Term Project
ME 346 Molecular Dynamics
December 15, 2005
Manish Chandra (manishc@stanford.edu)
Erich Elsen (eeelsen@stanford.edu)
Will Fong (willfong@stanford.edu)
Background:
A variety of materials have been used to produce nanowires. However, in this
project we will focus our attention on silicon nanowires. It is currently experimentally
difficult to torque nanowires10, but knowing the shear properties may become important
as nanowires are intertwined into longer and thicker wires or embedded into other
materials. There are already some applications making use of the torsional modes of
nanowires1. Two different boundary conditions are investigated as well as the effect of
different radii and lengths.
The authors are aware of one prior paper of worth
investigating the behavior of nanowires under torsion3.
Problem Description & Implementation:
y
Fig. 1: A silicon nanowire,
showing torsion about the
z-direction.
z
x
Investigating nanowires under torsion in a molecular simulation requires applying a
different boundary condition from the usual periodic boundary conditions. The usual
periodic boundary conditions will not match the ends correctly (they will be twisted by
different amounts) causing large simulation artifacts. Another possibility is to simply fix
the ends and then apply a rotation to the middle, this alleviates the periodic boundary
problem, however this introduces its own artifacts, in that essentially the periodic
boundary conditions have been removed and there may be large end effects. The ideal
technique would still preserve periodicity in the z-direction with correctly matched ends.
The technique to meet these requirements is described in the following. When the
nanowires are created in MD++ their length is the extent of the simulation cell, so to
isolate the nanowires from the normal periodic boundary conditions imposed by MD++,
the simulation cell was expanded so that the nanowire would no longer interact with any
of its images in any direction. Then image atoms are created at each end of the wire,
which are copied from the opposite end of the wire.
Fig. 2: A silicon nanowire,
showing the application of
periodic boundary
conditions.
After this operation a torque can be applied by rotating the atoms according to a linearly
varying angular displacement (with rotation angle = 0 in the middle of the rod). That is
the angle of rotation is linear from z = 0 to L/2 with negative z values correspond to
negative rotations.
Now, from this point on, two different approaches were tried. The first will be referred to
hereafter as the “true” periodic boundary condition and the other will hereafter be
referred to as the “pseudo-periodic” boundary condition.
The “pseudo-periodic”
condition simply fixed the atoms that had been copied at this point and would then either
let the atoms in the interior relax (if at 0K) or run (if at T>0K). Only after a full
relaxation or run finished would the position of these fixed, copied atoms be updated.
Here updated means that new copies are made from the original atoms. The “true”
periodic boundary condition essentially made the copied atoms slaves to the atom they
had been copied from, so that for every move the original atoms made, the copied atom
made an identical move.
A further change in the code was necessary for conjugate relaxation to converge when
using the “true” boundary conditions. The code used to calculate energy simply be
adding up the energy associated with each atom, however in our case, because some
atoms are copied, this leads to a double counting (or triple counting in the three body
term of the potential) of the energy of these atoms. To do this, before each energy term is
added to the total, a check is made to see if any of the atoms involved in the calculation
were marked as “copied.” If this was the case, then none, 1/3, 1/2, or 2/3 of the energy is
added to the total, depending on if it is the two or three body potential and how many of
the atoms are copied.
Last in order to determine if and when the nanowire fails we need to have a measure of
the torsional energy of the nanowire. Since we are applying an angle of rotation to the
wire instead of a torque, we will need to take a Virial approach to compute the average
torque. See the appendix for details regarding this Virial torque (or moment).
Results:
The silicon nanowires used in the simulations were configured to have their z-axis along
the [1 1 0] direction. For simplicity the temperature was taken to be 0K. Two lengths
(65 and 130 Angstroms) and two thicknesses (radius = 12 and 24 Angstroms) were used
in the study of torsional stiffness and modes of failure. At each step the nanowire is
rotated 0.02 radians at each end.
The first test involved the use of pseudo-periodic and true periodic boundary conditions
for the long thick nanowire.
Looking at the following figures the only noticeable
difference is that the maximum moment for the pseudo-periodic case is slightly larger
than the true periodic case.
One possible explanation for this behavior is that the
presence of the fixed ends in the pseudo-periodic case during the conjugate relaxation
require more torque for the same angular rotation.
Moment vs Angular Rotation for Pseudo PBC (Long Thick)
2500
Moment (ev-oA)
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500
0
0
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Angular Rotation (De g)
Moment vs Angular Rotation for True PBC (Long Thick)
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Moment (ev-oA)
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0
0
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Angular Rotation (Deg)
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The next investigation involved the effect of length. Here the comparison is made for the
thick nanowires that are short and long. We see that the longer nanowire exhibits higher
torsional strength than the shorter one. It is interesting to note that the moment values of
two nanowires at failure are identical and that the rotation angle of failure for the long
nanowire is double the angle for the short wire.
Moment vs Angular Rotation for PBC (Thick)
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1800
1600
Moment (ev- oA)
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0
0
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Angular Rotation (Deg)
Long
Short
Below are plots of the evolution of potential energy as a function of angular rotation for
the two wires. Notice that there is a corresponding drop in potential energy when the
nanowire fails. However inexplicably there is also a drop before failure, one that has no
effect on the moment. It is conjectured that this may be caused a dislocation that do not
influence the moment.
Potential Energy vs Angular Rotation (Short)
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0
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120
140
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-36200
Energy (ev)
-36300
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Angle (Deg)
Potential Energy Vs Angular Rotation (Long)
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-72800
0
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-72900
Energy (ev)
-73000
-73100
-73200
-73300
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-73500
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-73700
-73800
Angle (Deg)
In addition to length, the torsional properties due to the thickness of nanowires were also
examined. For this instance, the comparison is made between short nanowires that are
thin and thick. From the two figures below it can be concluded that thinner nanowires
have higher torsional strength.
Moment vs Rotation Curve (Thin)
250
Moment (ev-0A)
200
150
100
50
0
0
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Angle (Deg)
Moment vs Rotation Curve (Thick)
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We will now shift the focus of the simulations to the mode of failure. All of the
subsequent figures were created using the post-processing application AtomEye. The
coloring is by coordinate number, a number that reflects the number of neighbors an atom
has.
The first mode we encountered was the wedge formation for the long thick
nanowire. In this mode some dislocations occur in the nanowire as shown by the red
patch. However these dislocations do not manifest themselves on the surface.
On the next rotation step, a wedge of the nanowire slips causing it to jut out from the end
as shown in the figure below. This behavior can be rationalized by thinking of the
nanowire as a bundle of strands. If the strands are strong enough then slipping is a
reasonable mode of dissipating the buildup in torsional energy.
A second mode of failure occurred when simulating short thin nanowires. In the figure
below, notice that there are no visible defects in the nanowire.
Then in the next rotation step, the nanowire gives in the center without warning. This
behavior is more consistent with our expectations since it was thought that the failure of a
nanowire would be the formation of a weak point or fracture rather than the slipping of a
wedge.
Finite Temperature Simulation:
Work has begun on simulating nanowires at nonzero temperatures. As a first case the
temperature was selected to 300K.
However there appears to be a bug in the
implementation of the periodic boundary conditions as the following nonphysical figure
as obtained from the simulation.
Conclusions & Future Work
Our simulations have provided us with a glimpse into the torsional properties of silicon
nanowires, namely torsional stiffness. We have seen that at T = 0K two modes of failure
under torsion can occur depending on the length and thickness of the nanowire. The
formation of a wedge slippage was the failure mode in the long thick case whereas a
pseudo-fracture occurred for the short thin case. Although these results are promising,
further work needs to be done to check the validity of these failures as well as to get a
better understanding of their manifestation. Another interesting simulation to run would
be to introduce a series of dislocations into the nanowire and to see if the failure of the
nanowire is related to the dislocations.
Last, and perhaps most important, further
simulations need to be completed in the finite temperature regime. The conclusions
drawn from these nonzero temperature cases will give insight into real-world
applications.
References:
1. Papdakis, S., J., et al., “Resonant Oscillators with Carbon-Nanotube Torsion
Springs.” Phys. Rev. Lett. 93, 2004.
2. da Fonseca, A., F., et al., “Mechanical Properties of Nanowires.”
http://arxiv.org/PS_cache/cond-mat/pdf/0507/0507317.pdf.
3. Kang, W., J., “Twist of hypothetical Silicon nanotubes.” Modeling Simul. Mater.
Sci. Eng. 12, 2004. p 1-12.