Weierstrass Institute for Applied Analysis and Stochastics Electrochemical processes and porous media: mathematical and numerical modeling Jürgen Fuhrmann Alfonso Caiazzo, Klaus Gärtner, Hartmut Langmach, Alexander Linke, Hong Zhao Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de RICAM Workshop · Linz · 2011-10-05
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Electrochemical processes and porous media: mathematical ... · Electrochemical processes and porous media: mathematical and numerical modeling Jürgen Fuhrmann Alfonso Caiazzo, Klaus
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Weierstrass Institute forApplied Analysis and Stochastics
Electrochemical processes and porousmedia: mathematical and numericalmodeling
Jürgen FuhrmannAlfonso Caiazzo, Klaus Gärtner, Hartmut Langmach,Alexander Linke, Hong Zhao
Conversion of chemical energy stored in compounds within the cell intoelectrical energy
Different variants (low/high temperature, solid/liquid electrolyte . . . )
Fuel cells
Invented ≈ 1840 by Schönbein, Groves Conversion of chemical energy stored in hydrogen, methanol,
carbohydrates . . . into electrical energy Continuous supply of reactants, removal of products Different variants (low/high temperature, solid/liquid electrolyte . . . )
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 2 (46)
Liquid or solid (aquatic solution, molten salt, polymer membranes) Ionic conductors: electrons are blocked, charge carriers are ions of different
type, e.g. protons (H+).
Mixed conductors show properties of both
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 5 (46)
Charge transport: Nernst Planck Poisson system
Transport of i-th dissolved species (i = 1 . . .n)due to diffusion, electromigration, advection indilute solution⇒ Nernst-Planck equation:
~Ni =−
electromigration︷ ︸︸ ︷ziuiFci∇φ −
diffusion︷ ︸︸ ︷Di∇ci +
advection︷︸︸︷ci~v
∂tci +∇ ·~Ni = ji
Distribution of charged species⇒ self-consistent electric field⇒ Poisson equation:
−∇ ·ε∇φ = Fn
∑i=1
zici
Variables
n number of speciesφ electrostatic potentialci species concentration~Ni molar fluxzi chargeui mobilityDi diffusion coefficientε electrostatic permeabilityF Faraday constantji Reaction~v Substrate velocity
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 6 (46)
Assumptions behind Nernst-Planck
Dilute solution theory:
Ignore interactions (collisions) between different dissolved species ci.Otherwise: Stefan-Maxwell terms, “concentrated solution theory”
Velocity field not influenced by moving ions.Otherwise: contribution to momentum balance
Fluid density not influenced by concentration changesOtherwise: variable density flow
Special case: semiconductor device equations.
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 7 (46)
Bulk electroneutrality
ε << F ⇒ electroneutrality in bulk (away from interfaces and boundaries)n
∑i=1
zici = 0
Sum up equations multiplied by zi:
∂t
(n
∑i=1
zici
)−∇ ·
(n
∑i=1
z2i uiFci∇φ +
n
∑i=1
ziDi∇ci +~vn
∑i=1
zici
)=
n
∑i=1
zi ji
Express e.g. c1:
z1D1c1 =−n
∑i=2
D1zici
No bulk reactions⇒
∇ ·
(n
∑i=1
Fz2i uici∇φ +
n
∑i=2
zi(Di−D1)∇ci
)= 0
Equal diffusion coefficients or small concentration gradients⇒Ohm’s law:
∇ ·κ∇φ = 0
(κ = F
n
∑i=1
z2i uici : conductivity
)Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 8 (46)
Nernst Planck Ohm system
Transport of i-th dissolved species (i = 2 . . .n)due to diffusion, electromigration, advection indilute solution⇒ Nernst-Planck equation:
~Ni =−
electromigration︷ ︸︸ ︷ziuiFci∇φ −
diffusion︷ ︸︸ ︷Di∇ci +
advection︷︸︸︷ci~v
∂tci +∇ ·~Ni = ji
Self-consistent electric field from Ohm’s Law:
∇ ·κ∇φ = 0
Variables
n number of speciesφ electrostatic potentialci species concentration~Ni molar fluxzi chargeui mobilityDi diffusion coefficientκ conductivityF Faraday constantji Reaction~v Fluid velocity
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 9 (46)
Special case: solid electrolyte/electrode
c1: mobile charge carriersc2: immobile charges in solid lattice (u2 = D2 = 0)~v = 0⇒ c2 = const
Electroneutrality,z1 =−z2⇒ c1 = c2
small ∇c2⇒ κ = z21u1Fc1
Electrodes (graphite, metal): c1 ↔ free e−
Polymer electrolytes in fuel cells: c1 ↔ free H+.
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 10 (46)
Inert electrode-electrolyte interface
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-
-
-
--
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--
--
-
-
-
-
++
+
++
+++
++
++
+
+
- +
- +
- +
- +
- +
- +
- +Electrode
Electrons
Fixed charges at lattice
Electrolyte
Free ions
Electrodes: electron concentration as c1 ⇒ “electron potential” φs
(Acidic) electrolytes: concentration of protons (H+) as c1 ⇒ “proton potential” φl
Local electroneutrality violated in boundary layer
Potential jump at interface
Electrons and protons attracting each other may accumulate on both sides ofthe interface creating a double layer
cH2 : hydrogen concentrationθH : Share of catalyst sites occupied by hydrogenθ = 1−θH : Share of free catalyst sitesTypical way to include these processes into global model (for given δφ ).
~NH2 = D∇cH2 + cH2~v advection-diffusion in Ω
∂tcH2 +∇ ·~NH2 = 0 continuity in Ω
~NH2 ·~n+ r1 = 0 normal flux equals reaction on Γ⊂ ∂Ω
∂tθH − r1 + r2 = 0 evolution of catalyst coverage on Γ⊂ ∂Ω
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 14 (46)
Oxygen Reduction Reaction (ORR)
Multistep cathodic reaction in H2-PEMFC and DMFC
Oxygen reaction is split into several steps:
O2 + sk±1 O2,ad
O2,ad +H+ + e−k±2 HO2,ad
HO2,ad +H+ + e−k±3 H2O2,ad
H2O2,ad +2H+ +2e−k±4 2H2O+ s
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 15 (46)
Methanol Oxidation Reaction (MOR)
Anodic reaction in DMFC
CH3OH +2s1k±1 (CH2−OH)ad +Had
(CH2−OH)ad + s1k±2 (CH−OH)ad +Had
(CH−OH)ad + s1k±3 (C−OH)ad +Had
(C−OH)ad + s1k±4 (C−O)ad +Had
(C−O)ad +OHadk±5 (COOH)ad + s2
(COOH)ad +OHadk±6 CO2 +H2O+ s1 + s2
Hadk±7 H+ + s1 + e−
H2O+ s2k±8 OHad +H+ + e−
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 16 (46)
Open questions in catalysis modeling
Essentially, the way of modeling these reactions is guided by heuristics. Theproposed reaction chains follow one particuala hypothesis about the pathways.
Some missing effects:
Catalyst surface defects are preferred adsorption sites
Different crystal directions have different reaction speeds
Multisite adsorption processes
Restructuring of catalyst surface due to reaction
Elementary processes at electrodes may be more complex
Dissociation in presence of liquid water⇒free H3O+ ions available for conduction
⇒ conductivity depends on water content
⇒ water management needs to guarantee itspresence
Problems associated with membrane
Reactant crossover
Swelling
free radical attacks from reactionintermediates
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 20 (46)
Two phase flow
Cathode
supply of O2 (gaseous, low solubility) remove H2O in gaseous and liquid form
Anode (DMFC)
supply of reactant dissolved in water removal of CO2 in dissolved and gaseous form
Liquid H2O is needed in order to maintain membrane conductivity
Phases need to move in opposite directions
Pores kept open for gas flow by admixture of teflon
Model by standard ansatz with modified capillary pressure/saturation curve
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 21 (46)
Mixed wettability: Bernoulli function based ansatz for s(pc)
0
0.2
0.4
0.6
0.8
1
-40 -30 -20 -10 0 10 20 30 40
Effe
ctiv
e Sa
tura
tion
se
Capillary Pressure pc
1:21:1
Bernoulli
Measured values for drying andwetting branches of se(pc) forquartz sand/teflon mixture withdifferent compositions (triangles);Least squares fit to Bernoullifunction based ansatz (lines)
Two-phase flow model taking into accountmixed wettability.
0
0.5
1
0 500 1000 1500
Vol
tage
(V)
Current Density (A/m2)
Meas. 0.5 mol/lMeas. 1 mol/lMeas. 2 mol/l
Sim., 0.5 mol/lSim., 1 mol/lSim., 2 mol/l
Measured and calculatedpolarization curves at 60oC.
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 27 (46)
Divisek/Fuhrmann/Gärtner/Jung, J. Electrochem. Soc. (2003)
Influence of anodic saturation curve
0
0.2
0.4
0.6
0.8
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Satu
rati
on (m
3 /m3 )
Capillary Pressure (bar)
p1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar
0
0.5
1
0 500 1000 1500
Vol
tage
(V)
Current Density (A/m2)
Meas. 0.5 mol/lp1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar
Capillary pressure vs. saturation polarization curves
Best anode performance for more wettable material
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 28 (46)
J. Fuhrmann, K Gärtner, in: Device and Materials Modeling in PEM Fuel Cells, vol. 113 of Topics in Applied Physics, 2009
Influence of cathodic saturation curve
0
0.2
0.4
0.6
0.8
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Satu
rati
on (m
3 /m3 )
Capillary Pressure (bar)
p1/2=0.5 barp1/2=1.0 barp1/2=1.5 bar
0
0.5
1
0 500 1000 1500
Vol
tage
(V)
Current Density (A/m2)
measured, 0.5 mol/lmost wettable
medium wettableleast wettable
Capillary pressure vs. saturation polarization curves
Best cathode performance for less wettable material
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 29 (46)
J. Fuhrmann, K Gärtner, in: Device and Materials Modeling in PEM Fuel Cells, vol. 113 of Topics in Applied Physics, 2009
Flow cell modeling
Model simple xperimental devices to improve understanding of partial processes.
A+ s B+ s C + s Dissolved in fluidk±A k±B k±C Adsorption/Desorption
Aads
k±AB
Bads
k±BC
Cads Adsorbed at catalyst surface
H+ + e− H+ + e− Released ions
Thin-layer flow cell
cA inqin
Γin
cAout
cBout
cCout
Γout
Mass Spec
I + –
catalyst surface
Ω
Anode
Cathode
How does thefraction of thedesorbedintermediate Bdepend on thecatalystconcentration ?
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 30 (46)
Flow cell modeling
Model simple xperimental devices to improve understanding of partial processes.
A+ s B+ s C + s Dissolved in fluidk±A k±B k±C Adsorption/Desorption
Aads
k±AB
Bads
k±BC
Cads Adsorbed at catalyst surface
H+ + e− H+ + e− Released ions
Thin-layer flow cell
cA inqin
Γin
cAout
cBout
cCout
Γout
Mass Spec
I + –
catalyst surface
Ω
Anode
Cathode
How does thefraction of thedesorbedintermediate Bdepend on thecatalystconcentration ?
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 30 (46)
Model multistep surface reaction in a flow cell
Rate expressions:
rA = k+A cAθf−k−A θA
rAB = k+ABθA−k−ABθB
rB = k+B cBθf− k−B θB
rBC = k+BCθB−k−BCθC
rC = k+C cCθf− k−C θC
Algebraic conditions for adsorbedspecies:
dθA/dt−rA− rAB = 0
dθB/dt−rB + rAB− rBC = 0
dθC/dt−rC + rBC = 0
θX – fraction of catalyst sites occupied by adsorbed species
θ f = 1−θA−θB−θC – fraction of free catalyst sites
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 31 (46)
Solute transport coupled with surface reaction
Stationary diffusion and transport of species X = A,B,C in velocity field~v:
∂cX
∂ t−div(DX gradcX −~vcX ) = 0
Boundary conditions:
cX = cX ,in on Γin Dirichlet
(DX gradcX − cX~v) ·~n =−cX~v ·~n on Γout Outflow
(DX gradcX − cX~v) ·~n = ce f fcat rX on Γcat Electrode reaction
∂cX
∂~n= 0 on Γ\ (Γin∪Γout∪Γcat) No flow
Outflow of species X :
Xout =∫
Γout
(DX gradcX − cX~v) ·~n ds
Relative amount of reaction intermediates:
Iout = Bout/(Bout +Cout)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 32 (46)
H2O2 yield in catalytic oxygen reduction
0 0.2 0.4 0.6 0.8 1catalyst coverage
0
5
10
15
20
25
30
rela
tive
amou
nt o
f re
actio
n in
term
edia
te/%
O2(sol) H2O2(sol) H2O
O2(ads) H2O2(ads) H2O(ads)
Catalyst loading effect:H2O2 yield decreases with increasingcoverage of active Pt nanodisks on planarglassy carbon substrate (nanostructuredPt/GC model electrodes)
Top: Experimental, bottom: simulation.
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 33 (46)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 35 (46)
Two Point Flux Voronoi Finite Volume Method
~xK~xL
~xM
K L
Integral over space-time REV control volume
Newton/Leibniz, Gauss, quadrature
Approximation of inter-volume fluxes
0 =1
tn− tn−1
tn∫tn−1
∫K
(∂tb(~x,u)+∇ ·~j(~x,u)+ r(~x,u)
)dxdt
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)−∫
∂K
~j(~x,un) ·~nds
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)
− ∑L nb. of K
|∂K∩∂L||~xK −~xL|
g(~xK ,~xL,unK ,un
L)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 36 (46)
Macneal, Quart. Math. Appl. 1953
Two Point Flux Voronoi Finite Volume Method
~xK~xL
~xM
K L
Integral over space-time REV control volume
Newton/Leibniz, Gauss, quadrature
Approximation of inter-volume fluxes
0 =1
tn− tn−1
tn∫tn−1
∫K
(∂tb(~x,u)+∇ ·~j(~x,u)+ r(~x,u)
)dxdt
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)−∫
∂K
~j(~x,un) ·~nds
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)
− ∑L nb. of K
|∂K∩∂L||~xK −~xL|
g(~xK ,~xL,unK ,un
L)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 36 (46)
Macneal, Quart. Math. Appl. 1953
Two Point Flux Voronoi Finite Volume Method
~xK~xL
~xM
K L
Integral over space-time REV control volume
Newton/Leibniz, Gauss, quadrature
Approximation of inter-volume fluxes
0 =1
tn− tn−1
tn∫tn−1
∫K
(∂tb(~x,u)+∇ ·~j(~x,u)+ r(~x,u)
)dxdt
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)−∫
∂K
~j(~x,un) ·~nds
0 = |K|
(b(~xK ,un
K)−b(~xK ,un−1K )
tn− tn−1 + r(~xK ,unK)
)
− ∑L nb. of K
|∂K∩∂L||~xK −~xL|
g(~xK ,~xL,unK ,un
L)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 36 (46)
Macneal, Quart. Math. Appl. 1953
Exponential fitting flux for linear problems
Scalar, linear convection diffusion flux in given velocity field v(~x):
j = D∇u+uv(~x)
Flux projection onto control volume normal~xK~xL:
vKL :=1
|∂K∩∂L|
∫∂K∩∂L
v(s) ·(~xK −~xL)ds
Upwind numerical flux:
g(uK ,uL) := D(
B(vKL
D
)uK −B
(−vKL
D
)uL
)
(B(ξ ) = ξ
e−ξ−1 : Bernoulli function)
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 37 (46)
Allen/Southwell, Quart. J. Mech. Appl. Math. 1955; Il’in, Mat. Zametki 1969; Scharfetter/Gummel, IEEE Trans. El. Dev. 1969
Generalization to nonlinear problems
Scalar, nonlinear convection diffusion flux in given velocity field v(~x):
j = D(u)∇u+F(u)v(~x)
Define g(uK ,uL) := G from the solution of a local Dirichlet problem for the projectionof the equation onto the edge~xK~xL:Let w(ξ ) := u(~xL +ξ (~xK −~xL)):
D(w)w′+F(w)vKL = Gw(0) = uL
w(1) = uK
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 38 (46)
Eymard/Fuhrmann/Gärtner, Numer. Math. 2006
Maximum principle
Stationary convection diffusion problem:
j = D∇u+uv(~x), ∇ · j = 0
Fundamental property of solution: “no unphysical oscillations”
Divergence free velocity field (∇ ·v = 0)⇒ continuous maximum principle:solution in a given point x is bounded by its values in a surroundig of x.
⇒ discrete maximum principle for discrete problem:solution in point xi is bounded by values in neigboring points x j.
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 39 (46)
Divergence free discrete fluxes
How to guarantee discrete divergence free condition when coupling to flowproblems?
From pointwise divergence free velocity field v(x) by exact calculation of
vKL =1
|K∩L|
∫K∩L
v(s) ·(~xK −~xL)ds
Analytical expressions (Hagen - Poiseuille, von Karman . . . ) Pointwisde divergence free finite elements
Finite volume scheme for flow problem leading to discrete divergence free fluxes
Darcy (porous media) flow: v = K∇p, vKL = K(pK − pL) Compatible finite volume solution for Navier-Stokes: work in progress
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 40 (46)
Divergence free finite elements
div(velocity space)⊂ pressure space
Lowest order Scott Vogelius elements:(Pd ,P−(d−1)) (d:space dimension)
Stable on macro triangulations
Maintain two independent discretizations for transport (FV) and for flow (FE)
(FV): For every triangle T , calculate contributions σKL;T = ∂K∩∂L∩T to∂K∩∂L
(FE): Calculate velocity projections vKL;T from continuous FE velocity field
(FV): Assemble vKL from vKL;T
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 41 (46)
Burman/Linke, App. Num. Math 2008; Fuhrmann/Linke/Langmach, Appl. Num. Math. 2011
Compatible finite volume method
After Nicolaides - extension of MAC scheme:
Choice of unknowns:
Pressures pK on simplex mesh nodes
Velocities qKL along simplex mesh edges ≡ normal to Voronoi box faces
Discrete operators divh,gradh, roth, vector calculus (up to boundary terms):
divh = gradTh
divh gradh = rotTh roth−gradh divh
rotTh gradh = 0
divh roth = 0
Discrete Stokes system: rotTh roth v−gradh p = 0
divh v = 0
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 42 (46)
Eymard/Fuhrmann/Linke, Proc. FVCA6, 2011
pdelib2/fvsys API
Problem description
Grid
Accumulation terms b(~xK ,uK)
Reaction terms r(~xK ,uK)
Fluxes between control volumes g(~xK ,~xL,uK ,uL)
Species sets varying between subdomains and boundary parts
Solution strategy
Adaptive time step selection
Nonlinear solver: Newton’s method
Linear solver: PARDISO (direct), AMG with point block smoothers
Implementation
OpenMP based parallelization
Portable to Linux/Unix/Mac/Windows
OpenGL online graphics, FLTK GUI, Lua scripting language
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 43 (46)
www.pdelib.org
Advantages of Voronoï Methods
Focus on conservation character of mathematical model
Straigthforward use of two point finite difference formulae, upwinding
Convergence results for many cases
Simplex based assembly loop as in FEM: data assembled from simplicialcontributions⇒ known techniques for parallelization etc. apply
Heterogeneities represented by mesh⇒ no averaging across interior boundaries
M-property + discrete maximum principle for Laplacian and properly upwindedconvection problems in 2/3D⇒ conservation of qualitative properties (positivity, dissipativity)⇒ physically meaningful a priori bounds
Implementation tolerant to Delaunay violations:Algebraic expression for “|∂K∩∂L|” may become negative, but still can becalculated
Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 44 (46)
Challenges for Voronoï Methods
Simple scheme but considerable efforts in mesh generation.
No higher order version⇒ high accuracy solutions need very fine meshes.
Challenges
Reliable boundary conforming Delaunay meshing for general domains
Anisotropic problems −→ mesh alignment to anisotropy direction