1. Kinetic Monte Simulation:
Comparison with Cyclic-Voltammetry
Aaron Hastings

2. 2
• Lattice-Gas Model
• Kinetic Monte Carlo
• Metropolis-Hastings Algorithm
• Cyclic Voltammetry
Overview:

3. 3
• Governing Hamiltonian
• (used to calculate energy
𝐻 = −
𝑖<𝑗
ϕ𝑖𝑗 𝑐𝑖 𝑐𝑗 − μ
𝑖=1
𝑁
𝑐𝑖
• 𝑐𝑖 => occupation variable
• =1 (occupied)
• =0 (unoccupied)
Lattice-Gas Model:

4. 4
• Interaction Constants
ϕ𝑖𝑗 =
−∞, 𝑟𝑖𝑗 = 1
2
3
2ϕ 𝑛𝑛𝑛
𝑟𝑖𝑗
3 , 𝑟𝑖𝑗 ≥ 2
• Electrochemical Potential
μ = μ0 + 𝑘 𝑏 𝑇𝑙𝑛
𝐶
𝐶0
− 𝑒γ𝐸
Lattice-Gas Model:

5. 5
• Form a random lattice (128 x 128)
• Possibilities for lattice sites:
• Occupied = 1
• Unoccupied = 0
Kinetic Monte Carlo:

6. 6
• Apply periodic boundary conditions
• Simulates an infinite lattice
Kinetic Monte Carlo:

7. 7
• Apply periodic boundary conditions
• Simulates an infinite lattice
Kinetic Monte Carlo:

8. 8
• Choose a random lattice site, i
• Possibilities for moves at i:
• If occupied (=1)
• Adsorption
• If unoccupied (=0)
• List of nine options
Kinetic Monte Carlo:

9. 9
• Lattice site positions
• Blue => chosen site
• Green => nearest neighbors
• Purple => next-nearest neighbors
Kinetic Monte Carlo:

10. 10
• Site i is unoccupied:
• If nearest neighbor sites are unoccupied:
• Adsorption to the site is attempted
Kinetic Monte Carlo:

11. 11
• Site i is occupied:
• If nearest neighbor sites are occupied:
• Desorption occurs with 100% probability
• Otherwise, check if next-nearest neighbor sites are
unoccupied:
• Propose a diffusion to:
• Any one of the 4 nearest neighbors
• Any one of the 4 next-nearest neighbors
Kinetic Monte Carlo:

12. 12
• Form a weighted list of probabilities:
𝑅 𝐹 𝐼 = 𝑣 exp −∆ 𝜆 𝛽 exp −
∆𝐻 ∗ 𝛽
2
• ∆ 𝜆 => “Bare” barrier associated with process 𝜆
• β => 1/(kbT)
• ∆𝐻 => Energy change for the move
Kinetic Monte Carlo:

13. 13
• Visual example of weighted list:
Kinetic Monte Carlo:
0 1
Probability

14. 14
• Attempt a move:
• Generate a random number between (0,1):
• Check where it falls on the weighted list:
• Accept the move
Kinetic Monte Carlo:

15. 15
• Repeat for potential, µ:
• -200meV ≤ µ ≤ 600meV (increasing µ)
• 600meV ≥ µ ≥ -200meV (decreasing µ)
• Total number of attempts:
𝐿2
𝜌
2 600 − −200
• L => Lattice dimension
• 𝜌 => Scan rate (3*10^-5 to 0.1meV/MCSS)
Kinetic Monte Carlo:

16. 16
• Run eight simulations at each scan rate
• Average the data
• θ => Lattice Coverage
𝜃 = 𝑁−1
𝑖=1
𝑁
𝑐𝑖
• Take a numerical derivative
• Apply a Savitzky-Golay filter in MATLAB
• dθ/dµ
Cyclic Voltammetry:

17. 17
• Smoothed numerical derivative is
proportional to current density, j:
𝑗 =
𝛾2 𝑒2
𝐴 𝑠
𝑑𝜃
𝑑𝜇
𝑑𝐸
𝑑𝑡
• Differential adsorption capacitance per
unit area:
𝐶 =
𝑗
𝑑𝐸
𝑑𝑡
=
𝛾2 𝑒2
𝐴 𝑠
𝑑𝜃
𝑑𝜇
Cyclic Voltammetry:

18. 18
Coverage vs Voltage:
• Results after being averaged
eight times, then smoothed
• Effect as ρ is decreased
• (Increased number of attempted
moves)
Photo credit: [1]

19. 19
• Laptop trial:
• 30 x 30 lattice with ρ = 0.5
• Witnessed similar trends
• Supercomputer trial:
• 64 x 64 lattice with ρ = 1*10-3
Coverage vs Voltage:

20. 20
Coverage vs Potential:

21. 21
Cyclic Voltammetry:
• To study redox processes
• To determine electron transfer
kinetics
• To determine diffusion
coefficients
Photo credit: [1]

22. 22
• Submitted a few trials to OSC
• (Ohio Supercomputer Center)
• Need to further debug/optimize our code
• Use DFT to establish parameters for
Uranium
• (Density Functional Theory)
• Determine boundary conditions for
Diffusion Equation at the surface
Future Plans:

23. 23
1. Abou, Hamad I, P.A Rikvold, and G Brown. "Determination of the
Basic Timescale in Kinetic Monte Carlo Simulations by
Comparison with Cyclic-Voltammetry Experiments." Surface
Science. 572 (2004). Print.
2. Abou, Hamad I, Th Wandlowski, G Brown, and P.A Rikvold.
"Electrosorption of Br and Cl on Ag(1 0 0): Experiments and
Computer Simulations." Journal of Electroanalytical Chemistry.
(2003): 554-555. Print.
References:

24. 24
Thank you.
Any questions?