Title: Phase Field Modeling of Electrochemistry
1Phase Field Modeling of Electrochemistry
J. E. Guyer, W. J. Boettinger, J. A. Warren G.
B. McFadden
- Features common to electrochemistry melt
growth - Review of
- Sharp interface model of electroplating
- Interface structure
- Phase field model
- 1-D Equilibrium Results
NASA
Bob Sekerka
2- Mullins-Sekerka widely quoted and adapted to
predict roughening of plated surface e.g. Aogaki
et al.(1980-1995) - A theory of dendritic growth in electrolytes,
D. R Hamilton, Electrochimica Acta 8 (1963) 731. - Ivantsov / maximum growth rate hypothesis to fit
velocity vs. overpotential data of Barton
Bockris (Ag dendrites growing from AgNO3 in
NaNO3-KNO3 molten salt eutectic). - currentvelocity undercoolingoverpotential
Cu from aqueous solution Barkey, Oberholtzer
Wu, PRL 75(1995) 2980
3Plating in Narrow Trenches and Vias for
Electronic Applications
pores
D. Josell, D. Wheeler, W.H. Huber and T.P.
Moffat, Superconformal Electrodeposition in
Submicron Features Physical Review Letters 87,
016102 (2001).
D. Josell, B. Baker, C. Witt, D. Wheeler and T.P.
Moffat, Via Filling by Electrodeposition
Superconformal Silver and Copper and Conformal
Nickel, Journal of the Electrochemical Society,
in press
4Some possible uses of phase-field modeling in
electrochemistry
- Plating in submicron features
- High curvatures, high electric field gradients
- Avoid approximations
- Pulse plating
- Alloy Plating
- Alloy Dissolution
- Reveal new insight into relationship between
interface charge distribution / adsorption and
kinetics
5Electrochemistry
6Double Layer
electrode
electrolyte
7Length Scales in Electrochemistry
- Thickness of Electrode-Electrolyte interface
- Charge separation distance (Debye layer) related
to concentrations and dielectric constant - Long range concentration decay length due to
diffusion/convection in electrolyte
Helmholtz model Gouy-Chapman model Gouy-Chapman-St
ern model
8Models of Interface Charge Distribution
- Helmholtz Model
- Sharp Electrode-Electrolyte interface
- No Debye layer (? 0)
- Voltage jump at interface (dipole layer)
- Constant differential capacitance
- Gouy-Chapman Model
- Sharp Electrode-Electrolyte interface
- Finite Debye length (double layer)
- Voltage continuous across interface
- Parabolic differential capacitance
- Gouy-Chapman-Stern Model
- Linear voltage adjacent to electrode
- Parabolic differential capacitance with wings
9Interface Properties
- Surface energy ? depends on
- the voltage ??
- Electrocapillary equation
10Differential Capacitance
Cd / (µF/cm2)
Cd / (F/m2)
increasing NaF concentration
experimental data Ag electrode aqueous NaF
electrolyte G. Valette, J. Electroanal. Chem.
138 (1982) 37
Df/V
Comparison of our results with sharp interface
models
11Phase Field Model
- Add a new phase variable,
- and equation
- Solve over entire domain
- Phase field equation
- Poisson's equation
- Transport equations
- No boundary conditions at interface
- Treat complex interface shape / topology changes
- Avoid approximations
Diffuse Interface
Electrode
Electrolyte
- Phase-Field Model
- Diffuse Electrode-Electrolyte interface
- Finite Debye length
- Differential capacitance appears realistic
- Long-range diffusion possible
12 Example of Components in Phases
- Electrode is solid solution of Cu2 and
interstitial e-. - Electrolyte is aqueous solution of Cu2, SO4-2
and H2O.
- Mole Fractions
- Molar Volume
- Concentrations
- Assume
- Constraint
13Free Energy
14Equilibrium
- Minimization of free energy subject to
- Solute conservation
- Poisson's equation
-
15Choice of Thermodynamics
- Ideal solution
- Interpolation and double-well functions
g(?)
p(?)
16Phase Equilibria
- XiL,Ref, XiS,Ref chosen to obtain equilibrium
between - a liquid solution of CuSO4 in H2O metal (Cu2
2e-1)
zero charge plane
- DyRef chosen to set interface charge
distribution, for Ref Xs
17Interface Properties
- Surface Charge definitions
(a result, for our model)
- Can also define adsorptions
18Choice of other Parameters
- Equilibrium
- Double Well Heights,
- Gradient Energy Coefficient,
- Dielectric constant
- Related to
- thickness of f transition
- surface energy
- Numerical Calculations (First cut)
- Finite difference scheme
- Evolution of dynamical equations to steady state
- Insensitive to initial guess (but slow)
19Equilibrium Profiles
20Concentration Profiles
21Voltage Decay Length in Electrolyte
- Reproduces Gouy-Chapman Result
- Dilute electrolyte Exponential Decay
Exponential fits
22Traditional Double-Layer Theory (Gouy-Chapman)
- Boltzmannn Distribution
- Poisson Equation
(more generally, there is a first integral )
23Double Layer (cont.)
- Electrolyte voltage profile
- is the voltage/concentration decay length
(Debye length) - Surface energy, surface charge, differential
capacitance, etc. all related to voltage across
interface, i.e., - Nernst relation
24Numerical Technique Spectral Element Method
25Spectral Resolution and Adaptive Strategy
Fix N 16
In each panel, monitor aN aN-1
If max(aN,aN-1) gt ? bisect panel and repeat
until function is well-resolved on all panels
26Numerical Method
27Resolution of Double Layer
Phase field with uniform panels
Chebyshev coefficients with uniform panels
Charge distribution with uniform panels
Chebyshev coefficients with uniform panels
28Adaptivity
Charge distribution with uniform panels
Chebyshev coefficients with uniform panels
Charge distribution with two refinement levels
Chebyshev coefficients with two refinement levels
29Spectral Computation
surface energy
surface charge
differential capacitance
30Electrocapillarity
Bard Faulkner, Electrochemical Methods 2nd
Ed., Wiley Sons, New York (2001) after
D.C. Grahame, Chem. Rev. 41 (1947)
441
31Differential Capacitance
Cd / (µF/cm2)
Cd / (F/m2)
increasing NaF concentration
Df/V
experimental data Ag electrode aqueous NaF
electrolyte G. Valette, J. Electroanal. Chem.
138 (1982) 37
Our results
32Sharp Interface Limit
33Numerical Solution in Outer Variables
- Interface width
- 0.1
- ?/2, ?/4, ?/8, ?/16
34Numerical Solution in Inner Variables
Interface width ?, ?/2, ?/4, ?/8, ?/16
35Outer and Inner Solutions
36Matched Asymptotic Expansion
37Matched Asymptotic Expansion
38Surface Charge
39Sharp Interface Limit
Interface width ?, ?/2, ?/4, ?/8, ?/16 ? 0
(sharp)
40Conclusions
- Equilibrium 1-D solutions of the model exhibit
double layer behavior.Consistent with
Gouy-Chapman model, and incorporate - Decay length of electrostatic potential
- Interface energy (electrocapillary curves),
surface charge and differential capacitance all
look reasonable
41Current and Future Work
- Continue study of sharp-interface limits
- Kinetic studies underway
- Explore behavior for We ? 0, non-constant ?(?)
- Explore effects of curvature (cylindrical
electrode) - Adaptive Mesh in 2-D
- Alloy Plating/Corroding
- Additives, Adsorption and inhibitors