Phase Field Modeling of Electrochemistry

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Phase 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
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• 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
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Plating 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
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Some 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

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Electrochemistry
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Double Layer
electrode
electrolyte




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Length Scales in Electrochemistry

1. Thickness of Electrode-Electrolyte interface
2. Charge separation distance (Debye layer) related
   to concentrations and dielectric constant
3. Long range concentration decay length due to
   diffusion/convection in electrolyte

Helmholtz model Gouy-Chapman model Gouy-Chapman-St
ern model
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Models 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

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Interface Properties

• Surface energy ? depends on
• the voltage ??

• Electrocapillary equation

• Surface charge

• Differential capacitance

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Differential 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
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Phase 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

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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

• Ion Charge zi

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Free Energy
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Equilibrium

• Minimization of free energy subject to
• Solute conservation
• Poisson's equation

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Choice of Thermodynamics

• Ideal solution
• Interpolation and double-well functions

g(?)
p(?)
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Phase 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

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Interface Properties

• Surface Energy

• Surface Charge definitions

• Differential Capacitance

(a result, for our model)

• Can also define adsorptions

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Choice 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)

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Equilibrium Profiles
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Concentration Profiles
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Voltage Decay Length in Electrolyte

• Reproduces Gouy-Chapman Result
• Dilute electrolyte Exponential Decay

Exponential fits
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Traditional Double-Layer Theory (Gouy-Chapman)

• Boltzmannn Distribution
• Poisson Equation

(more generally, there is a first integral )
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Double 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

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Numerical Technique Spectral Element Method
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Spectral 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
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Numerical Method
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Resolution of Double Layer
Phase field with uniform panels
Chebyshev coefficients with uniform panels
Charge distribution with uniform panels
Chebyshev coefficients with uniform panels
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Adaptivity
Charge distribution with uniform panels
Chebyshev coefficients with uniform panels
Charge distribution with two refinement levels
Chebyshev coefficients with two refinement levels
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Spectral Computation
surface energy
surface charge
differential capacitance
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Electrocapillarity
Bard Faulkner, Electrochemical Methods 2nd
Ed., Wiley Sons, New York (2001) after
D.C. Grahame, Chem. Rev. 41 (1947)
441
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Differential 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
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Sharp Interface Limit
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Numerical Solution in Outer Variables

• Interface width
• 0.1
• ?/2, ?/4, ?/8, ?/16

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Numerical Solution in Inner Variables
Interface width ?, ?/2, ?/4, ?/8, ?/16
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Outer and Inner Solutions
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Matched Asymptotic Expansion
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Matched Asymptotic Expansion
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Surface Charge
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Sharp Interface Limit
Interface width ?, ?/2, ?/4, ?/8, ?/16 ? 0
(sharp)
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Conclusions

• 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

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Current 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