Weierstrass Institute for
Applied Analysis and Stochastics
Numerical Strategies for Electrolyte
Simulations
J. Fuhrmann
Mohrenstraße 39 · 10117 Berlin · www.wias-berlin.de
PEM2013 · NTNU Trondheim · 2013-10-03
Coupled flow and ion transport without electroneutrality assumption
Relevant in a significant number of applications:
PEM nanopores
nanofluidic devices
ion channels in biomembranes
porous electrode microstructures
overlimiting current in electrodialysis
Search for numerical approach which
takes into account recent work on excluded volume models
incorporates the knowledge from semiconductor device simulation
(unconditionally stable Scharfetter-Gummel scheme)
guarantees mass conservation due to discrete divergence condition
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 2
Problem formulation: Nernst-Planck-Poisson system
Mixture of N species; species N : neutral solvent; diagonal Onsager coefficients
To be closed by constitutive relationship for cα
−∇ε0 εr ∇φ = q
∂t cα + ∇ · (cα v + Nα ) = 0
α = 1...N −1
Dα
Mα
Nα = −
cα ∇µα −
∇µN + zα F∇φ
RT
MN
Dα
cα (∇µ̃α + zα F∇φ )
=−
RT
φ
cα
µα
µ̃α = µα −
Nα
v
q
zα
Mα
Dα
Mα
MN
µN
V
mol/m3
J/mol
J/mol
mol/(m2 · s)
m/s
As/m3
1
kg/mol
m2 /s
α = 1...N −1
α = 1...N −1
electrostatic potential
molar density (concentration)
chemical potential (α = 1 . . . N )
effective chemical potential (α = 1 . . . N − 1)
molar diffusion flux
barycentric velocity
space charge density
charge number
molar mass
diffusion coefficient
deGroot/Mazur 1962; Dreyer/Guhlke/Müller PCCP 2013
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 3
Problem formulation: Navier-Stokes equations
Incompressible flow of mixture: evolution of barycentric velocity v
∂t (ρv) + ∇ · (ρv ⊗ v) − η∆v + ∇p = −q∇φ
∂t ρ + ∇ · (ρv) = 0
N−1
ρ = MN c +
∑ (Mα − MN )cα
α=1
v
p
η
ρ
c = ∑Nα=1 cα
m/s
Pa
Pas
kg/m3
mol/m3
barycentric velocity
pressure
viscosity
density
(constant) summary concentration
deGroot/Mazur 1962; Dreyer/Guhlke/Müller PCCP 2013
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 4
Constitutive relationship for µ : dilute solution theory
ideal dilute solution
ions as point charges
µN = 0
µα = µ̃α = µα◦ + RT ln
cα
c
(α = 1 . . . N − 1).
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 5
Constitutive relationship for µ : dilute solution theory
ideal dilute solution
ions as point charges
µN = 0
µα = µ̃α = µα◦ + RT ln
cα
c
(α = 1 . . . N − 1).
Unphysically high concentrations in polarization boundary layer!
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 5
Constitutive relationship for µ : excluded volume models
Account for energy needed for species to enter already crowded configuration:
µN = 0
µα = µ̃α = µα◦ + RT ln
1
cα
+ ln
c
1−Φ
(α = 1 . . . N − 1)
Φ: Summary volume fraction of the dissolved species:
N−1
Φ=
∑ vα cα ,
α=1
vα : partial molar volume
E.g. estimate from spherical packing:
4
vα = 8NA πrα3 .
3
E.g. all molecules share one one lattice (Bikerman model)
1
vα = v = ,
c
cN
1−Φ =
c
Bikerman, Phil. Mag. 1942; Paunov, J. Coll. Interf. Sci. 1996; Biesheuvel/v.Soestbergen, J. Coll. Interf. Sci. 2007
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 6
Constitutive relationship for µ : pressure correction due to Dreyer et al.
Derivation from nonequilibrium thermodynamics leads to a constitutive
relationship which links pressure, concentrations and chemical potential:
cα
1
µα = µα◦ + (p − p◦ ) + RT ln
c
c
cα
Mα (p − p◦ ) Mα
1
◦
µ̃α = µα + RT ln
+ 1−
+
RT ln
c
MN
c
MN
1−Φ
(α = 1 . . . N)
(α = 1 . . . N − 1)
Gibbs Duhem + Thermodynamic equilibrium ⇒
1
cN
1
(p − p◦ ) = −RT ln
= RT ln
c
c
1−Φ
⇒ Equivalent to Bikerman model (v = 1c ):
in thermodynamical equilibrium
for equal molar masses
Limitations (currently under consideration in Dreyer group):
constant summary concentration c
no notation yet for different molar volumes
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 7
Dreyer/Guhlke/Müller PCCP 2013
Mechanical equilibrium
How to use pressure correction wihtout Navier-Stokes ?
∂t v = 0
∇v = 0
⇒ ∇p = −q∇φ
Taking the divergence gives as second order equation for p:
−∆p = ∇ · q∇φ
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 8
Constitutive relationship for µ : Fermi statistics
See semiconductor device theory:
cα = cG
.
G : (approximations of) Fermi-Dirac integrals of different orders:
G (η) ≈ F j (η) =
µα − µα◦
RT
1
Γ( j + 1)
Z ∞
0
ξj
dξ
1 + exp(ξ − η)
Special case: Fermi integral of order −1
F−1 (η) =
1
1 + exp(−η)
⇒
µ̃α = µα = µα◦ + RT ln
cα
1
+ RT ln
c
1 − ccα
(α = 1 . . . N − 1)
Resembles the excluded volume expression restricted to species α .
No interaction term between different species
Kornyshev 1981, Landstorfer 2011
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 9
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Dilute solution (Boltzmann) → classical Nernst Planck
NNP
α
zα F
= −Dα ∇cα +
cα ∇φ
RT
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Dilute solution (Boltzmann) → classical Nernst Planck
NNP
α
zα F
= −Dα ∇cα +
cα ∇φ
RT
Excluded volume model (Bikerman if vβ = v = 1c ):
N−1
Nα = Dα
∇cα − cα
1−
∑ vβ cβ
β =1
!
N−1
F
∑ vβ ∇cβ − cα zα RT ∇φ
β =1
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
!
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Dilute solution (Boltzmann) → classical Nernst Planck
NNP
α
zα F
= −Dα ∇cα +
cα ∇φ
RT
Excluded volume model (Bikerman if vβ = v = 1c ):
N−1
Nα = Dα
∇cα − cα
1−
∑ vβ cβ
β =1
!
N−1
F
∑ vβ ∇cβ − cα zα RT ∇φ
β =1
Pure pressure correction :
zα F
1
Nα = −Dα ∇cα +
cα ∇p +
cα ∇φ
cRT
RT
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
!
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Dilute solution (Boltzmann) → classical Nernst Planck
NNP
α
zα F
= −Dα ∇cα +
cα ∇φ
RT
Excluded volume model (Bikerman if vβ = v = 1c ):
N−1
Nα = Dα
∇cα − cα
1−
!
∑ vβ cβ
β =1
N−1
F
∑ vβ ∇cβ − cα zα RT ∇φ
β =1
Pure pressure correction :
zα F
1
Nα = −Dα ∇cα +
cα ∇p +
cα ∇φ
cRT
RT
Fermi statistics:
F
Nα = −Dα dα ∇cα + cα zα
∇φ
RT
1
cα
G =F−1
=
dα =
1 − ccα
cG 0 G −1 ccα
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
!
Concentration based formulation
µα → −∞ for small concentrations, especially in depletion regions
Dilute solution (Boltzmann) → classical Nernst Planck
NNP
α
zα F
= −Dα ∇cα +
cα ∇φ
RT
Wrong in boundary layer.
Excluded volume model (Bikerman if vβ = v = 1c ):
N−1
Nα = Dα
∇cα − cα
1−
!
∑ vβ cβ
β =1
Pure pressure correction :
Complicated flux coupling.
N−1
F
∑ vβ ∇cβ − cα zα RT ∇φ
β =1
!
Insufficient pressure regularity in Navier-Stokes.
zα F
1
Nα = −Dα ∇cα +
cα ∇p +
cα ∇φ
cRT
RT
Fermi statistics:
F
Nα = −Dα dα ∇cα + cα zα
∇φ
RT
1
cα
G =F−1
=
dα =
1 − ccα
cG 0 G −1 ccα
Degenerates for cα → c.
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 10
Activity based formulation
Expression of effective chemical potential similar to dilute solution case defines
activity :
µ̃α = µ̃α◦ + RT ln aα
Activity coefficient γα :
aα = γα
1
γα
cα
c
⇒ cα = c
aα
= cβα aα
γα
: inverse activity coefficient
βα =
Nernst-Planck-Poisson transforms to:
∂t (cβα aα ) + ∇ · (cβα aα v + Nα ) = 0
α = 1...N −1
−∇ε0 εr ∆φ = q
F
Nα = −Dα cβα ∇aα + aα zα
∇φ
RT
N−1
q = cF
∑ zα βα aα
α=1
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 11
α = 1...N −1
Inverse activity coefficients
Dilute solution:
βα = 1
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
Inverse activity coefficients
Dilute solution:
βα = 1
Volume exclusion models
βα = β =
1
1 + c ∑N−1
i=1 vi ai
vi = 1c
=
1
N−1
1 + ∑i=1
ai
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
Inverse activity coefficients
Dilute solution:
βα = 1
Volume exclusion models
βα = β =
vi = 1c
1
1 + c ∑N−1
i=1 vi ai
=
1
N−1
1 + ∑i=1
ai
Pure pressure correction model
βα = exp
p − p◦
cRT
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
Inverse activity coefficients
Dilute solution:
βα = 1
Volume exclusion models
βα = β =
1 + c ∑N−1
i=1 vi ai
=
1
N−1
1 + ∑i=1
ai
Pure pressure correction model
βα = exp
vi = 1c
1
p − p◦
cRT
Full pressure correction model (weighted with volume exclusion):
Mα
p − p◦ 1− MN
βα = exp
cRT
N−1
1−
∑ vi βi cai
i=1
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
! Mα
MN
Inverse activity coefficients
Dilute solution:
βα = 1
Volume exclusion models
βα = β =
1 + c ∑N−1
i=1 vi ai
p − p◦
cRT
Full pressure correction model (weighted with volume exclusion):
Mα
p − p◦ 1− MN
βα = exp
cRT
1
N−1
1 + ∑i=1
ai
=
Pure pressure correction model
βα = exp
vi = 1c
1
N−1
1−
∑ vi βi cai
i=1
Fermi statistics:
βα (aα ) =
G (ln aα )
aα
G =F−1
=
1
1 + aα
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
! Mα
MN
Inverse activity coefficients
Dilute solution:
βα = 1
Volume exclusion models
No cross-coupling of gradients.
βα = β =
1 + c ∑N−1
i=1 vi ai
1
N−1
1 + ∑i=1
ai
=
Pure pressure correction model No pressure gradient.
βα = exp
vi = 1c
1
p − p◦
cRT
Full pressure correction model (weighted with volume exclusion):
Mα
p − p◦ 1− MN
βα = exp
cRT
N−1
1−
∑ vi βi cai
i=1
No pressure gradient.
Fermi statistics:
βα (aα ) =
G (ln aα )
aα
G =F−1
=
1
1 + aα
No degeneration.
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 12
! Mα
MN
Equilibrium case: Nonlinear Poisson equation
Mechanical and thermodynamical equilibrium: Nα = 0:
∇µ̃α = −zα F∇φ
ψα : (constant) quasi-Fermi potential
Nonlinear Poisson equation:
⇒ µ̃α = zα F(ψα − φ )
N−1
−∇ε0 εr ∇φ = Fc
∑ zα βα aα
α=1
aα = exp
zα F
(ψα − φ )
RT
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 13
Generalizations of Poisson-Boltzmann
“classical” Poisson-Boltzmann:
N−1
−∇ε0 εr ∇φ = Fc
∑
α=1
zα exp
zα F
(ψα − φ )
RT
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 14
Generalizations of Poisson-Boltzmann
“classical” Poisson-Boltzmann:
N−1
−∇ε0 εr ∇φ = Fc
∑
zα exp
α=1
zα F
(ψα − φ )
RT
Excluded volume model (Bikerman):
−∇ε0 εr ∇φ = Fc
∑N−1
α=1 zα exp
zα F
RT (ψα
1 + c ∑N−1
α=1 vα exp
−φ)
zα F
RT (ψα
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 14
−φ)
Generalizations of Poisson-Boltzmann
“classical” Poisson-Boltzmann:
N−1
−∇ε0 εr ∇φ = Fc
∑
zα exp
α=1
Excluded volume model (Bikerman):
−∇ε0 εr ∇φ = Fc
zα F
(ψα − φ )
RT
∑N−1
α=1 zα exp
zα F
RT (ψα
1 + c ∑N−1
α=1 vα exp
−φ)
zα F
RT (ψα
−φ)
Pure pressure correction (Dreyer et al.):
N−1
zα F
p − p◦
−∇ε0 εr ∇φ = q = exp −
cF ∑ zα exp
(ψα − φ )
cRT
RT
α=1
∆p = −∇ · q∇φ
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 14
Generalizations of Poisson-Boltzmann
“classical” Poisson-Boltzmann:
N−1
−∇ε0 εr ∇φ = Fc
∑
zα exp
α=1
Excluded volume model (Bikerman):
−∇ε0 εr ∇φ = Fc
zα F
(ψα − φ )
RT
∑N−1
α=1 zα exp
zα F
RT (ψα
1 + c ∑N−1
α=1 vα exp
−φ)
zα F
RT (ψα
−φ)
Pure pressure correction (Dreyer et al.):
N−1
zα F
p − p◦
−∇ε0 εr ∇φ = q = exp −
cF ∑ zα exp
(ψα − φ )
cRT
RT
α=1
∆p = −∇ · q∇φ
Poisson-Fermi (G = F−1 ):
−φ)
.
= Fc ∑
zα F
(ψα − φ )
α=1 1 + exp RT
N−1
−∇ε0 εr ∇φ =
zα exp
zα F
RT (ψα
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 14
Numerics: Voronoi boxes + time grid
Definition
Voronoï box K around xK ∈ point set P :
Set of points x ∈ Ω which are closer to xK
than to any other point xL of P . The set of
Voronoï boxes is called Voronoï diagram
Voronoï diagram and Delaunay
triangulation are dual to each other:
Vertices of the Voronoï boxes are
triangle cirumcenters.
Green: Voronoï boxes
Black: Delaunay triangles.
Voronoï box boundaries are straight
line segments orthogonal to the
corresponding triangle edges.
Similar construction in 3D
Time grid in [0, T ]:
0 = t0 < t1 < · · · < tn−1 < tn < · · · < T
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 15
Numerics: Voronoi finite volume method
Flux equation:
n
Zt Z
0=
(∂t c + ∇ · N) dx dt
t n−1 K
n
n
Zt Z
=
Zt Z
∂t c dxdt +
t n−1 K
Z
=
K
n
N · n ds
t n−1 ∂ K
n−1
(c − c
Zt
) dxdt +
∑
Z
L neighbor of K n−1
∂ K∩∂ L
t
Approximation step (NKL : flux projection onto edge xK xL )
|K|
n
cnK − cn−1
n
K
+
=0
∑ |∂ K ∩ ∂ L|NKL
t n − t n−1 L neighbor
of K
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 16
N · nKL ds
Numerics: Voronoi finite volume method
Flux equation:
n
Zt Z
0=
(∂t c + ∇ · N) dx dt
t n−1 K
n
n
Zt Z
=
Zt Z
∂t c dxdt +
t n−1 K
Z
=
Zt
) dxdt +
∑
cnK − cn−1
n
K
+
=0
∑ |∂ K ∩ ∂ L|NKL
t n − t n−1 L neighbor
of K
Poisson equation (EKL : electric field projection onto edge xK xL ):
∑
L neighbor of K
n
Z
L neighbor of K n−1
∂ K∩∂ L
t
Approximation step (NKL : flux projection onto edge xK xL )
|K|
n−1
(c − c
K
n
N · n ds
t n−1 ∂ K
n
|∂ K ∩ ∂ L|EKL
= |K|qnK
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 16
N · nKL ds
Numerics: Edge fluxes
Electric field projection:
n
EKL
= εε0
φKn − φLn
.
xK − xL
Flux field projection using ansatz from semiconductor devices
ξ
zF
(B(ξ ) = exp(ξ )−1 : Bernoulli function, Z = RT
):
n
NKL
= β̄KL D(B(Z(φLn − φKn ))anK − B(Z(φKn − φLn ))anL )
Consistent to the thermodynamic equilibrium:
For any given constant value of ψ , assuming NKL = 0 we arrive at
aK
B(Z(φK − φL ))
exp(Z(φL − φK )) − 1
=
=−
aL
B(Z(φL − φK ))
exp(Z(φK − φL )) − 1
exp(ZφL ) exp(−ZφK ) − exp(−ZφL )
=−
·
exp(ZφK ) exp(−ZφL ) − exp(−ZφK )
exp(ZφL )
exp(Z(ψ − φK ))
=
=
exp(ZφK )
exp(Z(ψ − φL ))
a=,K
=
a=,L
Scharfetter/Gummel, IEEE Trans. Electron. Dev 1969
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 17
1D, Binary electrolyte, Nonlinear Poisson
Ω = [0, L].
Boundary conditions:
φ |x=0 = φ0 ,
φ |x=L = φ∞ = 0,
−∇p · n|x=0 = q∇φ · n|x=0 ,
p|x=L = p∞ = 0.
Quasi-Fermi potentials ψα : obtained from given concentration values
cα |x=L = cα,∞ << c such that q|x=L = F ∑N−1
α=1 zα cα,∞ = 0.
Symmetric 1:1 electrolyte with bulk solution molarity cα,∞ = c∞ for α = 1, 2
Summary concentration c set to the molarity of water c = 55.508 · mol/dm3 .
(Double layer) charge:
Z L
Qdl = Qdl (φ0 ) =
q dx
0
Differential (double layer) capacitance:
Cdl =
Q (φ0 + δ ) − Qdl (φ0 )
dQdl
≈ dl
,
dφ0
δ
δ << φ0
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 18
Gouy-Chapman (+Stern) differential double layer capacitance
Dilute solution with uncorrected Nernst-Planck equation ⇒ “Gouy-Chapman theory”
Guoy-Chapman
700
Guoy-Chapman-Stern
0.001M
0.01M
0.1M
1M
600
120
500
100
Cdl/µFcm-2
Cdl/µFcm-2
0.001M
0.01M
0.1M
1M
140
400
300
80
60
200
40
100
20
0
0
0
50
100
150
∆φ/mV
200
250
300
0
200
400
600
800
1000
∆φ/mV
Numerically obtained differential double layer capacitances curves according to the
Gouy-Chapman model (left) and the Gouy-Chapman-Stern model (right; OHP at
x = 0.5nm ) for aqueous 1:1 electrolytes at 25◦ C matching the classical results
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 19
Binary 1:1 electrolyte: Bikerman vs Fermi vs Pressure correction
Bikerman vs. pressure correction
600
400
Bikerman, 0.001M
Bikerman, 0.01M
Bikerman, 0.1M
Bikerman, 1M
Fermi, 0.001M
Fermi, 0.01M
Fermi, 0.1M
Fermi, 1M
500
400
Cdl/µFcm-2
500
Cdl/µFcm-2
Bikerman vs. Fermi
600
Bikerman, 0.001M
Bikerman, 0.01M
Bikerman, 0.1M
Bikerman, 1M
PC, 0.001M
PC, 0.01M
PC, 0.1M
PC, 1M
300
300
200
200
100
100
0
0
0
200
400
600
800
1000
0
200
400
∆φ/mV
600
800
1000
∆φ/mV
Comparison of the differential double layer capacitance curves between the
Bikerman model with vα = 1c and the pure pressure correction model (left)
resp. the Fermi-Dirac model of index -1 (right).
Bikerman and pressure correction are equivalent in equilibrium due to
Gibbs-Duhem
Bikerman and Fermi (-1) are indistinguisheable as in β = 1+a11 +a2 either both
activities are small or due to different charge one is large and the other is small.
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 20
Profiles
Concentration profile for c60
c=55.5
50
40
Bikerman
Dreyer et al.
Fermi
Gouy-Chapman
Gouy-Chapman-Stern
400
φ/mV
c/(mol/dm3)
Voltage Profile
500
Bikerman
Dreyer et al.
Fermi
Gouy-Chapman
Gouy-Chapman-Stern
30
20
300
200
100
10
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x/nm
0.6
0.8
1
x/nm
Comparison of negative ion concentration (left) and potential profile (right) for the
different models. The Debye length is λD = 3.04nm.
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 21
Spherical coordinates
Differential double layer capacitance for spherical electrode
0.01 mol/dm3 binary electrolyte, Bikerman model with excluded volume =1/csolute
600
r=1nm
r=2nm
r=4nm
r=8nm
planar
600
580
500
560
540
520
Cdl/µFcm-2
400
500
200 220 240 260 280 300
300
100
80
200
60
40
20
100
0
0
20
40
60
80 100
0
0
200
400
600
800
Δφ/mV
Differential double layer capacitances for different electrode radii
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 22
1000
Dynamic calculations
Double Layer Charging Current
100
50
-
c Bikerman
c PC
c GC
40
3
c/(mol/dm )
80
I/(A/cm2)
Evolution of Maximum Concentration
60
c=55.5
IBikerman
IPC
IGC
60
40
30
20
20
10
0
1e-10
1e-08
1e-06
0.0001
0.01
1
0
1e-10
1e-08
1e-06
t/s
0.0001
0.01
t/s
Evolution of current (left) and maximum negative ion concentration (right).
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 23
1
Electrolytic diode, binary 1:1 electrolyte, v = 0
εε0 ∂n φ = −σ
εε0 ∂n φ = σ
c1,2 = cbulk
φ =0
p=0
c1,2 = cbulk
φ = φbias
p=0
If not stated otherwise, homogeneous Neumann boundary conditions:
∂n φ = 0, ∂n p = 0, N1,2 · n = 0, σ = 500µAs/m2
IV curve for fluidic diode
1M solution
7000
6000
Guoy-Chapman
Bikerman v=1/c
5000
I/mA
4000
3000
2000
1000
0
-1000
-4
-2
0
2
4
Δφ/V
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 24
Electrolytic diode: distributions (Bikerman model, 0.01M solution,σ = 150µAs/m2 )
φbias = −2V
φbias = 2V
φ
c+
c−
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 25
Electrolytic diode: maximum concentrations + pressure (σ = 150µAs/m2 )
Maximum concentration
Grid refinement level 4, 0.1M solution
Maximum concentration
Grid refinement level 6, 0.1M solution
160
180
140
160
140
Guoy-Chapman
100
Bikerman
Dreyer et al.
120 Guoy-Chapman
Bikerman
100
Dreyer et al.
I/mA
I/mA
120
80
80
60
c=55.5
60
c=55.5
40
40
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
Δφ/V
Maximum pressure
Grid refinement level 6
465
460
455
p/MPa
0
Δφ/V
450
0.1M
1M
445
440
435
430
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Δφ/V
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 26
0.5
1
1.5
2
Coupling to Navier-Stokes
Stationary distribution of single neutral species in Navier-Stokes velocity field v
Mass balance
∇·N = 0
N = −D∇c + cv
Fick’s law + convection
Problem: coupling to Navier-Stokes while maintaing concentration maximum
principle.
Stabilized finite elements do not guarantee this property!
Voronoi finite volumes with discrete Scharfetter-Gummel (Il’in) flux:
v v KL
KL
cK − B −
cL
NKL = D B
D
D
vKL : some projection of v onto edge xK xL
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 27
Discrete Minimax Principle
Divergence free discrete flux:
|∂ L ∩ ∂ K|
|σ |vσ = 0
vKL +
∑
|xK − xL |
L∈neighbors(K)
σ ∈outbound(K)
∑
(DIV0)
Fuhrmann/Linke/Langmach APNUM 2011
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 28
Discrete Minimax Principle
Divergence free discrete flux:
|∂ L ∩ ∂ K|
|σ |vσ = 0
vKL +
∑
|xK − xL |
L∈neighbors(K)
σ ∈outbound(K)
∑
(DIV0)
Lemma: If (DIV0) is valid, for any solution (cK )K∈K :
1. Global discrete minimax principle:
0 ≤ cK ≤ cinlet
∀K ∈ K
Fuhrmann/Linke/Langmach APNUM 2011
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 28
Discrete Minimax Principle
Divergence free discrete flux:
|∂ L ∩ ∂ K|
|σ |vσ = 0
vKL +
∑
|xK − xL |
L∈neighbors(K)
σ ∈outbound(K)
(DIV0)
∑
Lemma: If (DIV0) is valid, for any solution (cK )K∈K :
1. Global discrete minimax principle:
0 ≤ cK ≤ cinlet
∀K ∈ K
2. Local discrete minimax principle:
min
L∈neighbors(K)
cL ≤ cK ≤
max
L∈neighbors(K)
cL
∀K ∈ K
0
Fuhrmann/Linke/Langmach APNUM 2011
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 28
Discrete Minimax Principle
Divergence free discrete flux:
|∂ L ∩ ∂ K|
|σ |vσ = 0
vKL +
∑
|xK − xL |
L∈neighbors(K)
σ ∈outbound(K)
(DIV0)
∑
Lemma: If (DIV0) is valid, for any solution (cK )K∈K :
1. Global discrete minimax principle:
0 ≤ cK ≤ cinlet
∀K ∈ K
2. Local discrete minimax principle:
min
L∈neighbors(K)
cL ≤ cK ≤
max
L∈neighbors(K)
cL
∀K ∈ K
0
3. The system matrix has the M-Property
Fuhrmann/Linke/Langmach APNUM 2011
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 28
Divergence free discrete fluxes
How to guarantee this condition when v comes from the solution of a flow problem ?
Continuous divergence free velocity field v:
1 R
Exact calculation of vKL = |K∩L|
K∩L v(s) · (xK − xL )ds:
Exact solutions: Hagen - Poiseuille et al
Pointwise divergence free finite elements (e.g. Scott Vogelius for
Navier-Stokes)
Burman/Linke, App. Num. Math 58(2008)11,1704-1719
Postprocessed Crouzeix-Raviart mixed finite elements (A. Linke 2013, submitted)
Finite volume scheme for flow problem including discrete divergence free fluxes
Finite volume solution for Navier-Stokes (Eymard/Fuhrmann/Linke 2011, submitted)
insufficient convergence properties
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 29
Divergence free Scott-Vogelius finite elements
div(velocity space) ⊂ pressure space
Lowest order Scott Vogelius elements:
(Pd , P−(d−1) ) (d:space dimension)
Stable on macro triangulations
Arnold/Qin, Proc. IMACS 1992
Burman/Linke, App. Num. Math 2008
Linke, PhD Thesis, FU Berlin 2008
Maintain two independent discretizations for transport (FV) and for flow (FE)
(FV): For every simplex S, calculate simplicial contributions σKL;S = ∂ K ∩ ∂ L ∩ S
to ∂ K ∩ ∂ L
(FE): Calculate velocity projections qKL;S from continuous FE velocity field
(FV): Assemble qKL from qKL;S
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 30
Cylindrical DEMS flow cell
working chamber
working electrode
Ar flux
electrolyte inlet
teflon spacer
electrolyte outlet
reference and
counter electrode
+ counter electrode
MS compartment
six capillaries
six capillaries connecting
electrochemical and mass
spectrometrical compartments
capillary for electrolyte inlet
electrolyte inlet
reference and
counter electrodes
mass spectroscope
Ar flux
teflon spacer
electrolyte outlet
+ counter electrode
holes for screws
Jusys/Massong/Baltruschat, J. Electrochem. Soc. 1999
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 31
Numerical solution ansatz
Navier-Stokes solution by Taylor-Hood or
Scott Vogelius finite elements
(implemented in Alberta)
Voronoi finite volume solution implemented in
pdelib (WIAS)
Grid generation by “extrusion” of a 2D
Delaunay mesh (created by triangle) in z
direction in order to guarantee proper
anisotropic alignment at anode
Calculation on 61 of the cell geometry, together
with stubs of inlet and outlet channels
x
inlet
x
capillary
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 32
Comparison to experiment in a 3D cylindrical cell
Limiting Current Ilim / mA
1
H=50µm
H=75µm
H=100µm
Measured
0.17u1/3
0.1
1
10
100
3
Flow Rate u /(mm /s)
Measured and calculated values for the limiting current using fitted data from channel
flow cell experiment for different values of working chamber height.
No “simple” asymptotic expression available.
Fuhrmann/Linke/Langmach/Baltruschat, Electrochim. Acta 2009
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 33
Flow regimes
6.5
0.5mm3 /s
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
10mm3 /s
80mm3 /s
Concentration isolevels for different inlet flow rates
Fuhrmann/Linke/Langmach/Baltruschat, Electrochim. Acta 2009
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 34
Scott Vogelius vs. Taylor Hood (coarse grid)
Scott Vogelius
Taylor Hood
Taylor-Hood velocity field does not guarantee discrete divergece condition ⇒
violation of maximum principle
Main measured effects concentrated in boundary layer ⇒ tolerable errors with Taylor
Hood, if grid is fine enough.
Fuhrmann/Linke/Langmach/Baltruschat Electrochim. Acta 2009
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 35
Outlook
Implement Nernst-Planck-Poisson-Navier-Stokes coupling using divergence free
FEM ansatzes of A. Linke based on postprocessed Crouzeix-Raviart
updated version of Nernst-Planck including different molar volumes
nanopore calculation (model air electrode in DEMS cell, PEM)
nanofluidic devices, electrodialysis, biomembranes etc.
degenerate semiconductors (OLED, nitride)
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 36
Acknowledgement
German Federal Ministry of Education and Research (BMBF)
Research initiative on energy storage
Collaborative project “Perspectives for rechargeable magnesium air batteries”
H. Baltruschat (Bonn), Th. Bredow (Bonn), J. Behm (Ulm), M. Wachtler (Ulm)
Grant 03EK3027D
Numerical Strategies · PEM2013 · NTNU Trondheim · 2013-10-03 · Page 37