd law of thermodynamics (cf. [Bothe,Dreyer:2014])
principle of material frame indifference
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General Framework
non-equilibrium thermodynamics distinguishes
general balance laws
material dependent constitutive equations
constitutive equations restricted by
2nd law of thermodynamics (cf. [Bothe,Dreyer:2014])
principle of material frame indifference
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Electrolyte bulk model
constituents: A0, A1, · · · , AN
atomic mass: m0,m1, · · · ,mN
charge number: z0, z1, · · · , zN
+ −
−
−
+
+
+
+
+
+−
−
−
+−
−−
+
−
+
fields of matterρ0, · · · , ρN mass density
ρv momentum
ρu internal energy
electromagnetic fieldsE
= −∇ϕ
electric field
B magnetic field
partial mass balances ∂tρα + div(ραv + Jα) = 0 for α = 0, 1, · · · , N
quasi-static momentum −div(Σ) = ρb
electrostatic Poisson eqn. −ε0(1 + χ)∆ϕ = nF
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Electrolyte bulk model
constituents: A0, A1, · · · , AN
atomic mass: m0,m1, · · · ,mN
charge number: z0, z1, · · · , zN
+ −
−
−
+
+
+
+
+
+−
−
−
+−
−−
+
−
+
fields of matterρ0, · · · , ρN mass density
ρv momentum
ρu internal energy
electromagnetic fieldsE = −∇ϕ electric field
B magnetic field
partial mass balances ∂tρα + div(ραv + Jα) = 0 for α = 0, 1, · · · , N
quasi-static momentum −div(Σ) = ρb
electrostatic Poisson eqn. −ε0(1 + χ)∆ϕ = nF
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Electrolyte bulk model
constituents: A0, A1, · · · , AN
atomic mass: m0,m1, · · · ,mN
charge number: z0, z1, · · · , zN
+ −
−
−
+
+
+
+
+
+−
−
−
+−
−−
+
−
+
fields of matterρ0, · · · , ρN mass density
ρv momentum
ρu internal energy
electromagnetic fieldsE = −∇ϕ electric field
B magnetic field
partial mass balances ∂tρα + div(ραv + Jα) = 0 for α = 0, 1, · · · , N
quasi-static momentum −div(Σ) = ρb
electrostatic Poisson eqn. −ε0(1 + χ)∆ϕ = nF
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Entropy production
[Guhlke:2015]
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Constitutive modeling
free energy ρψ = ρψ(T,ρ0,ρ1,··· ,ρN ,) + χ|E|2
µα =∂ρψ
∂ρα, p := −ρψ +
N∑α=0
ραµα (Gibbs-Duhem)
Jα = −N∑β=1
Mαβ
(∇(µβ−µ0T
)− 1
T
(zβe0mβ− z0e0
m0
)E)
, J0 = −N∑α=1
Jα
Σ = −p+ ε0(1 + χ)(E ⊗E − 12 |E|
21)
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Constitutive modeling
free energy ρψ = ρψ(T,ρ0,ρ1,··· ,ρN ,) + χ|E|2
µα =∂ρψ
∂ρα, p := −ρψ +
N∑α=0
ραµα (Gibbs-Duhem)
Jα = −N∑β=1
Mαβ
(∇(µβ−µ0T
)− 1
T
(zβe0mβ− z0e0
m0
)E)
, J0 = −N∑α=1
Jα
Σ = −p+ ε0(1 + χ)(E ⊗E − 12 |E|
21)
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Free energy
ρψ =
N∑α=0
ραgRefα (T )︸ ︷︷ ︸
reference values
+kT
N∑α=0
nα lnnαn︸ ︷︷ ︸
entropy of mixing
+ρψM︸ ︷︷ ︸elastic energy
elastic law: p = pRef +K (∑N
α=0 vRefα nα − 1)
vRefα : specific volume of Aα
incompressible limit K →∞: ρψM = (p− pRef )(∑N
α=0 vRefα nα − 1)
µα = gRefα + kTmα
ln nαn + vRef
αmα
(p− pRef )
Jα = −Mα
(∇nα − mα
m0
nαn0∇n0 + vRefα
(1− mαv
Ref0
m0vRefα
)∇p+ zαe0∇ϕ
)
Nernst-Planck: Jα = −Mα (∇nα + zαe0∇ϕ),
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Free energy
ρψ =
N∑α=0
ραgRefα (T )︸ ︷︷ ︸
reference values
+kT
N∑α=0
nα lnnαn︸ ︷︷ ︸
entropy of mixing
+ρψM︸ ︷︷ ︸elastic energy
elastic law: p = pRef +K (∑N
α=0 vRefα nα − 1)
vRefα : specific volume of Aα
incompressible limit K →∞: ρψM = (p− pRef )(∑N
α=0 vRefα nα − 1)
µα = gRefα + kTmα
ln nαn + vRef
αmα
(p− pRef )
Jα = −Mα
(∇nα − mα
m0
nαn0∇n0 + vRefα
(1− mαv
Ref0
m0vRefα
)∇p+ zαe0∇ϕ
)
Nernst-Planck: Jα = −Mα (∇nα + zαe0∇ϕ),
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Free energy
ρψ =
N∑α=0
ραgRefα (T )︸ ︷︷ ︸
reference values
+kT
N∑α=0
nα lnnαn︸ ︷︷ ︸
entropy of mixing
+ρψM︸ ︷︷ ︸elastic energy
elastic law: p = pRef +K (∑N
α=0 vRefα nα − 1)
vRefα : specific volume of Aα
incompressible limit K →∞: ρψM = (p− pRef )(∑N
α=0 vRefα nα − 1)
µα = gRefα + kTmα
ln nαn + vRef
αmα
(p− pRef )
Jα = −Mα
(∇nα − mα
m0
nαn0∇n0 + vRefα
(1− mαv
Ref0
m0vRefα
)∇p+ zαe0∇ϕ
)
Nernst-Planck: Jα = −Mα (∇nα + zαe0∇ϕ),
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Solvated ions
solvent moleculesanion
cation
specific volumina:
A: vRefA
∗ + κA vRefS
C: vRefC
∗ + κC vRefS
S: vRefS
q =
∫ ∑α
zαnα dx
at ϕL − ϕR = 0.25V :
experiment: |q| ≈ (0− 50) µCcm2
Nernst-Planck: |q| ≈ 600 µCcm2
simple mixture: |q| ≈ 100 µCcm2
solvated ions: |q| ≈ 28 µCcm2
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Solvated ions
solvated anion
solvated cationbounded solvent
molecules
free solvent molecules
specific volumina:
A: vRefA∗ + κA vRefS
C: vRefC∗ + κC vRefS
S: vRefS
q =
∫ ∑α
zαnα dx
at ϕL − ϕR = 0.25V :
experiment: |q| ≈ (0− 50) µCcm2
Nernst-Planck: |q| ≈ 600 µCcm2
simple mixture: |q| ≈ 100 µCcm2
solvated ions: |q| ≈ 28 µCcm2
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Solvated ions
solvated anion
solvated cationbounded solvent
molecules
free solvent molecules
specific volumina:
A: vRefA∗ + κA vRefS
C: vRefC∗ + κC vRefS
S: vRefS
q =
∫ ∑α
zαnα dx
at ϕL − ϕR = 0.25V :
experiment: |q| ≈ (0− 50) µCcm2
Nernst-Planck: |q| ≈ 600 µCcm2
simple mixture: |q| ≈ 100 µCcm2
solvated ions: |q| ≈ 28 µCcm2
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Solvated ions
solvated anion
solvated cationbounded solvent
molecules
free solvent molecules
specific volumina:
A: vRefA∗ + κA vRefS
C: vRefC∗ + κC vRefS
S: vRefS
q =
∫ ∑α
zαnα dx
at ϕL − ϕR = 0.25V :
experiment: |q| ≈ (0− 50) µCcm2
Nernst-Planck: |q| ≈ 600 µCcm2
simple mixture: |q| ≈ 100 µCcm2
solvated ions: |q| ≈ 28 µCcm2
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1 Ion transport in electrolyte bulkShortcomings of Nernst-Planck model
General continuum model for the bulk
2 Electrochemical double layerSurface balances and adsorption
Free energy models for metals
Double-layer capacity
3 Lippmann equation for liquid mercury electrodeMatched asymptotic analysis
Numerical results
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Surface balances
volume: A0, A1, · · · , AN surface: A0, A1, · · · , AN , · · · , ANS
bulk Ω±
∂tρα + div(ραv + Jα) = 0
−Σ = ρb
−ε0(1 + χ)∆ϕ = nF
surface S
∂tρsα + [[(ρα(v −w) +Jα) · ν]] = 0
−[[Σ ·ν]] = ρsbs
+ 2kMγsν
−ε0[[(1 + χ)∂nϕ]] = ns
F
ρψ = ρψ(T,ρ0,ρ1,··· ,ρN) + χ|E|2 ρsψs= ρ
sψs(Ts,ρs0,ρs1,··· ,ρ
sNS
)
µα =∂ρψ
∂ραµsα=
∂ρsψs
∂ρsα
p := −ρψ +
N∑α=0
ραµα γs:= ρ
sψs−
NS∑α=0
ρsαµsα
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Surface balances
volume: A0, A1, · · · , AN surface: A0, A1, · · · , AN , · · · , ANS
bulk Ω±
∂tρα + div(ραv + Jα) = 0
−Σ = ρb
−ε0(1 + χ)∆ϕ = nF
surface S
∂tρsα + [[(ρα(v −w) +Jα) · ν]] = 0
−[[Σ ·ν]] = ρsbs
+ 2kMγsν
−ε0[[(1 + χ)∂nϕ]] = ns
F
ρψ = ρψ(T,ρ0,ρ1,··· ,ρN) + χ|E|2 ρsψs= ρ
sψs(Ts,ρs0,ρs1,··· ,ρ
sNS
)
µα =∂ρψ
∂ραµsα=
∂ρsψs
∂ρsα
p := −ρψ +
N∑α=0
ραµα γs:= ρ
sψs−
NS∑α=0
ρsαµsα
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Free energy of metal
incompressible simple mixture of metal ions M and electrons e
vRefe := 0, p = pe + pM
goal: ρψ = ρMψM (ρM ) + ρeψe(ρe), ρsψs(ρs
0, · · · , ρsN , ρ
se, ρsM )
ρMψM = pRef + vRefM nM , µe =(
38π
)2/3 h2me
n2/3e (Drude-Sommerfeld)
coupling to surface
Constitutive laws (fast adsoption limit)
0 =µ±α−µ±0
T −µsα−µ
s0
Ts
, 0 =µ±0T −
µs0
Ts
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Free energy of metal
incompressible simple mixture of metal ions M and electrons e
vRefe := 0, p = pe + pM
goal: ρψ = ρMψM (ρM ) + ρeψe(ρe), ρsψs(ρs
0, · · · , ρsN , ρ
se, ρsM )
ρMψM = pRef + vRefM nM , µe =(
38π
)2/3 h2me
n2/3e (Drude-Sommerfeld)
coupling to surface
Constitutive laws (fast adsoption limit)
0 =µ±α−µ±0
T −µsα−µ
s0
Ts
, 0 =µ±0T −
µs0
Ts
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Free energy of metal surface
Volume mixtureCation
SolventAnion
Solvation shell
Bound solvent
Cation
Partial solvation shell
Vacancy
SolventAnion
Surface mixtureadsorption sides
nsM = n
s
RefM
vacancies
nsV = n
sM −
∑α nsα
µsα = µ
s
Refα + kT
mαln
(nsα
nsV
)µsM = µ
s
RefM − aRef
MmM
γs
+ kTmM
ln
(nsV
nsM
)
[Kohn,Lang:1971]
metal – vacuum: ϕM − ϕV ∼ 7V
metal – surface: ϕM − ϕsV ∼ 2V
µse = µ
s
Refe (!) depending on surface orientation
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Free energy of metal surface
Volume mixtureCation
SolventAnion
Solvation shell
Bound solvent
Cation
Partial solvation shell
Vacancy
SolventAnion
Surface mixtureadsorption sides
nsM = n
s
RefM
vacancies
nsV = n
sM −
∑α nsα
µsα = µ
s
Refα + kT
mαln
(nsα
nsV
)µsM = µ
s
RefM − aRef
MmM
γs
+ kTmM
ln
(nsV
nsM
)
[Kohn,Lang:1971]
metal – vacuum: ϕM − ϕV ∼ 7V
metal – surface: ϕM − ϕsV ∼ 2V
µse = µ
s
Refe (!) depending on surface orientation
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Free energy of metal surface
Volume mixtureCation
SolventAnion
Solvation shell
Bound solvent
Cation
Partial solvation shell
Vacancy
SolventAnion
Surface mixtureadsorption sides
nsM = n
s
RefM
vacancies
nsV = n
sM −
∑α nsα
µsα = µ
s
Refα + kT
mαln
(nsα
nsV
)µsM = µ
s
RefM − aRef
MmM
γs
+ kTmM
ln
(nsV
nsM
)
[Kohn,Lang:1971]
metal – vacuum: ϕM − ϕV ∼ 7V
metal – surface: ϕM − ϕsV ∼ 2V
µse = µ
s
Refe (!) depending on surface orientation
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Capacity
differential capacity: C :=dQ
dEQ := q
s+ q
G. Valette, J. Electroanal. Chem. 122 (1981), 285-297 W. Dreyer, C. Guhlke, M. Landstorfer, Electrochim. Acta, submitted,(2015)
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Electrochemical double layer
D. Kolb, Surface Science, 500(1-3),722-740, (2002)
W. Dreyer, C. Guhlke, M. Landstorfer, Electrochem. Commun., 43,75-78, (2014)
O. Stern, Z. Elektrochem. angew. phys. Chem., 20(21-22),508-516, (1924)
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Electrochemical double layer
D. Kolb, Surface Science, 500(1-3),722-740, (2002) W. Dreyer, C. Guhlke, M. Landstorfer, Electrochem. Commun., 43,75-78, (2014)
O. Stern, Z. Elektrochem. angew. phys. Chem., 20(21-22),508-516, (1924)
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1 Ion transport in electrolyte bulkShortcomings of Nernst-Planck model
General continuum model for the bulk
2 Electrochemical double layerSurface balances and adsorption
Free energy models for metals
Double-layer capacity
3 Lippmann equation for liquid mercury electrodeMatched asymptotic analysis
Numerical results
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Challanges
Lippmann dγdE = −Q
Butler-Volmer Rs
= R0fs
exp(αf
e0kT ηS
)R0bs
exp(−αb e0kT ηS
)thermodynamic variable:E not ϕ
sharp boundary layers, Debye length: λLRef
−(1 + χ)ε0∂xxϕ = nF −λ2(1 + χ)∂ξξϕ = nF
λ =
√ε0kT
e20n
Ref (LRef )2, λδ =
ns
Ref
nRefLRef
LRef = 1cm, nRef = 6.022 · 1025 m−3, ns
Ref = 7.3 · 1018 m−2
λ ≈ 1.54 · 10−8 , λδ ≈ 1.21 · 10−5 , λ|b| ≈ 2.38 · 10−8 .
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Challanges
Lippmann dγdE = −Q
Butler-Volmer Rs
= R0fs
exp(αf
e0kT ηS
)R0bs
exp(−αb e0kT ηS
)thermodynamic variable:E not ϕ
sharp boundary layers, Debye length: λLRef
−(1 + χ)ε0∂xxϕ = nF −λ2(1 + χ)∂ξξϕ = nF
λ =
√ε0kT
e20n
Ref (LRef )2, λδ =
ns
Ref
nRefLRef
LRef = 1cm, nRef = 6.022 · 1025 m−3, ns
Ref = 7.3 · 1018 m−2
λ ≈ 1.54 · 10−8 , λδ ≈ 1.21 · 10−5 , λ|b| ≈ 2.38 · 10−8 .
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Challanges
Lippmann dγdE = −Q
Butler-Volmer Rs
= R0fs
exp(αf
e0kT ηS
)R0bs
exp(−αb e0kT ηS
)thermodynamic variable:E not ϕ
sharp boundary layers, Debye length: λLRef
−(1 + χ)ε0∂xxϕ = nF −λ2(1 + χ)∂ξξϕ = nF
λ =
√ε0kT
e20n
Ref (LRef )2, λδ =
ns
Ref
nRefLRef
LRef = 1cm, nRef = 6.022 · 1025 m−3, ns
Ref = 7.3 · 1018 m−2
λ ≈ 1.54 · 10−8 , λδ ≈ 1.21 · 10−5 , λ|b| ≈ 2.38 · 10−8 .
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Rescaled equations
Assumptions
equilibrium: ∂t· = 0, v = 0
δ ∼ O(1)
kM · λLRef 1
bulk Ω±
∇(µα + zαϕ) = 0
∂xΣ = λρb
−λ2(1 + χ)∂xxϕ = nF
surface S
(µα + zαϕ))± = µsα + zαϕ
s
[[Σ · ν]] = λδ(ρsbs
+ 2kMγs)
−[[λ(1 + χ)∂xϕν]] = δns
F
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Formal Asymptotic Expansion
r(s , s )
S
Ω−
Ω+
1 2
ν
x = r(s , s )1 2 + z ν
rescale in layer
u(r, z) := u(r + λzν)
formal asymptotics expansions for 0 < λ 1
in bulk: u = u(0) + λu(1) +O(λ2)
in layer: u = u(0) + λu(1) +O(λ2)
matching conditions:
u(0)(z)− u(0),±(x(0)S ) = o(1/|z|)
∂zu(0)(z) = o(1/|z|)
in higher order
u(1)(z)− z ∂νu(0),± − u(1),± = o(1/|z|)
∂zu(1)(z)−∂νu(0),± = o(1/|z|)
zu
S
(x)uλ
λ( )~
λ
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Formal Asymptotic Expansion
r(s , s )
S
Ω−
Ω+
1 2
ν
x = r(s , s )1 2 + z ν
rescale in layer
u(r, z) := u(r + λzν)
formal asymptotics expansions for 0 < λ 1
in bulk: u = u(0) + λu(1) +O(λ2)
in layer: u = u(0) + λu(1) +O(λ2)
matching conditions:
u(0)(z)− u(0),±(x(0)S ) = o(1/|z|)
∂zu(0)(z) = o(1/|z|)
in higher order
u(1)(z)− z ∂νu(0),± − u(1),± = o(1/|z|)
∂zu(1)(z)−∂νu(0),± = o(1/|z|)
zu
S
(x)uλ
λ( )~
λ
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Inner equations
∇u = λ−1∂zuν + |τ1|−1∂1u τ1 + |τ2|−1∂2u τ2 +O(λ) ,
div(u) = λ−1∂zu · ν + divτ (u) +O(λ) ,
−∆u = −λ−2∂zzu− λ−12kM∂zu+O(1) ,
Σνν = −p+ 1+χ2 |∂zϕ|
2 +O(λ2)
layer
λ−1(∂z p+ nF ∂zϕ) +O(λ) = λρb
−(1 + χ)(∂zzϕ+ λ 2kM∂zϕ) +O(λ2) = nF
S
−[[Σνν ]] = λ δ 2kM γs
+ λ2 δ ρsbs· ν +O(λ2)
−[[(1 + χ)∂zϕ]] = δns
F +O(λ)
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Leading order
layer: Σ(0)νν = −p(0) + 1+χ
2 |∂zϕ(0)|2 bulk: Σ
(0)νν = −p(0)
layer
−∂zΣ(0)νν = 0 ,
−(1 + χ)∂zzϕ(0) = nF,(0).
S
−[[Σ(0)νν ]] = 0 ,
−[[(1 + χ)∂zϕ(0)]] = δ n
s
F,(0)
Σ(0)νν (−∞) = Σ
(0)νν (∞) =⇒ [[[p(0)]]] = 0
∫∞−∞ ∂zzϕ
(0) dz = ∂zϕ(0)|0−∞ + ∂zϕ
(0)|∞0 = −[[∂zϕ(0)]]
q+ :=∫∞
0 nF,(0) dz, q− :=∫ 0−∞ n
F,(0) dz
global electro-neutrality of double layer: δ ns
F,(0) + q− + q+ = 0
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Leading order
layer: Σ(0)νν = −p(0) + 1+χ
2 |∂zϕ(0)|2 bulk: Σ
(0)νν = −p(0)
layer
−∂zΣ(0)νν = 0 ,
−(1 + χ)∂zzϕ(0) = nF,(0).
S
−[[Σ(0)νν ]] = 0 ,
−[[(1 + χ)∂zϕ(0)]] = δ n
s
F,(0)
Σ(0)νν (−∞) = Σ
(0)νν (∞) =⇒ [[[p(0)]]] = 0
∫∞−∞ ∂zzϕ
(0) dz = ∂zϕ(0)|0−∞ + ∂zϕ
(0)|∞0 = −[[∂zϕ(0)]]
q+ :=∫∞
0 nF,(0) dz, q− :=∫ 0−∞ n
F,(0) dz
global electro-neutrality of double layer: δ ns
F,(0) + q− + q+ = 0
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Higher order
Σ(1)νν = −p(1) + (1 + χ)∂zϕ
(0)∂zϕ(1)
layer
∂z p(1) + nF,(0)∂zϕ
(1) + nF,(1)∂zϕ(0) = 0 ,
−(1 + χ)(∂zzϕ
(1) + 2kM ∂zϕ(0))
= nF,(1)
S
−[[Σ(1)νν ]] = δ 2kM γ
s
∂zΣ(1)νν = −2kM (1 + χ)(∂zϕ
(0))2
γ+ :=∫∞
0 (1 + χ)(∂zϕ(0))2 dz , γ− :=
∫ 0−∞(1 + χ)(∂zϕ
(0))2 dz
Young-Laplace [[[p(1)]]] = 2kM (γs
+ γ+ − γ−)
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Higher order
Σ(1)νν = −p(1) + (1 + χ)∂zϕ
(0)∂zϕ(1)
layer
∂z p(1) + nF,(0)∂zϕ
(1) + nF,(1)∂zϕ(0) = 0 ,
−(1 + χ)(∂zzϕ
(1) + 2kM ∂zϕ(0))
= nF,(1)
S
−[[Σ(1)νν ]] = δ 2kM γ
s
∂zΣ(1)νν = −2kM (1 + χ)(∂zϕ
(0))2
γ+ :=∫∞
0 (1 + χ)(∂zϕ(0))2 dz , γ− :=
∫ 0−∞(1 + χ)(∂zϕ
(0))2 dz
Young-Laplace [[[p(1)]]] = 2kM (γs
+ γ+ − γ−)
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Higher order
Σ(1)νν = −p(1) + (1 + χ)∂zϕ
(0)∂zϕ(1)
layer
∂z p(1) + nF,(0)∂zϕ
(1) + nF,(1)∂zϕ(0) = 0 ,
−(1 + χ)(∂zzϕ
(1) + 2kM ∂zϕ(0))
= nF,(1)
S
−[[Σ(1)νν ]] = δ 2kM γ
s
∂zΣ(1)νν = −2kM (1 + χ)(∂zϕ
(0))2
γ+ :=∫∞
0 (1 + χ)(∂zϕ(0))2 dz , γ− :=
∫ 0−∞(1 + χ)(∂zϕ
(0))2 dz
Young-Laplace [[[p(1)]]] = 2kM (γs
+ γ+ − γ−)
Lippmann Equation · March 3. 2015 · Page 25 (29)
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Lippmann equation
interfacial tension γ := γs
+ γ+ − γ−
Ω+ metal, apply potential E := ϕs− ϕ−
µse = const. → constant layer in metal =⇒ d
dE γ+ = 0
γ− = (1 + χ)∫ 0−∞(∂zϕ
(0))2 dz
= (1 + χ)∫ 0E
√p(ϕ+ ϕ−) dϕ [DGM13]
ddE γ
− = −(1 + χ)√p|−S = −q− [DGL2014pre]
ddEγ
s= d
dE
(ρsψs−∑nsαµsα
)= d
dE
(ρsψs−∑
α∈Ω− nα(µα|−S + zαE))
= −qs
[Guhlke2015]
double layer charge: Q := qs
+ q− Lippmann: ddEγ = −Q
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Lippmann equation
interfacial tension γ := γs
+ γ+ − γ−
Ω+ metal, apply potential E := ϕs− ϕ−
µse = const. → constant layer in metal =⇒ d
dE γ+ = 0
γ− = (1 + χ)∫ 0−∞(∂zϕ
(0))2 dz
= (1 + χ)∫ 0E
√p(ϕ+ ϕ−) dϕ [DGM13]
ddE γ
− = −(1 + χ)√p|−S = −q− [DGL2014pre]
ddEγ
s= d
dE
(ρsψs−∑nsαµsα
)= d
dE
(ρsψs−∑
α∈Ω− nα(µα|−S + zαE))
= −qs
[Guhlke2015]
double layer charge: Q := qs
+ q− Lippmann: ddEγ = −Q
Lippmann Equation · March 3. 2015 · Page 26 (29)
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Lippmann equation
interfacial tension γ := γs
+ γ+ − γ−
Ω+ metal, apply potential E := ϕs− ϕ−
µse = const. → constant layer in metal =⇒ d
dE γ+ = 0
γ− = (1 + χ)∫ 0−∞(∂zϕ
(0))2 dz
= (1 + χ)∫ 0E
√p(ϕ+ ϕ−) dϕ [DGM13]
ddE γ
− = −(1 + χ)√p|−S = −q− [DGL2014pre]
ddEγ
s= d
dE
(ρsψs−∑nsαµsα
)= d
dE
(ρsψs−∑
α∈Ω− nα(µα|−S + zαE))
= −qs
[Guhlke2015]
double layer charge: Q := qs
+ q− Lippmann: ddEγ = −Q
Lippmann Equation · March 3. 2015 · Page 26 (29)
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Lippmann equation
interfacial tension γ := γs
+ γ+ − γ−
Ω+ metal, apply potential E := ϕs− ϕ−
µse = const. → constant layer in metal =⇒ d
dE γ+ = 0
γ− = (1 + χ)∫ 0−∞(∂zϕ
(0))2 dz
= (1 + χ)∫ 0E
√p(ϕ+ ϕ−) dϕ [DGM13]
ddE γ
− = −(1 + χ)√p|−S = −q− [DGL2014pre]
ddEγ
s= d
dE
(ρsψs−∑nsαµsα
)= d
dE
(ρsψs−∑
α∈Ω− nα(µα|−S + zαE))
= −qs
[Guhlke2015]
double layer charge: Q := qs
+ q− Lippmann: ddEγ = −Q
Lippmann Equation · March 3. 2015 · Page 26 (29)
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Numerical results
Interfacialtension
γInt[1
0−3N/m
]
Potential E (vs. UR) [V]
cNaClO4=0.005
cNaClO4=0.01
cNaClO4=0.02
cNaClO4=0.04
cNaClO4=0.1
−0.4 −0.2 0 0.2 0.4 0.6160
170
180
190
200
210
220
230
240
Salt concentration: c = 0.01M
Interfacialtension
γInt[1
0−3N/m
]
Potential E (vs. UR) [V]
NaF
NaClO4
KPF6
−0.4 −0.2 0 0.2 0.4 0.6160
170
180
190
200
210
220
230
240
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Numerical results
Interfacialtension
γInt[1
0−3N/m
]
Potential E (vs. UR) [V]
cNaClO4=0.005
cNaClO4=0.01
cNaClO4=0.02
cNaClO4=0.04
cNaClO4=0.1
−0.4 −0.2 0 0.2 0.4 0.6160
170
180
190
200
210
220
230
240
Salt concentration: c = 0.01M
Interfacialtension
γInt[1
0−3N/m
]
Potential E (vs. UR) [V]
NaF
NaClO4
KPF6
−0.4 −0.2 0 0.2 0.4 0.6160
170
180
190
200
210
220
230
240
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Validity of asymptotic results
rS = 25nm , c = 0.1 M
Bound
arylayertension
γE BL[1
0−3N/m
]
Cell voltage E [V] (vs. SCE)
asymp. approximationε0(1 + χ)
´
(rS/r)E2r dr (numercial solution)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−250
−200
−150
−100
−50
0
rS = 5nm , c = 0.04 M
Bound
arylayertension
γE BL[1
0−3N/m
]
Cell voltage E [V] (vs. SCE)
asymp. approximationε0(1 + χ)
´
(rS/r)E2r dr (numercial solution)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−250
−200
−150
−100
−50
0
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Validity of asymptotic results
rS = 25nm , c = 0.1 M
Bound
arylayertension
γE BL[1
0−3N/m
]
Cell voltage E [V] (vs. SCE)
asymp. approximationε0(1 + χ)
´
(rS/r)E2r dr (numercial solution)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−250
−200
−150
−100
−50
0
rS = 1nm , c = 0.04 M
Bound
arylayertension
γE BL[1
0−3N/m
]
Cell voltage E [V] (vs. SCE)
asymp. approximationε0(1 + χ)
´
(rS/r)E2r dr (numercial solution)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−250
−200
−150
−100
−50
0
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Summary & Conclusions
pressure dependence, momentum balance necessary
thermodynamically consistent complete model derived
incompressible mixture model for metal surface; µse = const.
qualitative & quantitative prediction of differential capacity
jump conditions by formal asymptotic analysis
electric field contributes to interfacial tension
Lippmann equation recovered within asymptotic limit
asymptotic not valid in nm range
Thank you for your attention! Questions?
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Summary & Conclusions
pressure dependence, momentum balance necessary
thermodynamically consistent complete model derived
incompressible mixture model for metal surface; µse = const.
qualitative & quantitative prediction of differential capacity
jump conditions by formal asymptotic analysis
electric field contributes to interfacial tension
Lippmann equation recovered within asymptotic limit
asymptotic not valid in nm range
Thank you for your attention! Questions?
Lippmann Equation · March 3. 2015 · Page 29 (29)