Design Methodology:
Design Overview:
Classical engineering design makes use of bulk material
properties, such as Young's modulus, heat capacity, heat conductivity,
diffusion coefficients, etc. These
properties have more fundamental energy and force components at the atomistic
level, such as electrostatic and van der Waals interactions, as well as
covalent intramolecular interactions.
These properties are embedded in classical force fields, mathematical
expressions for the energy and forces within a molecular system given the
charges and positions of all atoms. In
turn, the parameters in the force field depend on the nuclear and electronic
structure of the molecules, which can be estimated using Quantum Mechanics ^{[17,18]}.
In order to have an accurate description of a nanodevice
it is desirable to include as much molecular detail as possible in the analysis
of its behavior. However, for the
reasons described above it is not practical to conduct all design steps at the
molecular level. Hence, we present a
hybrid methodology that combines molecular simulations with classical
engineering. The molecular simulations
provide the elastic and/or electrostatic properties of each component of the
system considered individually, estimated from the force field, while the
classical analysis provides the behavior of the assembled system based on those
properties. In summary the steps
followed in the present design are the following:
Molecular Simulation Steps:
Classical Engineering Steps:
Design Details:
Molecular Simulation Steps:
Figure
5: Analysis of monolayer cohesive energy (normalized
to the number of adsorbed molecules) for three different geometrical
arrangements of four organic acids (acrylic, hexanoic, dodecanoic and
eicosanoic) on a Si(111) surface at 50% coverage. The results show increasing monolayer stability (more negative
cohesive energy values) with increasing carbon chain length. As shown on the graph, the linear
arrangement was more stable at shorter chain lengths, while the hexagonal
arrangement was more stable for longer chain lengths. The three geometrical arrangements considered for the Si(111)
surface are shown in figure 6.
Figure
6: Three
geometrical arrangements considered in the determination of monolayer cohesive
energy on the Si(111) surface at 50% coverage.
Figure
7: 15x3x2 nm silicon cantilever (LxWxH). The top surface is the Si(100). The strain energy at different levels of
deflection was obtained by fixing the atoms on one end of the cantilever,
displacing those on the right end to their final position and allowing the
system to relax. Figure 8 shows
the strain energy curve for this system as a function of curvature (1/Rc, where
Rc is the radius of curvature).
Figure
8: Strain energy as a function of curvature
(1/Rc) for the cantilever shown in figure 7. This curve has been corrected to represent a system where the
plane of zero deformation is at the bottom face of the cantilever. The average Young's modulus for silicon
calculated from the above strain energy curve is 76.7 GPa versus the
experimental bulk value of 47 GPa.
Figure
9: 17,17 carbon nanotube at a curvature of
0.0031/Ang (beyond the point of buckling).
Figure 10: Strain energy as a function of curvature (1/Rc) for the nanotube shown in figure 9. The graph shows two distinct strain energy functions below and above the point of buckling. The average calculated value of the Young's modulus was 1719 Gpa. The experimental value from vibrational frequencies measurement is 1250 Gpa ^{[10]} and the experimental value from tensile molecular simulations is 640 – 673 GPa ^{[1]}.
Figure
11: Partial view of the model used to construct
the crimping energy curve, showing the tip of the cantilever, part of the 17,17
nanotube and part of the Si(100) surface on which the system rests.
Figure
12: Strain energy as a function of the internal
opening of the carbon nanotube for the model shown in figure 11. Note the sharp change in the slope of the
curve at approximately 1.81 Angstroms of internal opening. The amount of energy required to crimp the
nanotube below this value increases exponentially according to the force field
shortrange van der Waals functions ^{[17]}. For the classical analysis of the present device it was assumed
that no fluid flows through the nanotube below this value. As shown in figure 14, this is a
reasonable assumption based on the energy required for a molecule to flow from
one side of the crimped section to the other.
Figure
13: Strain
energy (due to crimping of the nanotube) Vs. curvature of the cantilever for
the model shown in figure 11.
The yaxis values used to construct this graph correspond to the energy
values of the graph in figure 12 (crimping energy as a function of
internal opening). The xaxis value for
each point corresponds to the curvature that the cantilever would have to have
in order to crimp the nanotube up to that point. The curvature of the
cantilever was determined through geometrical calculations based on its
displacement towards the nanotube and assuming that it has a uniform curvature
throughout its length.
Figure
14: Incremental energy of the "inline" valve
system due to the presence of a single molecule moving through the
crimped section (throat) of the 17,17 nanotube when the minimum opening (skin
to skin) is 1.81 Angstrom. The results
show that the incremental energy of the system due to the presence of a single
molecule at the valve throat is above the total crimping energy for the system
(see figures 12 and 13), hence it is reasonable to consider this
as the closed position of the valve.
Note: the zero energy of the graph corresponds to the lowest energy of
each molecule during the trajectory analyzed.
Figure
15: Electrostatic energy as a function of cantilever
curvature for a 22.5x6x2 (LxWxH) cantilever completely deprotonated without
including solvent effects.
The electrostatic energy calculations in this analysis,
include all the net charges on the surface.
No cutoffs or spline functions were used for the reasons discussed
above.
Classical Engineering Steps:
Figure
16: Left
and center, nanotube junctions considered for the initial design. Right, chosen method of assembly (front
view), whereby the silicon block is perforated and the nanotube is inserted
through the opening. Note that the
perforation does not need to be as small as shown on this illustration. In fact, it may be much larger than the tube
to facilitate the assembly process, as long as the relative position of the
silicon assembly and the tube are fixed in the final design.
Figure
17: Performance chart of a cantilever valve,
showing the partial energy contributions and the total energy of a system with
a 15x3x2 nm cantilever (LxWxH). This
corresponds to a "freeend" design with a 17,17 SWCNT. It is assumed that the available
electrostatic energy is only 10% of the maximum (i.e. each carboxylic acid
group has a charge of approximately 0.32 e).
As shown, the system is not able to reach the point of buckling of the
SWCNT.
Figure
18: Performance chart of a cantilever valve,
showing the partial energy contributions and the total energy of a system for a
22.5x6x2 nm cantilever (LxWxH). This
corresponds to a "freeend" design with a 17,17 SWCNT. It is assumed that the available
electrostatic energy is only 10% of the maximum (i.e. each carboxylic acid
group has a charge of approximately 0.32 e).
As shown, the system is capable of deflecting the SWCNT beyond its point
of buckling.
Figure
19: Performance chart of a cantilever valve,
showing the partial energy contributions and the total energy of a system for a
45x9x2 nm cantilever (LxWxH). This
corresponds to an "inline" design with a 17,17 SWCNT. It is assumed that the available
electrostatic energy is only 10% of the maximum (i.e. each carboxylic acid
group has a charge of approximately 0.32 e).
As shown, the system is capable of crimping the SWCNT to the point that
interrupts the flow through it. Note
that the curvature of the cantilever of an "inline" valve will always be
limited by the surface against which the SWCNT is crimped. As shown in figure 12, there is a
disproportionate increase in the strain energy of the system if it the internal
opening of the SWCNT is reduced beyond the "closed" position, in this case 1.81
Angstroms. This is because the motion of the cantilever beyond this point
starts causing significant deformation of the surface below the SWCNT.
Table 1: Force
field energy expression ^{[17,18]}
Total Energy 
E = E _{bond
stretch} + E _{angle bend} + E _{torsion} + E _{inversion} + E _{van
der Waals} + E _{electrostatic} 
Bond Stretch Energy (Universal) 
_{} 
Angle Bend Energy (Universal) 
_{} 
Torsion Energy (Dreiding) 
_{} 
Inversion Energy (Dreiding) 
_{} 
Van der Waals Energy (Universal) 
_{} 
Electrostatic Energy (Classical)* 
_{} 
*Energies in kcal/mol, charges in electronic units and
distances in Angstroms 
Table 2: Force
field atom types
H_ 
Hydrogen 
H___A 
Acid Hydrogen 
C_3 
Tetrahedral carbon (sp^{3}) 
C_R 
Resonant carbon 
C_2 
Planar carbon (nonresonant sp^{2}) 
O_3 
Tetrahedral oxygen 
O_R 
Resonant oxygen 
O_2 
Planar oxygen (nonresonant sp^{2}) 
Si3 
Tetrahedral silicon 
Table 3: Force
field bond stretch parameters
Atom 1

Atom 2 
K_{r} 
R_{o} 
C_3 
H_ 
662.9963 
0.7080 
C_3 
C_3 
699.5920 
1.5140 
C_R 
H_ 
715.3873 
1.0814 
C_R 
C_3 
739.8881 
1.4860 
C_R 
C_R 
925.3104 
1.3793 
C_2 
H_ 
709.4702 
1.0844 
C_2 
C_3 
735.4249 
1.4890 
O_3 
H_ 
1120.7078 
0.9903 
O_3 
C_3 
1078.4241 
1.3938 
O_R 
H_ 
1049.6934 
1.0121 
O_R 
C_2 
1085.0881 
1.391 
O_2 
C_2 
1610.4076 
1.2195 
O_2 
C_R 
1153.3079 
1.3630 
O_3 
H___A 
500.0000 
1.0000 
Si3 
H_ 
345.6964 
1.4930 
Si3 
C_3 
453.3563 
1.8669 
Si3 
Si3 
321.4845 
2.3650 
Table 4: Force
field angle bend parameters
Atom 1

Atom 2 
Atom 3 
_{} 
_{} 
H_ 
C_3 
H_ 
75.2779 
109.4710 
C_3 
C_3 
H_ 
117.2321 
109.4710 
C_3 
C_3 
C_3 
214.2065 
109.4710 
C_2 
C_3 
H_ 
121.1966 
109.4710 
C_2 
C_3 
C_3 
219.5725 
109.4710 
O_3 
C_3 
H_ 
160.9632 
109.4710 
O_3 
C_3 
C_3 
284.0680 
109.4710 
Si3 
C_3 
H_ 
89.6088 
109.4710 
Si3 
C_3 
C_3 
181.9182 
109.4710 
C_R 
C_R 
H_ 
103.1658 
120.0000 
C_R 
C_R 
C_R 
188.4421 
120.0000 
C_R 
C_3 
H_ 
121.6821 
109.4710 
C_R 
C_3 
C_3 
220.2246 
109.4710 
C_3 
C_2 
H_ 
98.7841 
120.0000 
C_3 
C_R 
O_2 
242.4495 
120.0000 
O_R 
C_2 
C_3 
229.9906 
120.0000 
O_2 
C_2 
H_ 
139.6784 
120.0000 
O_2 
C_2 
C_3 
240.9266 
120.0000 
O_2 
C_2 
O_R 
315.2170 
120.0000 
O_2 
C_R 
O_2 
333.7212 
120.0000 
C_3 
O_3 
H_ 
165.6001 
104.5100 
H_ 
O_3 
H_ 
113.0577 
104.5100 
C_2 
O_R 
H_ 
142.0707 
110.3000 
H_ 
Si3 
H_ 
32.4318 
109.4710 
C_3 
Si3 
H_ 
57.6239 
109.4710 
Si3 
Si3 
H_ 
48.9079 
109.4710 
Si3 
Si3 
C_3 
102.7429 
109.4710 
Si3 
Si3 
Si3 
98.4346 
109.4710 
H___A 
O_3 
H___A 
120.0000 
109.4710 
Table 5: Force field
torsion parameters
Atom 1

Atom 2
(center1) 
Atom 3
(center 2) 
Atom 4 
V 
n 
d 
Any 
C_3 
C_3 
Any 
2.0000 
3 
1 
Any 
C_R 
C_R 
Any 
25.0000 
2 
1 
Any 
C_2 
C_3 
Any 
2.0000 
3 
1 
Any 
O_3 
C_3 
Any 
2.0000 
3 
1 
Any 
O_R 
C_2 
Any 
25.0000 
2 
1 
Any 
Si3 
C_3 
Any 
2.0000 
3 
1 
Any 
Si3 
Si3 
Any 
2.0000 
3 
1 
Any 
C_R 
C_3 
Any 
2.0000 
3 
1 
Table 6: Force
field inversion parameters
Atom 1
(center) 
Atom 2 
Atom 3 
Atom 4 
_{} 
_{} 
C_R 
Any 
Any 
Any 
6.0000 
0.0000 
C_2 
O_2 
Any 
Any 
50.0000 
0.0000 
Table 7: Force
field diagonal van der Waals parameters and combination rules
Atom

R_{o} 
D_{o} 
H_ 
2.8859 
0.043999 
H___A 
0.8999 
0.009999 
C_3 
3.8510 
0.104999 
C_R 
3.8510 
0.104999 
C_2 
3.8510 
0.104999 
O_3 
3.5000 
0.059999 
O_R 
3.5000 
0.059999 
O_2 
3.5000 
0.059999 
Si3 
4.2950 
0.402000 
Spline
function (3^{rd} degree polynomial):
cuton 11.0000 A Cutoff 14.0000 A 

Combination
rule for parameters of different atoms: 