Final

Theoretical Study of the Rupture of Graphene Membranes in a
Strong Electric Field
Krystle M. Reiss
Chemistry Department, Alma College, Alma, Michigan 48801
(Dated: April 20, 2016)

Contents
Acknowledgments 2
I. Abstract 3
II. Introduction 3
III. Pristine and Defective Membranes under Electric Fields 6
A. Membrane Construction and Parameters 7
B. Membrane Defects 8
C. Electric Fields 10
D. Results and Discussion 10
IV. Concerted Ion Bombardment 13
A. Parameters 13
B. Results and Discussion 14
V. Sequential Ion Bombardment 15
A. Parameters 16
B. Results and Discussion 16
VI. Conclusion and Future Work 18
References 21
A. Sample ﬁles and scripts 22
1. Pulling Energies from out.dftb 22
2. Initializing DFTB 23
3. Building Geometry Files 24
4. Sequential Bombardment 25
5. Finding Broken Bonds 27
6. Parameter File for Sequential Bombardment 29
7. Input Geometry for Sequential Bombardment 30
1

Acknowledgments
I would like to extend my thanks to the Alma College Provost’s Office for their support of
this project and the National Science Foundation, the University of Tennessee, Knoxville,
and Oak Ridge National Lab for hosting the 2015 CSURE REU Program.
I would also like to thank Dr. Jacek Jakowski and Dr. Kwai Wong for mentoring me during
the first steps of this project during my summer at Oak Ridge.
I also need to thank my roommates, Carrie Berkompas, Sarah Jack, and Christine Wiersma,
for keeping me (mostly) sane and for looking at many more pictures and videos of graphene
than any of them likely care to remember.
For being my sounding board when scripts wouldn’t work and the first in line to celebrate
with me when they did, I would like to thank Jacob Blazejewski.
I would especially like to thank Dr. James W. Mazzuca for his guidance and constant support
and for reading through the many drafts of this thesis. This project would not have been
possible without him.
2

I. ABSTRACT
Experimental results have shown that a graphene membrane submerged in a 1 M KCl
aqueous solution will rupture when a 3 V/nm field is applied. The cause of this rupture
could not be determined in a laboratory setting, leading to the use of molecular dynam-
ics calculations to discover why the ruptures were occurring. The density functional tight
binding electronic structure method and the Velocity Verlet algorithm were used to evaluate
various possible factors. It was determined that the field alone could not have caused the
tears, nor did defects in the membrane. However, it was shown that the ions present in the
solution surrounding the membrane could tear the membrane upon impact if they possessed
approximately 0.7 EH of kinetic energy.
II. INTRODUCTION
Graphene, the two-dimensional form of graphite, is a fairly new material with many
fascinating properties. Stronger than its equivalent weight in steel and very elastic, graphene
is composed of a highly conjugated system of carbon atoms, giving it 150 times the electron
mobility of silicon. This means that graphene is an extremely good conductor. However,
graphene is not yet a viable replacement for silicon switches in electronic devices, as graphene
has no band gap [1]. Silicon is a semiconductor, meaning that its band gap is just small
enough for electrons to cross it if an electric field of suitable magnitude is applied. When no
electric field is applied, silicon’s electrons are unable to cross the gap. Graphene’s lack of a
band gap makes it metallic in nature and electrons can move between HOMO and LUMO
energy levels without the application of an electric field.
In the preliminary results from an experiment conducted by Dr. Ivan Vlassiouk of Oak
Ridge National Laboratory in 2015, circular graphene membranes submerged in a 1 M KCl
aqueous solution ruptured when a 3 V/nm electric field was applied. These ruptures were
often catastrophic, with the membrane sometimes splitting entirely in half. This was surpris-
ing considering graphene’s high conductivity. There was no correlation between membrane
size and rupture. Though the membranes were perpendicular to the field overall, wrinkles
and defects could have caused regions of the membranes to be parallel to the field. This
could have caused regions of the membranes to be under higher stress from increased polar-
3

ity. Further stress may have arisen from ion bombardment. The field that was applied was
extremely strong, 3 V/nm, and could have caused the dissociated potassium and chloride
ions to be fired at the membranes at high enough velocities to cause a rupture. Because
the tears were so catastrophic, it was not possible to image any defects in the membranes
afterwards.
As the cause of rupture could not be determined from experimental observations, molec-
ular dynamics calculations were employed to study the system. Specifically, the density
functional tight binding (DFTB) method was used to calculate the electronic structure of
the system. DFTB calculates energy using the following equation:
E =
i
< Ψi|H|Ψi > +
1
2 α,β
γα,β∆qα∆qβ + Erep (1)
The first term in this equation is the input density of the valence electrons. The second
term is the self-consistent charge term which optimizes the electron density of the system.
The last term is the repulsive term which corrects the first electron density term. This third
term is entirely dependent on the distance between nuclei and is based on empirical data.
To solve Newton’s equations of motion and propagate the position of particles through
time, the Taylor expansion,
q(t + ∆t) = q(t) + v(t)∆t +
1
2!
a(t)(∆t)2
+
1
3!
d3
q(τ)
dt3
|τ=t(∆t)3
(2)
is often a starting point, with q(t) as the position, v(t) the velocity, and a(t) the acceleration.
Verlet devised a method of summing Taylor expansions of opposite time steps (±∆t) trun-
cated at the third order term. The odd-order derivatives in the equation above disappear
due to the opposite signs of the odd powers of ±∆t. This leaves
q(t + ∆t) = 2q(t) + a(t)(∆t)2
− q(t − ∆t) (3)
which is known as the Verlet algorithm. Thus any particle’s next position can be determined
from its current position, its previous position, and its acceleration. However, for simulations
in which temperature control is desired, propagation according to velocity is more beneficial
as particles’ velocities will be scaled according to a predefined coupling to a thermostat. The
4

leapfrog algorithm, a modiﬁed version of Verlet’s method, resolves this problem. Here, the
Taylor expansion is truncated at the second order term and half time-steps are utilized.
q t +
1
2
∆t ±
1
2
∆t = q t +
1
2
∆t ± v (t + ∆t)
1
2
∆t +
1
2!
a(t + ∆t)
1
2
∆t
2
(4)
Subtracting equations relating to opposite half time-steps, the position and velocity prop-
agation equations are simpliﬁed to
q(t + ∆t) = v t +
1
2
∆t ∆t + q(t) (5)
and
v t +
1
2
∆t = v t −
1
2
∆t + a(t)∆t (6)
Notice that the position and velocity are calculated one half-step out of phase. This means
that at each position q, the velocity v is averaged from the velocity a half step ahead and
behind q [2].
The algorithm used in these calculations is the Velocity Verlet algorithm, another variation
of the Verlet algorithm.
q(t + ∆t) = q(t) + v(t)∆t +
a(t)∆t2
2
(7)
v(t + ∆t) = v(t) +
[a(t) + a(t + ∆t)]∆t
2
(8)
This algorithm maintains superb precision in solving Newton’s equations of motion while
remaining relatively simple. Here, position and velocity are calculated in phase with each
other, unlike in the leapfrog algorithm [3].
Additionally, the system can be coupled to a theoretical heat bath to control the temper-
ature of the system. To achieve this, the heat bath scales the velocities of particles in the
system based on the coupling constant. A simple example is the Berendsen thermostat [4].
Here velocity is scaled according to a coupling constant λ:
v t +
1
2
∆t ← λv t +
1
2
∆t (9)
5

λ = 1 +
∆t
τT
T0
T(t − 1
2
∆t)
− 1
1
2
(10)
where T is the current temperature, T0 is the goal temperature, and τT is the time constant
that determines the strength of the coupling. A small coupling constant means that the bath
will have minimal effect on the system and may perform poorly in controlling temperature. A
coupling constant that is too large may cause jagged spikes and drops in the kinetic energy of
the system, creating a wholly unrealistic simulation. Many other types of thermostats exist,
including the Nose-Hoover thermostat. The way in which this thermostat scales velocity is
more complex than the Berendsen thermostat but serves the same purpose.
All of these tools (electronic structure methods, Newton’s equations of motion, and ther-
mostats) are combined to calculate the dynamics of chemical systems. On large systems like
graphene membranes, it is important to find a balance between precision and efficiency. An
ab initio electronic structure method would be entirely impractical for propagating a system
of this size through more than a few time-steps, though it would likely provide more accurate
results. Similarly, because this simulation focuses on bond breakage, a molecular mechanics
method where bonds are treated as springs would be far too simple for the purpose of these
calculations. DFTB provides a good balance between precision and simplicity, as does the
Velocity Verlet algorithm.
III. PRISTINE AND DEFECTIVE MEMBRANES UNDER ELECTRIC FIELDS
Before evaluating any of the effects of the surrounding solution, calculations were first
conducted using the two most basic elements of the system: a graphene membrane and an
electric field. A meaningful study required the use of multiple membrane sizes and field
strengths in order to determine if the field caused significant stress in the membrane. As
pristine (defect-free) graphene membranes are virtually nonexistent, it was necessary that
several common defects also be tested under an electric field. These calculations would form
a baseline for the effects of an electric field on a graphene membrane.
6

A. Membrane Construction and Parameters
All molecular dynamics calculations were performed using DFTB+ 1.2 [5]. These calcu-
lations were performed over 16 cores on the Beacon supercomputer at Oak Ridge National
Laboratory in Oak Ridge, Tennessee. Simulations were performed at 300 K using the Velocity
Verlet driver and the Nose-Hoover thermostat [3, 6]. The time-step size was 1.0 femtosec-
ond. The mio and halorg DFTB parameter sets were employed to calculate the electronic
and repulsive contributions of atoms in the system [7, 8]. Simulations were run for 5000
time-steps unless interrupted by Beacon’s twenty-four hour wall-clock time limit. Graphene
membranes containing 218 and 508 carbon atoms were created using the TubeGen 3.4 online
nanotube structure generator [9]. The larger 1018-carbon membrane was generated with the
Carbon Nanostructure Builder plugin within VMD [10]. All membrane edges were saturated
with hydrogens in IQmol [11]. Each membrane size was constructed with two varieties of
hydrogen termination along the zigzag edge as seen in Fig 1. In the first variation, the zigzag
FIG. 1: Along the zigzag edge, carbons could be either completely (left) or only partially (right)
saturated with hydrogen atoms.
edge was completely saturated with hydrogens. The second variation favored maintaining
graphene’s flexibility and two-dimensional geometry over complete saturation.
Within the 24-hour wall-clock time limit, less than 400 time-steps were completed for
the 1018-carbon membrane. It was determined that it was impractical to perform full MD
simulations on so many atoms, so the largest membrane was abandoned. Rupture occurred
in the original experiment regardless of the size of the membrane. Therefore, the effects of
membrane size on rupture is not the primary goal of this investigation.
The two smaller membranes were able to complete enough time-steps to provide usable
data. Between 1500 and 2500 time-steps were completed for the 508-carbon membranes and
the full 5000 time-steps were completed for the 218-atom membrane. Wrinkled membranes
7

were also created to analyze the effect that regions not perpendicular to the field had on
the membrane’s stability. These wrinkled membranes were created by performing a 5000
time-step simulation at 2000 K. The excessive heat allowed the membrane to warp and fold,
creating the desired wrinkles.
To prevent translational motion, corners or entire edges of membranes were immobilized.
It was deemed that freezing entire edges was excessively restrictive and so this method of
immobilization was abandoned. Freezing only the carbons at the corners of the membranes
allowed for a reasonable amount of flexibility while still pinning the membrane in place.
Unless otherwise noted, all calculations were performed twice, once with an unconstrained
membrane and again with frozen corners.
B. Membrane Defects
There are a variety of possible defects that can be present in a graphene membrane.
Among the most common are vacancy-type defects and Stone-Wales defects [12]. These
defects can have major impacts on the membrane, including causing folding and disrupting
conjugation. Seen in Figure 2, vacancy-type defects are fairly simplistic, their presence
merely indicating that there are carbon atoms missing from the membrane structure. Shown
in Figure 3, Stone-Wales defects involve the 90 degree rotation of two carbon atoms around
their mutual bond to form a pair of heptagons and a pair of pentagons instead of the regular
hexagons seen in a pristine graphene structure.
FIG. 2: Shown are single vacancy defect (left), double vacancy defect (middle), and reconstructed
double vacancy defect (right). The vertical axis is parallel to the zigzag edge and the horizontal
axis is parallel to the armchair edge.
8

FIG. 3: In a Stone-Wales defect, two carbon atoms are rotated 90 degrees, within the plane of the
membrane, around their mutual bond.
Using IQmol, four types of these defects were created for testing. The first was a single
vacancy-type defect in which a single carbon was removed from the membrane. Similarly,
a double vacancy was created by removing two adjacent carbon atoms whose bond was
parallel to the armchair axis. A second variety of the double vacancy was created in which
the defective area of the membrane was reconstructed. A membrane containing a Stone-
Wales defect was also created. Figure 4 shows how defects in the membrane can cause
folding, which could cause polarities when a field is applied. Calculations were performed
FIG. 4: Defects in the membrane often caused folding. Above is the final geometry of a membrane
containing a double vacancy-type defect.
with defects near the center of the membrane and then again near the edge of the membrane
to determine if defect placement played a role in ruptures.
9

C. Electric Fields
To create a 3 V/nm field at the membrane as seen in the original experiment, point
charges with energies of 15.0 eV and -15.0 eV were placed 10.0 nm away on either side of
the plane of the membrane. As before, MD simulations were performed for 5000 time-steps
unless interrupted by Beacon’s wall-clock limit. All field simulations were performed on both
unconstrained and constrained membranes.
D. Results and Discussion
The presence of an electric field had little to no effect on the graphene membranes. As seen
in Figure 5, there were no remarkable irregularities in the potential energies of any membrane
-393
-392.5
-392
-391.5
-391
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-390
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
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0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
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0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
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-391.5
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-390.5
-390
0 1000 2000 3000 4000 5000
Energy(EH)
Step
Total Energy
Potential Energy
Kinetic Energy
FIG. 5: The electric field had little effect on the membrane, as evidenced by the virtually identical
energy profiles of identical membranes under no electric field (top left), under a 3 V/nm field (top
right), and under a 3 V/nm field with frozen corners (bottom left). The bottom right image shows
an unconstrained warped membrane under a 3 V/nm electric field.
10

variations under a 3 V/nm field, indicating that no rupture occurred. After converging to
a low-energy geometry, the membrane continued to undulate, which is shown in the regular
ripples in the energy graphs. It should be noted that in Figures 5 and 7, kinetic energy
has been scaled for easier graphing. This is explained in Appendix A1. Examination of
the membranes’ geometries throughout the simulations confirmed that no rupture occurred.
In the case of the wrinkled membranes, the extra strain, if any, caused by regions not
perpendicular to the field was not evidenced in the results. Additionally, unless forced
to maintain their wrinkled geometries, these membranes immediately resumed their planar
forms when returned to 300 K, both under the electric field and not. This is demonstrated
in Figure 6. The large change in geometry is evidenced in the initial drop in energy seen
FIG. 6: Unless constrained, wrinkled graphene will resume its two-dimensional form. Shown is the
original wrinkled membrane (left), the same membrane after 5000 time-steps (middle) and with
frozen corners (right)
in these membranes’ energy profiles, which is significantly steeper those seen with the flat
membranes.
As the 3 V/nm field did not rupture the membrane, an extremely strong 30 V/nm field
was also applied to each of the membranes. This was accomplished by amplifying the point
charges tenfold to 150.0 eV and -150.0 eV, respectively. As with the weaker field, the 30
V/nm field showed no evidence of putting excessive strain on the membrane. However, it
should be noted that unconstrained membranes were shown to rotate to align themselves
with the field over time. This can be seen in the second major drop in the two unconstrained
energy profiles in Figure 7. Membranes that were constrained, and were therefore incapable
of rotation, showed no notable features when compared to previous simulations in which no
field was present. Close examination of the membrane geometries confirmed that no rupture
occurred. Because the membrane did not rupture even when under a field ten times stronger
11

-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
-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
FIG. 7: Flat unconstrained membrane under 30 V/nm field (top left), flat membrane with corners
frozen under 30 V/nm field (top right), unconstrained warped membrane under 30 V/nm field
(bottom left), and warped membrane with frozen corners under 30 V/nm field (bottom right)
than what was used in the original experiment, it was determined that the rupture could
not be due to the field alone.
In the membranes containing defects, the removal and rearrangement of atoms caused the
membrane to bend. Initial MD simulations showed no apparent overall structural difference
between a single vacancy-type defect and a pristine graphene membrane. The double vacancy
and Stone-Wales defects caused significant folding of the membrane, comparable to that seen
in the wrinkled pristine membranes. Though this caused large portions of the membranes to
no longer be perpendicular to the field, no rupture was observed in any defective membrane,
even under a 30 V/nm field.
12

IV. CONCERTED ION BOMBARDMENT
As it was shown that electric fields and defects alone were not enough to cause the
rupture of a graphene membrane, bombardment by the dissociated ions in the solution was
investigated. As a strong electric field was being applied to the system, it can be assumed
that the motion of the potassium cations and chloride anions was directed by this field.
That is, chloride anions would have accelerated towards the positive end of the field and the
potassium cations towards the negative end. If they possessed great enough momentum, it
would have been possible for the ions to break the carbon-carbon bonds of the membrane
and cause a rupture.
A. Parameters
To simulate bombardment by chloride anions, chlorides were placed between the mem-
brane and the negative point charge based on a uniform random sampling (Fig 8). In the
XZ plane, chlorides were not permitted beyond the edge of the membrane (+/-11.0 ˚A in
either direction) to maximize the probability of contact with the membrane. Along the Y
−10
0
10
−10 0 10
ZPosition(Å)
X Position (Å)
Membrane 1
Membrane 2
Membrane 3
FIG. 8: Three examples of the uniform random sampling of chloride positions used for ion bom-
bardment
axis (perpendicular to the membrane), the chloride ions were placed between 5.0 and 15.0 ˚A
away from the membrane. They were given 1.0 EH (2625.5 kJ/mol) of kinetic energy in the
13

direction of the positive point charge and thus the membrane. The overall system was given
a neutral charge as placing a full -10.0 e charge on the system caused SCC convergence to
consistently fail. In addition, the original experiment’s system would have a neutral overall
charge as the negative charge of the dissociated chloride anions would have been neutralized
by potassium cations also in solution. The Nose-Hoover thermostat used in previous calcu-
lations was removed for all ion bombardment simulations as the velocity scaling performed
by the thermostat severely reduced any initial momentum given to the ions. Corner atoms
in the membrane were frozen in the same manner as previously described to prevent the
impact of the ions from being transferred into translational motion of the membrane rather
than bond breakage.
FIG. 9: Sample input geometry for a concerted bombardment trial
B. Results and Discussion
Concerted bombardment of 10 chloride atoms with 1.0 EH of kinetic energy yielded an
average of 18.56 broken bonds after 40 trials, as shown in Figure 10. This method proved
that it is possible to tear a graphene membrane via ion bombardment. An example of such
a tear is given in Figure 11. This method is physically unrealistic, however, as the chloride
ions were packed fairly close together (10 ions in 1000 ˚A3
). A more realistic simulation with
14

0
7
14
21
28
35
0 5 10 15 20 25 30 35 40BondsBroken
Trial
Bonds Broken
Running Mean
FIG. 10: Bonds broken during concerted bombardment at 1.0 EH
the ions spread farther apart was desired.
FIG. 11: Sample output geometry from a 1.0 EH concerted trial
V. SEQUENTIAL ION BOMBARDMENT
To create a more physically realistic system, initial concerted bombardment was replaced
with a more sequential approach. Here, rather than in a single closely-spaced group of ten,
ions were initiated in a series of ﬁve pairs. The goal of this style was to spread the ions apart,
creating a distribution closer to what would be seen in solution. The secondary accomplish-
ment of this style was that it created a more eﬃcient method for handling calculations that
15

failed to converge. If calculations failed for one of the five pairs, the simulation could restart
with a new chloride pair from the beginning of that step rather than starting over completely.
A. Parameters
In this new style of bombardment, pairs of chloride ions were initiated with a set amount
of kinetic energy and removed after rebounding off or puncturing the membrane. This
process was repeated until calculations had been successfully completed for five pairs of ions.
Initial chloride positions were taken from a uniform sampling as before, however the range of
starting distances from the membrane was reduced to between 5.0 and 10.0 ˚A. Calculations
that did not complete a minimum number of time-steps (determined by how long an ion
took to reach and rebound off the membrane from a distance of 10.0 ˚A) were restarted with
a new pair of chloride ions. These simulations were repeated using kinetic energies ranging
from 0.2 to 1.0 EH to evaluate the correlation between ion momentum and bond breakage.
All systems using this method were given a charge of -2.0 e to account for the chloride ions.
The positions and velocities of the carbon and hydrogen atoms in the graphene membrane
were preserved between chloride pair removal and replacement.
B. Results and Discussion
Sequential bombardment yielded better results than the concerted bombardment as un-
successful trials were automatically rerun with new chloride pairs until they completed the
minimum number of time-steps. Examples of the output geometries can be seen in Figure
12. This method of bombardment proved to increase the efficiency of the calculations sig-
nificantly as failed simulations could be restarted from the point of failure rather than from
the beginning. Additionally, calculating two chlorides at a time rather than ten decreased
the overall time for calculations, even though more total time-steps were being performed.
A running average was calculated as trials were completed. As shown in Figure 13, after
40 trials, the average number of broken bonds converged for each set of trials. Figure 14
and Table 1 show the relationship between the initial kinetic energy of the chloride ions
and the number of bonds broken. Each point is the result of 200 individual calculations
(40 membranes bombarded by 5 pairs of chlorides each) for a total of 1400 calculations
16

FIG. 12: Example output geometries from the 0.4 EH (top left), 0.6 EH (top middle), 0.7 EH (top
right), 0.8 EH (bottom left), 0.9 EH (bottom middle), and 1.0 EH (bottom right) cases
represented by the entire figure. As expected, there is a definite positive correlation between
initial kinetic energy and bond breakage, with no bonds being broken at the lowest energies
and a steady increase in bond breakage at higher energies. When the chloride ions were
given 0.2 EH of kinetic energy, no bonds were broken during any of the calculations. At
0.4 EH, there were 3 cases out of the 40 in which one bond was broken and a single case
that yielded two broken bonds. However, the majority were identical to the 0.2 EH case.
At 0.6 EH, the distribution of the number of broken bonds begins to increase, with a few
calculations yielding as many as five broken bonds, though the majority remain at zero. It is
not until kinetic energy reaches 0.7 EH that bonds are broken consistently. Presumably, if ion
bombardment were to continue beyond the initial five pairs, the membrane would eventually
tear completely in half as was seen in the original experiments, possibly even at the lower
energies where broken bonds were rarer.
17

0
1
2
3
4
5
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0
4
8
12
16
20
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0
6
12
18
24
30
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
0
6
12
18
24
30
0 5 10 15 20 25 30 35 40
BondsBroken
Trial
Bonds Broken
Running Mean
FIG. 13: Running averages were taken of the number of broken bonds at each energy. After forty
trials, these running averages converged. Represented are the 0.4 EH (top left), 0.6 EH (top right),
0.7 EH (middle left), 0.8 EH (middle right), 0.9 EH (bottom left), and 1.0 EH (bottom right) cases.
VI. CONCLUSION AND FUTURE WORK
It has been shown that strong electric ﬁelds and singular defects alone (and combined)
cannot cause the rupture of a graphene membrane. Even an electric ﬁeld with strength of 30
V/nm applied to a membrane bent by a double vacancy does not cause a tear. As graphene
has superior mechanical strength and high electron mobility, these results are not surprising.
18

0
5
10
15
0 0.2 0.4 0.6 0.8 1 1.2
BondsBroken(Mean)
Kinetic Energy (EH)
FIG. 14: Bond breakage increases with ion kinetic energy, with 0.7 EH seeming to be the threshold
energy for signiﬁcant tearing.
Energy (EH) Broken Bonds SD
0.2 0.00 0.00
0.4 0.13 0.40
0.6 1.23 1.75
0.7 2.40 2.05
0.8 6.46 3.90
0.9 10.58 4.32
1.0 13.90 3.65
TABLE I: Values of results presented in Figure 14
However, ion bombardment has been shown to cause membrane rupture. Beginning at
0.7 EH of kinetic energy, the membrane begins to tear consistently upon ion impact with
the number of broken bonds increasing roughly proportionally to the kinetic energy applied.
Based on this data, ion bombardment appears to be the cause of the membrane observed in
the original experiments, though a complete tear (ripping the membrane in half) has not yet
been observed. It is predicted that if more ions were to bombard the membrane, a complete
tear would eventually occur. Future work on this project may focus on continuing to make
the simulation more physically realistic. This would likely include calculating the shielding
19

effects of the solvent in the system. Additionally, in an effort to achieve a complete tear,
potassium cations should be added to the system. Because of the electric field, the cations
would push through the opposite side of the membrane from the chloride anions. This could
potentially cause more catastrophic tears, similar to what was observed in the laboratory.
20

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[5] B. Aradi, B. Hourahine, and Th. Frauenheim. Dftb+, a sparse matrix-based implementation
of the dftb method, 2007.
[6] 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.
[7] M. Elstner, D. Porezag, G. Jungnickel, M. Haugk J. Elsner, Th. Frauenheim, S. Suhai, , and
G. Seifert. Self-consistent-charge density-functional tight-binding method for simulations of
complex materials properties. Phys. Rev. B, 58:7260, 1998.
[8] Tomáˇs Kubaˇr, Zoltán Bodrog, Michael Gaus, Christof Köhler, Bálint Aradi, Thomas Frauen-
heim, and Marcus Elstner. Parametrization of the scc-dftb method for halogens. J. Chem.
Theor. Comput., 9:2939–2949, 2013.
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edu/research/tubegenonline.html. Accessed: 2016-02-04.
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Graphics, 14:33–38, 1996.
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and V. Meunier. Elastic, plastic, and fracture mechanisms in graphene materials. J. Phys.:
Condens. Matter, 27:1–18, 2015.
21

APPENDIX A: SAMPLE FILES AND SCRIPTS
1. Pulling Energies from out.dftb
This script reads the file out.dftb and copies the kinetic, potential, and total energies at each
time-step. These energies are pasted into a new file, which can be used to create energy
profile graphs. Additionally, line 14 scales the kinetic energies down to near where the total
and potential energies are. This allows for easier graphing. This results in a file titled
all energies that has five columns: time-step, potential energy, kinetic energy, total energy,
and scaled kinetic energy.
1 #!/bin/bash
2
3 steptime=‘echo "scale=4; 1/5000" | bc‘
4 step=$steptime
5
6 sed -n -e ’s/ˆ.*Total Mermin free energy: //p’ out.dftb > potential_energies
7 sed ’s/H//’ potential_energies > 1_potential_energies
8 sed ’1 iPotential Energy’ 1_potential_energies > pot_en
9
10 sed -n -e ’s/ˆ.*MD Kinetic Energy:
//p’ out.dftb > kinetic_energies
11 sed ’s/H//’ kinetic_energies > 2_kinetic_energies
12 sed ’1 iKinetic Energy’ 2_kinetic_energies > kin_en
13
14 awk ’{print ($1 - 900.5)}’ 2_kinetic_energies > scaled_kin
15 sed ’1 iScaled Kin Energy’ scaled_kin > sca_kin_en
16
17 a=$(wc kinetic_energies)
18 steptot=$(echo $a|cut -d’ ’ -f1)
19
20 sed -n -e ’s/ˆ.*Total MD Energy: //p’ out.dftb > total_energies
21 sed ’s/H//’ total_energies > 3_total_energies
22 sed ’1 iTotal Energy’ 3_total_energies > tot_en
23
24 for i in ‘seq 1 $steptot‘;
25 do echo "$i" >> steps
26 done
27 sed ’1 iStep’ steps > en_steps
28 rm potential_energies kinetic_energies total_energies 1_potential_energies 2_kinetic_energies 3_t
29
30 paste en_steps pot_en kin_en tot_en sca_kin_en> all_energies
31
32 rm pot_en kin_en tot_en steps sca_kin_en en_steps
22

2. Initializing DFTB
This script initializes multiple calculations serially. It is designed to utilize two cores, though
this can be easily changed by altering line 11.
1 #!/bin/bash
2 shopt -s expand_aliases
3 source /home/mariereiss/.bashrc
4
5 echo "Number of simulations?"
6 read RUNS
7
8 echo "Title number of first test?"
9 read RUN
10
11 export OMP_NUM_THREADS=2
12
13 for i in ‘seq 1 $RUNS‘; do
14
15 cd $RUN* #Move to simulation folder
16 date > list_times #List start time
17 dftb #Run DFTB
18 date >> list_times #List end time
19 cd .. #Move to parent folder
20 ((RUN+=1)) #Move to next title number
21
22 done
23

3. Building Geometry Files
This script builds the geometry ﬁles required for simultaneous bombardment calculations.
1 #!/bin/bash
4
6 read RUNS
7
8 echo "Title number of folder?"
9 read RUN
10
11 echo "Line of first chloride geometry?"
12 read LINE
13
14 echo "Test number?"
15 read NOM
16
18
19 mkdir $RUN-seqtrial$NOM #Create folder
20 cd $RUN* #Move to new folder
21 cat ../maplerandomtest/testcoord | tail -n +$LINE | head -10 > chlor_coord
22 #Copy 10 random ion coordinates to holder file
23
24 paste ../in_col chlor_coord > gen_chlor_coord
25 #Assign atom number and type to ions
26
27 rm chlor_coord
28 cat gen_chlor_coord >> geom.gen
29 #Append ion coordinates to geometry file
30
31 cd .. #Back into parent directory
32 ((RUN+=1))
33 ((LINE+=10)) #Move to next ten ion coordinates
34 ((NOM+=1))
35
36 done
24

4. Sequential Bombardment
This script is used to perform sequential ion bombardment calculations. It copies geom.gen
to the working directory and adds a pair of chlorides with random positions to the end of
that ﬁle. It also copies a velocity.dat ﬁle and adds the velocity corresponding to the desired
kinetic energy for the ions to the end.
1 #!/bin/bash
4
6 read RUNS
7
8 echo "Title number of folder?"
9 read RUN
10
11 echo "Number of steps per run?"
12 read STEPS #For 5 pairs of ion, type 6
13
14 echo "Line of first chloride geometry?"
15 read LINE
16
18
19 STEP=1
20 cd $RUN* #Move to main calculation folder
21 export OMP_NUM_THREADS=2
22 cp ../seq_sam/geom.gen geom.gen.tmp #Copy base geometry file
23 cp ../seq_sam/velocity.dat velocity.dat #Copy base velocity file
24 date > list_times
25
26 for j in ‘seq 1 20‘; do
27
28 cat ../maplerandomtest/seq_chlorides | tail -n +$LINE | head -2 > chlor_coord
29 #Copies a pair of chloride coordinates to a holder file
30
31 paste ../seq_sam/seq_col chlor_coord > gen_chlor_coord
32 #Adds atom number and type to ion coordinates
33
34 rm chlor_coord
35 mkdir $STEP #Creates subfolder for current step
36 cd $STEP #Move to subfolder
37 cp ../../seq_sam/dftb_in.hsd . #Copies input file to subfolder
38 cp ../velocity.dat . #Copies velocity from parent folder
39 cp ../geom.gen.tmp geom.gen #Copies geometry from parent folder
40 cat ../gen_chlor_coord >> geom.gen #Appends ion coordinates to geometry
41 dftb
42 test=$(grep "Geometry optimization step: " detailed.out | tail -1 | cut -c29-32)
43
44 if [ "$test" -lt "100" ]; then
45
46 cd ..
47 mv $STEP $STEP-failed-$j #Check if minimum steps completed
48
49 else
50
51 cat geo_end.xyz | tail -260 | head -258 > velocity.dat.tmp1
52 awk ’{print $6, $7, $8}’ velocity.dat.tmp1 > velocity.dat.tmp2
53 echo "0.00000000 121.72336910 0.000000" >> velocity.dat.tmp2
54 echo "0.00000000 121.72336910 0.000000" >> velocity.dat.tmp2
55 cp velocity.dat.tmp2 ../velocity.dat
56 #Update membrane velocity and append velocities for new ion pair
57
58 cp geo_end.gen geom.gen.tmp
25

59 sed -i ’$ d’ geom.gen.tmp
60 sed -i ’$ d’ geom.gen.tmp
61 cp geom.gen.tmp ../geom.gen.tmp
62 ((STEP+=1))
63 cd ..
64 #Copy membrane geometry to parent folder
65
66 fi
67
68 if [ "$STEP" -eq "$STEPS" ]; then
69
70 break #Break loop if desired number of ions achieved
71
72 else
73
74 ((LINE+=2)) #Select next two chloride geometries
75
76 fi
77
78 if [ "$LINE" -gt "998" ]; then
79
80 break 2 #Break loop if all ion coordinates used
81
82 fi
83
84 done
85
86 date >> list_times
87 cd ..
88 ((RUN+=1))
89 echo "$LINE"
90
91 done
92
93 ((LASTLINE=$LINE+1))
94 echo "Last line is $LASTLINE"
26

5. Finding Broken Bonds
This script finds difference between the number of carbon-carbon bonds before and after ion
bombardment. In the parent loop, the script searches the input geometry a bond between
each pair of carbon atoms. If it finds a bond (defined by a carbon-carbon distance of less
than 1.7 ˚A), the script then checks the output geometry to see if the bond is still present
(defined by a carbon-carbon distance less than 2.0 ˚A).
1 #!/bin/bash
2
3 echo "Folder number?"
4 read RUN
5 echo "Number of runs?"
6 read RUNS
7
8 for k in ‘seq 1 $RUNS‘;do
9
10 cd $RUN*
11 COUNT1=0
12 COUNT2=0
13
14 for i in ‘seq 1 217‘; do
15
16 ((START=$i+1))
17
18 for j in ‘seq $START 218‘; do
19
20 LINE1=$(($i+2))
21 LINE2=$(($j+2))
22 cat 1/geom.gen | tail -n +$LINE1 | head -1 > coords.dat
23 cat 1/geom.gen | tail -n +$LINE2 | head -1 >> coords.dat
24 #Copy a pair of carbon coordinates from input
25
26 XCOORD1=$(cat coords.dat | tail -n +1 | head -1 | cut -c11-30)
27 XCOORD12=$(echo $XCOORD1 | sed -e ’s/[eE]+*/*10ˆ/’)
28 XCOORD2=$(cat coords.dat | tail -n +2 | head -1 | cut -c11-30)
30 #Find x coordinates of each carbon
31
32 YCOORD1=$(cat coords.dat | tail -n +1 | head -1 | cut -c31-50)
33 YCOORD12=$(echo $YCOORD1 | sed -e ’s/[eE]+*/*10ˆ/’)
34 YCOORD2=$(cat coords.dat | tail -n +2 | head -1 | cut -c31-50)
36 #Find y coordinates of each carbon
37
38 ZCOORD1=$(cat coords.dat | tail -n +1 | head -1 | cut -c51-67)
39 ZCOORD12=$(echo $ZCOORD1 | sed -e ’s/[eE]+*/*10ˆ/’)
40 ZCOORD2=$(cat coords.dat | tail -n +2 | head -1 | cut -c51-67)
42 #Find z coordinate of each carbon
43
44 XDIS=$(echo "$XCOORD12 - $XCOORD22" | bc -l)
45 YDIS=$(echo "$YCOORD12 - $YCOORD22" | bc -l)
46 ZDIS=$(echo "$ZCOORD12 - $ZCOORD22" | bc -l)
47 #Find 1-D distances
48
49 XSQUARE=$(echo "$XDIS * $XDIS" | bc -l)
50 YSQUARE=$(echo "$YDIS * $YDIS" | bc -l)
51 ZSQUARE=$(echo "$ZDIS * $ZDIS" | bc -l)
52 #Square each distance
53
54 DIS1=$(echo "$XSQUARE + $YSQUARE + $ZSQUARE" | bc -l)
55 #Add all squared distances together
56
57 if [[ $(echo "if ($DIS1 <= 3) 1 else 0" | bc) -eq 1 ]]; then
27

58
59 ((COUNT1+=1)) #Check for bond in input
60
61 cat 5/geo_end.gen | tail -n +$LINE1 | head -1 > coords1.dat
62 cat 5/geo_end.gen | tail -n +$LINE2 | head -1 >> coords1.dat
63 #Copy carbon pair coordinates from output
64
65 XCOORD3=$(cat coords1.dat | tail -n +1 | head -1 | cut -c11-30)
67 XCOORD4=$(cat coords1.dat | tail -n +2 | head -1 | cut -c11-30)
69
70 YCOORD3=$(cat coords1.dat | tail -n +1 | head -1 | cut -c31-50)
72 YCOORD4=$(cat coords1.dat | tail -n +2 | head -1 | cut -c31-50)
74
75 ZCOORD3=$(cat coords1.dat | tail -n +1 | head -1 | cut -c51-67)
77 ZCOORD4=$(cat coords1.dat | tail -n +2 | head -1 | cut -c51-67)
79
80 XDIS1=$(echo "$XCOORD32 - $XCOORD42" | bc -l)
81 YDIS1=$(echo "$YCOORD32 - $YCOORD42" | bc -l)
82 ZDIS1=$(echo "$ZCOORD32 - $ZCOORD42" | bc -l)
83
84 XSQUARE1=$(echo "$XDIS1 * $XDIS1" | bc -l)
85 YSQUARE1=$(echo "$YDIS1 * $YDIS1" | bc -l)
86 ZSQUARE1=$(echo "$ZDIS1 * $ZDIS1" | bc -l)
87
88 DIS2=$(echo "$XSQUARE1 + $YSQUARE1 + $ZSQUARE1" | bc -l)
89
90 if [[ $(echo "if ($DIS2 < 4) 1 else 0" | bc) -eq 1 ]]; then
91
92 ((COUNT2+=1)) #Check for bond in output
93
94 fi
95 rm coords1.dat
96 fi
97
98 rm coords.dat
99
100 done
101
102 done
103 cd ..
104 BROKEN=$(($COUNT1-$COUNT2)) #Find difference between input and output
105 echo "$RUN $COUNT1 - $COUNT2 = $BROKEN" >> broken.9.dat #Record broken bonds
106 ((RUN+=1))
107 done
28

6. Parameter File for Sequential Bombardment
This ﬁle dictates the parameters of the calculation, including temperature, atom velocities,
charge, and ﬁeld strength.
1 Geometry = GenFormat {
2 <<< "./geom.gen"
3 }
4 Driver = VelocityVerlet {
5 Steps = 150
6 TimeStep [Femtosecond] = 1.0
7 Thermostat = None {}
8 OutputPrefix = "geo_end"
9 MovedAtoms = 4:98 101:118 121:215 H Cl
10
11 Velocities [AA/ps] = {
12 <<< "./velocity.dat"
13 }
14
15 }
16
17 Hamiltonian = DFTB {
18 SCC = Yes
19 SCCTolerance = 1.0E-6
20 MaxSCCIterations = 1000
21 MaxAngularMomentum = {
22 C = "p"
23 H = "s"
24 Cl = "p"
25 }
26
27 Charge = -2.0
28
29 Filling = Fermi {
30 Temperature [Kelvin] = 300
31 }
32 SlaterKosterFiles = {
33 C-C = "../../mio/params/C-C.skf"
34 H-C = "../../mio/params/H-C.skf"
35 C-H = "../../mio/params/C-H.skf"
36 H-H = "../../mio/params/H-H.skf"
37 Cl-H = "../../mio/params/Cl-H.skf"
38 H-Cl = "../../mio/params/H-Cl.skf"
39 C-Cl = "../../mio/params/C-Cl.skf"
40 Cl-C = "../../mio/params/Cl-C.skf"
41 Cl-Cl = "../../mio/params/Cl-Cl.skf"
42
43 }
44 ElectricField = {
45 PointCharges = {
46 CoordsAndCharges [Angstrom]= {
47 0 100 0 15.0
48 0 -100 0 -15.0
49 }
50 }
51 }
52 }
53
54 Options = {
55 WriteAutotestTag = Yes
56 AtomResolvedEnergies = Yes
57 }
58
59 ParserOptions = {
60 ParserVersion = 3
61 }
29

7. Input Geometry for Sequential Bombardment
This file lists the atom number, types, and coordinates. The first line states the number of
atoms in the file and the geometry type (C stands for cluster). The order of elements in the
second line specifies what the numbers in the second column mean (for example, 1 represent
carbon).
1 260 C
2 C H Cl
3 1 1 -0.1178400351E+02 -0.1343219345E+01 -0.9729946338E+01
4 2 1 -0.1039611338E+02 -0.1468043701E+01 -0.9872934589E+01
5 3 1 -0.9772759310E+01 -0.1855013368E+01 -0.1111117486E+02
6 4 1 -0.8439427255E+01 -0.1850813217E+01 -0.1132636364E+02
7 5 1 -0.7544695465E+01 -0.1433285923E+01 -0.1027995423E+02
8 6 1 -0.6071410110E+01 -0.1330985989E+01 -0.1046275125E+02
9 7 1 -0.5438917429E+01 -0.1047349951E+01 -0.1170282216E+02
10 8 1 -0.4103374287E+01 -0.7961626475E+00 -0.1188573037E+02
11 9 1 -0.3224930612E+01 -0.9057856901E+00 -0.1080851050E+02
12 10 1 -0.1760746050E+01 -0.9694696477E+00 -0.1097591516E+02
13 11 1 -0.1232694656E+02 -0.8430203545E+00 -0.8590698820E+01
14 12 1 -0.1146995476E+02 -0.6464595908E+00 -0.7396553537E+01
15 13 1 -0.1006901817E+02 -0.8459609185E+00 -0.7514294532E+01
16 14 1 -0.9503450242E+01 -0.1144657647E+01 -0.8798732415E+01
17 15 1 -0.8093054256E+01 -0.1267844220E+01 -0.8981525974E+01
18 16 1 -0.7249487972E+01 -0.1122405838E+01 -0.7841410500E+01
19 17 1 -0.5843028425E+01 -0.1171028185E+01 -0.7998507505E+01
20 18 1 -0.5214858259E+01 -0.1149685309E+01 -0.9319289391E+01
21 19 1 -0.3781121052E+01 -0.1116724258E+01 -0.9514255775E+01
22 20 1 -0.2932380444E+01 -0.1106190451E+01 -0.8346350965E+01
23 21 1 -0.1529836532E+01 -0.1094046706E+01 -0.8524678823E+01
24 22 1 -0.1204281358E+02 -0.2689622969E+00 -0.6163260871E+01
25 23 1 -0.1122242329E+02 -0.3104737618E+00 -0.4975048962E+01
26 24 1 -0.9790935430E+01 -0.5236615204E+00 -0.5100253255E+01
27 25 1 -0.9232250626E+01 -0.7872934475E+00 -0.6404312199E+01
28 26 1 -0.7810490361E+01 -0.9044319327E+00 -0.6584221787E+01
29 27 1 -0.6982189996E+01 -0.8162394187E+00 -0.5453188995E+01
30 28 1 -0.5544441694E+01 -0.7875014306E+00 -0.5571057079E+01
31 29 1 -0.4984077052E+01 -0.9512897722E+00 -0.6862455376E+01
32 30 1 -0.3514364944E+01 -0.1038810528E+01 -0.7089703065E+01
33 31 1 -0.2705081032E+01 -0.9327750274E+00 -0.5915399660E+01
34 32 1 -0.1253038953E+01 -0.9106798359E+00 -0.6108792616E+01
35 33 1 -0.1183018741E+02 -0.1598111959E+00 -0.3664472628E+01
36 34 1 -0.1101413676E+02 -0.2492339467E+00 -0.2531201523E+01
37 35 1 -0.9518429148E+01 -0.3720745641E+00 -0.2687580204E+01
38 36 1 -0.8965171080E+01 -0.4357784950E+00 -0.3980290635E+01
39 37 1 -0.7571552156E+01 -0.6096547679E+00 -0.4112698232E+01
40 38 1 -0.6730119402E+01 -0.3312205720E+00 -0.2983788891E+01
41 39 1 -0.5266194141E+01 -0.3107278034E+00 -0.3157703769E+01
42 40 1 -0.4687366343E+01 -0.5541067163E+00 -0.4436834807E+01
43 41 1 -0.3279364541E+01 -0.6187669756E+00 -0.4625087757E+01
44 42 1 -0.2470948374E+01 -0.3487112479E+00 -0.3501813601E+01
45 43 1 -0.1040289619E+01 -0.3077438864E+00 -0.3723985244E+01
46 44 1 -0.1157332486E+02 -0.2723564288E+00 -0.1224275138E+01
47 45 1 -0.1068350528E+02 -0.2994269110E+00 -0.1158936453E+00
48 46 1 -0.9257924465E+01 -0.1560311215E+00 -0.2356896915E+00
49 47 1 -0.8761806159E+01 -0.2635348790E+00 -0.1548327553E+01
50 48 1 -0.7295239323E+01 -0.2103632793E+00 -0.1644742171E+01
51 49 1 -0.6415544719E+01 -0.1548349863E+00 -0.5584931645E+00
52 50 1 -0.5000532522E+01 0.2072473036E-01 -0.7402318386E+00
53 51 1 -0.4414841896E+01 -0.8823123579E-01 -0.2045035651E+01
54 52 1 -0.3030830214E+01 -0.1171042866E+00 -0.2184903103E+01
55 53 1 -0.2203229059E+01 0.1345756509E+00 -0.1068720778E+01
56 54 1 -0.7620454010E+00 0.1968150475E+00 -0.1253626687E+01
57 55 1 -0.1128466175E+02 -0.1761632676E+00 0.1187621272E+01
58 56 1 -0.1047172941E+02 -0.1103638917E-01 0.2371624932E+01
30

59 57 1 -0.9033572641E+01 0.3300239011E-02 0.2224176314E+01
60 58 1 -0.8468290221E+01 -0.5805580523E-01 0.9152424836E+00
61 59 1 -0.7007051487E+01 -0.4285111859E-01 0.7139964417E+00
62 60 1 -0.6196181950E+01 0.1422543718E+00 0.1877654301E+01
63 61 1 -0.4792488622E+01 0.2511869260E+00 0.1695708549E+01
64 62 1 -0.4188650998E+01 0.1422430133E+00 0.4121406248E+00
65 63 1 -0.2798681655E+01 0.3021104512E+00 0.2409473416E+00
66 64 1 -0.1966165617E+01 0.4951379179E+00 0.1378713872E+01
67 65 1 -0.5499643693E+00 0.4666961220E+00 0.1214851021E+01
68 66 1 -0.1098881289E+02 0.4489416030E-01 0.3656990560E+01
69 67 1 -0.1015423430E+02 0.2429366188E+00 0.4782707970E+01
70 68 1 -0.8738710320E+01 0.2932900222E+00 0.4667072178E+01
71 69 1 -0.8198172926E+01 0.1363110519E+00 0.3354041556E+01
72 70 1 -0.6761384851E+01 0.2203183242E+00 0.3167964013E+01
73 71 1 -0.5901084427E+01 0.4720183462E+00 0.4302355967E+01
74 72 1 -0.4471701067E+01 0.3352380579E+00 0.4155959689E+01
75 73 1 -0.3957437939E+01 0.3464347268E+00 0.2813542512E+01
76 74 1 -0.2540643711E+01 0.4458657068E+00 0.2658677393E+01
77 75 1 -0.1666679258E+01 0.5910529859E+00 0.3828967971E+01
78 76 1 -0.2617988575E+00 0.6153236014E+00 0.3645948862E+01
79 77 1 -0.1068617013E+02 0.3567164201E+00 0.6092288479E+01
80 78 1 -0.9846707809E+01 0.4682733941E+00 0.7234264624E+01
81 79 1 -0.8443579135E+01 0.7156867205E+00 0.7058722401E+01
82 80 1 -0.7857926210E+01 0.3999615811E+00 0.5800896160E+01
83 81 1 -0.6460274819E+01 0.4992936490E+00 0.5645050239E+01
84 82 1 -0.5599949234E+01 0.6237277956E+00 0.6757588796E+01
85 83 1 -0.4201599996E+01 0.6634347555E+00 0.6580481129E+01
86 84 1 -0.3600862303E+01 0.4181990260E+00 0.5294593460E+01
87 85 1 -0.2245684868E+01 0.5489544086E+00 0.5160596032E+01
88 86 1 -0.1389283910E+01 0.6080306953E+00 0.6292187765E+01
89 87 1 0.7822923341E-02 0.6717284481E+00 0.6144458515E+01
90 88 1 -0.1038849246E+02 0.5754164754E+00 0.8528399866E+01
91 89 1 -0.9567502712E+01 0.7683315078E+00 0.9690833619E+01
92 90 1 -0.8178683833E+01 0.8851158712E+00 0.9516680142E+01
93 91 1 -0.7600941591E+01 0.7605142291E+00 0.8244953599E+01
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31

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32

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261 259 3 10.98881564 -9.997458100 7.21168649
262 260 3 -4.667314414 -6.439246724 7.41413796
33

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