modeling, simulation and characterization of atomic force microscopy …€¦ · ·...
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Modeling, simulation and characterization of atomic force microscopy measurements for ionic transport and impedance in PEM fuel cells
Peter M. PinskyDavid M. BarnettYongxing Shen
Department of Mechanical EngineeringDepartment of Materials Science and EngineeringStanford University
GCEP meeting, June 16, 2005
Broad scope of research
Modeling of ionic transport in proton exchange membrane fuel cells
Electrostatic atomic force microscope imaging
Simulation of impedance spectroscopy measurements
Backgroundfuel cell membranes
Local variations in ion concentration in the Nernst diffusion layer at the membrane surface in proton exchange membrane fuel cells (PEMFC) under current load conditions are poorly understood and may significantly influence mass transport across the membrane.
Increased understanding of the ion behavior at the Nernst diffusion layer of the membrane surface could enable new classes of solid polymer fuel cell membranes with increased mass transport.
Backgroundelectrostatic force microscopy
Long-range electrostatic forces between a sample and a noncontactAFM tip is used to extract surface potential or capacitance images
Since fuel cell membrane charge characteristics may be inhomogeneous, imaging these variations could prove crucial to understanding the functionality of membranes.
Our research is aimed to provide a better understanding of the relationship between the image obtained and the charge distributions present on the membrane.
Backgroundimpedance spectroscopy
Nanometer scale visualization and measurement of impedance − quantifying the response of a material to an applied varying voltage − is valuable for a wide variety of materials investigations, including fuel cell systems.
Prinz et al. have introduced an atomic force microscope-based impedance imaging technique with < 100 nm resolutionImpedance measured between AFM tip
and bulk electrode – spreading resistance ensures local characterization
Backgroundimpedance spectroscopy
Factors contributing to “electrochemical” impedance imaging results for ionic materials are poorly understood.
Modeling the physical processes involved in the impedance measurement could greatly enhance the usefulness of this technique.
Impedance images of the Nafionelectrolyte membrane as a function of
humidity (O’Hayre et al.)
Broad scope of research
Modeling of ionic transport in proton exchange membrane fuel cells
Electrostatic atomic force microscope imaging
Simulation of impedance spectroscopy measurements
Electrostatic atomic force microscope imaging
A starting point: consider the electrostatic force acting on a conductive tip above a conductive plane. We are developing:
A novel analytical (Green’s function) approach to determining the electrostatic force by solving for the charge distribution based on realistic tip geometry
Direct numerical simulation using the finite element method
Given the electrostatic potential distribution on the AFM tip and the sample surface/bulkSolve for the electrostatic potential φCalculate the charge distribution on the tip, the system capacitance and tip-sample force
/G∂ ∂ =n 0
( )2
0
1 'G δε
∇ = − −r r
0G =
/φ∂ ∂ =n 02 0φ∇ =
0φ =
0φ φ=
gap
Electrostatic atomic force microscope imaging
Electrostatic atomic force microscope imaging
Use of Green’s theorem gives:
tipS
0
( ) ( )dS
C
σ
φ=∫∫ r r
tip
0S
( , ') ( ) ( )G dSσ φ=∫∫ r r r rC Capacitanceσ Charge densityφ0 PotentialG Green’s function
0( ) ( )nφσ ε ∂
=∂
r r
Electrostatic atomic force microscope imaging
Semi-analytical solution via scale-independent variational principle:
tip
tip tip
2
S
S S
ˆ ( ') ( ')ˆ
ˆ ˆ( ) ( ') ( ') ( , ') ( )
dS
CdS dS G
σ
σ σ
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦=∫∫
∫∫ ∫∫
r r
r r r r r r
( )0
ˆ ˆ ˆ ˆ 0dC Cd ε
δ σ εδσε =
= + = ⇔ˆ
C C
σ σ=
=
Electrostatic atomic force microscope imaging
Finite element results for potentialNeed Dirichlet-to-Neumann radiation condition
Mesh Potential field
/φ∂ ∂ =n 0
Electrostatic atomic force microscope imaging
Finite element results for charge density on tip surface
Electrostatic atomic force microscope imaging
Computation of capacitive forceBased on the Maxwell stress tensor
012j ij iS
i j k k
F n dSx x x xφ φ φ φε δ
⎡ ⎤∂ ∂ ∂ ∂= −⎢ ⎥
∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦∫
Broad scope of research
Modeling of ionic transport in proton exchange membrane fuel cells
Electrostatic atomic force microscope imaging
Simulation of impedance spectroscopy measurements
Modeling ionic transport in fuel cell membranes
Ionic mass transfer in ion-selective membranes is not fully understood
Local concentration changes in the Nernst diffusion layer influences mass transfer behavior of membrane
Transient and steady state transport modeling are needed
Modeling ionic transport in fuel cell membranes
Mathematical model
c DqD c ct kT∂ ⎛ ⎞= −∇ − ∇ −⎜ ⎟∂ ⎝ ⎠
E
( )00
,r
qc cφε ε
−∇ = ∇ = −E E
Mass balance with Nernst-Planck model (drift-diffusion)
Charge conservation
No-flux boundary conditionDqD c ckT
= − ∇ − = 0J E
Modeling ionic transport in fuel cell membranes
1-d non-dimensional c-E steady-state PEMFC modelFinite difference (Jeremy Cheng and David Barnett)
1 1 1 1 1 12
20
2 2j j j j j j j
j j
c c c c c E EE c
x x xλ− + + − + −− + ⎡ − − ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞
− + =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
Drift-diffusion
Poisson’s equation
( )1 1 1 22
j
j j i ii
xE E c cβ+ − −
∆⎛ ⎞− = − − ⎜ ⎟⎝ ⎠
∑Non-dimensional parameters
* *0 0
0 0
4.2 7, 3r
Lc q q E LeE kT
β λε ε
= ≈ = ≈
Modeling ionic transport in fuel cell membranes
Detecting the Nernst boundary layer in 100 nm membrane
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
1.20E+00
0 0.5 1 1.5 2 2.5
x [Å]
c/co
[ ] Finite Difference
Analytical
0 0
1sinh ( )2
sinh
xqlc c
qε λ
−=
r
qccE
kTEDqccDj
εε 0
0 )(
0
−=⋅∇
=+∇−=
Modeling ionic transport in fuel cell membranes
1 21
3 3
( )00 0
cd
qdtε
⎡ ⎤⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎢ ⎥+ =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎣ ⎦
K KM c c 0φ φ FK M
Variational c-φ coupled form of BVP
Finite element approximation
0 , , , , ,
0 , , ,h
t i i i i
i i n
Dqwc d w Dc d w c dkT
qv d v cd vc d v d
φ
φ φε
Ω Ω Ω
Ω Ω Ω Γ
= Ω+ Ω+ Ω
′= − Ω+ Ω− Ω+ Ω
∫ ∫ ∫
∫ ∫ ∫ ∫
Next stepsAtomic force microscope imaging− Surface and/or bulk trapped charge distributions− Compute capacitive forces− Extend to 3-d− Application to experiments
Modeling ionic transport in fuel cell membranes− Consider boundary layer effects in time-varying electric
fields− Extend to 2-d and 3-d models (finite element approach)− Fully nonlinear coupled problem