modeling, simulation and characterization of atomic force microscopy …€¦ ·  ·...

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