1. Theoretical Study of the Rupture of Graphene Membranes in a
Strong Electric Field
Krystle Reiss
Alma College
Senior Honors Defense
Alma, MI
April 21, 2016
Under the supervision of Dr. James W. Mazzuca

2. Outline
1 Electronic structure and molecular dynamics
2 Types of membranes
3 Electric fields
4 Concerted bombardment
5 Sequential bombardment

3. Graphene Properties
• Stronger than steel
• Can be stretched 120%
• 150× more mobile than silicon
• No band gap
• Thinnest material on Earth
• 2500 m2
per gram

4. Experiment
• The experiment: Membranes in 1 M KCl solution under a 3 V/nm field
• Density functional tight binding
E =
i
< Ψi|H|Ψi > +
1
2
α,β
γα,β∆qα∆qβ + Erep (1)

5. Solving Newton’s equations of motion
• Verlet Method
q(t + ∆t) = 2q(t) + a(t)(∆t)2
− q(t − ∆t) (2)
• Leap Frog Method
q(t + ∆t) = v t +
1
2
∆t ∆t + q(t) (3)
v t +
1
2
∆t = v t −
1
2
∆t + a(t)∆t (4)
• Velocity Verlet
q(t + ∆t) = q(t) + v(t)∆t +
a(t)∆t2
2
(5)
v(t + ∆t) = v(t) +
[a(t) + a(t + ∆t)]∆t
2
(6)

6. Environmental condtions
• Berendsen thermostat
v t +
1
2
∆t ← λv t +
1
2
∆t (7)
λ = 1 +
∆t
τT
T0
T(t − 1
2
∆t)
− 1
1
2
(8)
• Nose-Hoover thermostat
• Electric field

7. Calculation parameters
• DFTB+ 1.2 on the Beacon supercomputer
• mio and halorg parameter sets
• Velocity Verlet driver using 1.0 femtosecond time-step
• Nose-Hoover thermostat set to 300 K
• 5000 time-steps (5 ns)

8. Membrane types: Edge saturation

9. Membrane types: Sizes
• 218 carbons
• 5000 steps
• 508 carbons
• 2000-2500 steps
• 1018 carbons
• < 400 steps

10. Warped membranes
Input
Unconstrained output Frozen corners output

11. Graphene defects
Single vacancy
Reconstructed vacancy
Double vacancy
Stone-Wales defect

12. Defect effects
• Single vacancies resembled pristine membranes
• Double vacancy and SW defect caused folding

13. Electric field construction
• +15 eV point charge at 10.0 nm
• -15 eV point charge at -10.0 nm
• Field is perpendicular to the plane of the membrane

14. Results: 3 V/nm
• Flat membrane
• No field
-393
-392.5
-392
-391.5
-391
-390.5
-390
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
• Flat membrane
• 3 V/nm field
-393
-392.5
-392
-391.5
-391
-390.5
-390
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
• Warped membrane
• 3 V/nm field
-393
-392.5
-392
-391.5
-391
-390.5
-390
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy

15. Results: 30 V/nm
Flat membrane
-393
-392.5
-392
-391.5
-391
-390.5
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
-393
-392.5
-392
-391.5
-391
-390.5
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
Warped membrane
-393
-392.5
-392
-391.5
-391
-390.5
-390
-389.5
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
-393
-392.5
-392
-391.5
-391
-390.5
-390
-389.5
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy

16. Concerted ion bombardment
• No rupture from fields or defects
• Potassium and chloride ions in solution
• Applied field causes ions to accelerate
• Ions push through membrane

17. Parameters
• 10 chlorides initiated 5-15 ˚A away
• 1.0 EH (2626 kJ/mol) of kinetic energy
• Random uniform distribution
• No thermostat
−10
0
10
−10 0 10
ZPosition(Å)
X Position (Å)
Membrane 1
Membrane 2
Membrane 3

18. Sample input

19. Results
18.56 broken bonds
0
7
14
21
28
35
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean

20. Results
Is this physically realistic?

21. Sequential bombardment
• Ions initiated in pairs 5-10 ˚A away
• Pairs replaced with new ions after impact
• Total of 5 pairs initiated
• Repeated for 40 trials
• Made system more closely resemble ions in solution
• Better handled calculations that failed to converge

22. Results
0.4 EH
0.8 EH
0.6 EH
0.9 EH
0.7 EH
1.0 EH

23. Results
0.4 EH
0.13 broken bonds
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0.6 EH
1.23 broken bonds
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean

24. Results
0.7 EH
2.40 broken bonds
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0.8 EH
6.46 broken bonds
0
4
8
12
16
20
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean

25. Results
0.9 EH
10.58 broken bonds
0
6
12
18
24
30
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
1.0 EH
13.90 broken bonds
0
6
12
18
24
30
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean

26. Results
0
5
10
15
0 0.2 0.4 0.6 0.8 1 1.2
BondsBroken(Mean)
Kinetic Energy (EH)

27. Conclusions
• No rupture from electric fields or defects
• Concerted bombardment tears membrane
• Sequential bombardment is more realistic
• Ions in solution can cause membrane to rupture

28. Future work
• Increase number of ions used
• Add potassium ions
• Completely tear membrane in half

29. References
• B. Aradi, B. Hourahine, and Th. Frauenheim. Dftb+, a sparse
matrix-based implementation of the dftb method, 2007.
• C. Daniels, A. Horning, A. Phillips, D.V.P. Massote, L. Liang, Z. Bullard,
B.G. Sumpter, and V. Meunier. Elastic, plastic, and fracture mechanisms
in graphene materials. J. Phys.: Condens. Matter, 27:1-18, 2015.
• C. Swope, H.C. Andersen, P.H. Berens, and K.R. Wilson. A computer
simulation method for the calculation of equilibrium physical clusters of
molecules: Application to small water clusters. J. Chem. Phys.,
76:637-649, 1982.
• Christopher J. Cramer. Essentials of Computational Chemistry: Theories
and Models. 2nd edition, 2013.
• G. J. Martyna, M. E. Tuckerman, D. J. Tobias, and M. L. Klein. Explicit
reversible integrators for extended systems dynamics. Molecular Phys.,
87:1117-1157, 1996.

30. Acknowledgements
• Dr. James W. Mazzuca
• Dr. Jacek Jakowski
• Dr. Kwai Wong
• Jacob Blazejewski

31. Questions?