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 nano-device 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:

 

1. Selection of force field parameters for the elements included in the design
2. Selection of a monolayer
3. Evaluation of the Young's modulus of silicon at length scale of the actuator cantilever
4. Selection of the SWCNT and evaluation of its strain energy function and point of mechanical failure (buckling) in the appropriate range of curvature (for "free-end" designs)
5. Evaluation of the SWCNT crimping energy as a function of the inner opening (for "in-line" designs)
6. Evaluation of the electrostatic energy of the monolayer as a function of cantilever curvature and dimensions for various levels of charge density (pH).

 

Classical Engineering Steps:

 

1. Valve assembly and geometry optimization
2. Evaluation of the mechanical properties of the charged cantilever as a function of curvature and dimensions
3. Evaluation of the mechanical properties (strain energy and forces) acting within the assembled device as a function of monolayer charge, device geometry and curvature of the components, and determination of ranges of operation as well as equilibrium geometries.

 

Design Details:

 

Molecular Simulation Steps:

 

1. Selection of force field parameters:  As described above, the quantum mechanical properties of molecular systems can be approximated with force fields.  Force field approximations are based on the assumption that the potential energy of a molecular system can be expressed as the sum of individual energy contributions, which can be reproduced with simple mathematical expressions.  Extensive documentation is available on the accuracy and applicability of existing force fields.  The present analysis was conducted using Universal Force Field ^[17] with two modifications: (a) the silicon-silicon bond length parameter was corrected in order to match the lattice parameters of the silicon crystal, and (b) the torsion and inversion terms were taken from the Dreiding generic force field ^[18].  Tables 1 through 7 summarize the atom types, force field energy expression, and force field parameters used in the characterization of the device described here.
2. Selection of the monolayer:  Various chain lengths of the carboxylic acids (C₃, C₆, C₁₂ and C₂₀) and surface geometrical arrangements (linear, zig-zag and hexagonal) were considered on the Si(111) surface at 50% coverage.  This was necessary in order to assess the magnitude of the forces on the cantilever beam due to the interactions between the closely packed molecules of the monolayer.  The calculations were repeated for the same percentage of coverage using the Si(100) surface and the C₃ monolayer (note that the density of molecules on the Si(100) surface is lower than on on the Si(111) surface at 50% coverage).  In each case the monolayer cohesive energies were calculated through molecular simulations.  Due to the more negative cohesive energy of longer carbon chains (which would oppose the deflection of the cantilever) and due to having its carboxylic acid groups closest to the surface, an acrylic acid monolayer (C₃) was selected.  Very little strain is stored in the short hydrocarbon tail of this acid.  The choice of a monolayer that brings the carboxylic groups close to the surface also facilitates the modeling of the system by restricting the motion of these groups, approximating future designs that contain a charged metal plate instead of the organic monolayers on the surface of the cantilever.  Finally, the Si(100) surface was selected versus the Si(111) due to the complex reconstruction that the latter undergoes, which makes it difficult to work with.  The results of the cohesive energy study for the Si(111) surface are shown in figure 5.  It is not necessary to evaluate different geometries of the Si(100) surface because there is only one possible uniform arrangement at 50% coverage.

 

 

 

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).

 

3. Evaluation of the Young's modulus of silicon at the length scale of the actuator cantilever:  A series of molecular mechanics simulations of 15x3x2 nm cantilevers (LxWxH) at various levels of curvature (1/Rc, where Rc is the radius of curvature) were conducted using the Si(100) and Si(111) hydrogen terminated surfaces and their strain energy curves were constructed as a function of the curvature.  Utilizing classical elasticity approximations and simple regression techniques, the Young's modulus of the material was calculated.  Knowledge of the Young's modulus allows the construction of cantilevers of various sizes and the classical analysis of their strain energies and deflection forces as a function of curvature.   Figure 7 shows a deflected Si(100) cantilever and figure 8 shows its strain energy curve as a function 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.

 

4. Selection of the SWCNT and evaluation of its strain energy function and point of mechanical failure (buckling) in the appropriate range of curvature (for "free-end" designs):  The selection of the SWCNT size should be based on the desired fluid flow rates and the size of the molecule that is flowing through it.  Based on molecular simulations, it is known that SWCNT's are stable with a circular cross section up to a radius of approximately 3 nm ^[1].  Larger radii yield collapsed tubes, which require non-zero internal pressure to keep them open. Estimates of the strain energy for the SWCNT as a function of curvature are necessary if the discharge end of the valve is allowed to move, as in the "free-end" design shown in figure 2.  It is also necessary to determine the value of the curvature at which the SWCNT buckles and completely interrupts the flow through it.  For this particular design a 17,17 SWCNT was selected, the radius of which is within the region where SWCNT's are stable with round cross sections (radius = 1.15 nm).  The molecular simulations showed that when bent, the 17,17 SWCNT deflects uniformly up to a curvature of approximately 0.0027/Ang. after which it buckles.  The classical Young's modulus calculation was performed by treating the SWCNT as a hollow pipe with a wall thickness equal to the van der Waals radius of a carbon atom.  Figure 9 shows a 17,17 SWCNT at the point of buckling and figure 10 shows the strain energy results as a function of curvature, including the point of buckling.

 

 

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.

 

5. Evaluation of the SWCNT crimping energy as a function of the inner opening (for "in-line" designs):  The energy of the SWCNT was determined for different levels of "crimping", from completely open to completely closed.  The size of the "crimped" section is comparable to the contact area the tube would have when the cantilever is fully deflected.   The energy curves as a function of the internal opening and cantilever curvature for the 17,17 SWCNT are summarized in figures 12 and 13.  Figure 11 shows a partial view of the model used to determine the crimping curve of the device.  It consists of a section of 17,17 SWCNT plus the tip of a Si(100) cantilever.  Note that the strain energy curve also includes the deformation experienced by the tip of the cantilever.

 

 

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 short-range 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 y-axis 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 x-axis 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.

 

 

6. Evaluation of the electrostatic energy of the monolayer as a function of cantilever curvature:  The last and largest energy contribution to the operation of the device is the electrostatic energy.  The electrostatic energy was modeled by placing charged atoms at the positions where the functional groups of the monolayer would be located for the various levels of curvature of the Si(100) cantilever, which was selected to construct the device.  Each carboxylic acid group was assigned one electron charge, which corresponds to complete deprotonation (at this stage we have not considered solvent effects, although we have analyzed partial deprotonation states).  The result of this analysis is a function that relates the electrostatic energy of the cantilever (with the assumptions listed above) to its curvature.  At this point one fundamental question arises: What is the net charge on each molecule of the monolayer and what is the effect of the surrounding environment on the electrostatic interactions between all molecular pairs?  Although this topic is not treated in this paper, the present analysis allows for the introduction of correction factors to the total electrostatic energy if additional knowledge regarding the total charge of the system or the effect of the solvent is available.    Charge scaling effects are proportional to q², where q is the net charge on each carboxylic acid group, and the correction for solvent effects can be introduced in the form of a dielectric constant.  This assumes that all carboxylic acid groups on the monolayer have the same charge, hence all energy terms considered in the calculation of the electrostatic energy are of the form Kq²/r, where K is a constant that includes the dielectric constant of the surrounding environment and r is the distance between pairs of charged molecules.  Figure 15 shows the electrostatic energy as a function of curvature for a 22.5x6x2 (LxWxH) cantilever completely deprotonated and without solvent effects.

 

 

Figure 14:  Incremental energy of the "in-line" 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:

 

1. Valve assembly, geometry and different attachment systems:  Different ways of assembling the valve were evaluated for chemical feasibility and ease of assembly.  Several of them included the attachment of the silicon cantilever to the SWCNT through SWCNT junctions such as the ones shown in figure 16, however the chemical feasibility of these design is quite low, and there are no manufacturing procedures that guarantee that such devices could be produced efficiently and with high yields with current chemical synthesis methods.  The chosen design consists of a perforated block of silicon through which the SWCNT is inserted. The insertion of the SWCNT through the block of silicon can be a difficult operation in itself depending on the size of the opening, but there is no need to make that opening as small as the SWCNT, as long as the relative position of the tube with respect to the cantilever is accurately fixed.  Alternatively, the SWCNT could be generated in situ through placement of catalytic metal particles inside the silicon cavity prior to etching the silicon block to its final dimensions.

 

 

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.

 

2. Evaluation of the mechanical properties of a charged cantilever as a function of cantilever curvature and dimensions:  The quantities of interest regarding the performance of the device are the energy and forces that the cantilever is capable of exerting at different levels of curvature within its operational range of deflection.  A high-performance cantilever will be one whose potential energy is high in the undeflected state, with a steep gradient towards high curvatures (in the charged state).  Thus, the cantilever is capable of exerting a large force in the direction of deflection.  The total energy of the cantilever was calculated as the sum of its strain energy and the electrostatic energy of the charges distributed on it, both of which had been previously calculated as a function of curvature using molecular simulations.  Clearly as the curvature of the cantilever increases the strain energy of the material increases while the electrostatic energy decreases until the point of equilibrium where these balance each other.  The force is the derivative of the total energy of the cantilever as a function of the displacement along the path of motion.  The 15x3x2 Si(100) cantilever shown in figure 7, for example, would be capable of a maximum downward force of approximately 7.5 nN if the monolayer of acrylic acid on its top surface (50% coverage) were completely deprotonated in vacuum.  This force would decrease to approximately half of that value if the tip were allowed to deflect 5 nm vertically (a third of the length of the cantilever).   The forces the same cantilever would be able to exert in a solution environment would of course be lower than that according to the dissociation properties of the monolayer and the pH of the solution.  It is important to note that the strain energies of the cantilevers must be calculated by stretching the top (charged) surface of the cantilever while the bottom surface remains at its original length, which is the effect that a compressive stress on the top surface would cause.  This mode of deflection is different than the deflection of a cantilever through the application of a force to its tip.  In the latter mode, the top surface stretches and the bottom surface compresses by similar amounts, while the plane of zero deformation runs through the middle of the cantilever (between the top and the bottom surfaces).  The former mode of deflection corresponds to strain energies four times greater than the latter mode according to ideal elasticity calculations. 
3. Evaluation of the mechanical properties of the assembled valve as a function of monolayer charge and curvature of the components:  As in the previous case, the total energy of the system can be found by adding the partial energy contributions of the components.  The total energy of the system is the sum of the total energy of the cantilever (charge and strain as previously calculated) plus the deformation (strain) energy of the SWCNT, whether due to deflection or crimping.  The strain energy of the SWCNT needs to be supplied by the charged cantilever.  In order to have a working system it is only required that the electrostatic energy available for deflection exceeds the strain energy of the deflected components at the closed position of the valve.  Figure 17 shows the partial energy contributions and the total energy of a system that uses a 15x3x2 nm cantilever (LxWxH), which deflects a "free-end" 17,17 SWCNT.  It was assumed that the available electrostatic energy is only 10% of the maximum corresponding to total deprotonation of the system in vacuum (this is equivalent to assuming that there is a net charge of -0.32e on each carboxylic acid group).  The total energy of the system shows a minimum at a curvature of 0.00085/Ang., which is not enough to deflect the SWCNT to the point of buckling (curvature of 0.0027/Ang.).  Thus this system is not capable of interrupting the flow through the SWCNT.  Using the methodology described above, the system was redesigned to include a cantilever that is twice as wide as the one used previously and 50% longer.  The redesign includes re-calculating the electrostatic and strain energies of the new (larger) cantilever.  The performance of the new system is shown in figure 18 (again, considering that only 10% of the maximum electrostatic energy is available).  In this case, the total energy of the system doesn't exhibit a minimum in the range of curvature evaluated and the cantilever is capable of deflecting the SWCNT past the point of buckling.  Figure 19 shows the same analysis for an "in-line" design.  This requires the use of the SWCNT "crimping" energy curve instead of the deflection energy curve.  As in the previous case, the 15x3x2 nm cantilever (LxWxH) was not capable of interrupting the flow through the SWCNT and the system had to be redesigned with a larger cantilever.   In this case however, the working design required a 45x9x2 nm cantilever (LxWxH), which has three times the monolayer surface area as the cantilever used in the "free-end" design.   Note that the length of the silicon block holding the cantilever needs to be added to the length of the cantilever to obtain the total length of the system.  If the length of this block is set at 10 nm, for example, then a feasible "free-end" design would be 32.5 nm long and a feasible "in-line" design would be 55 nm long.

 

 

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 "free-end" 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 "free-end" 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 "in-line" 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 "in-line" 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]

 

 

 

Table 2:  Force field atom types

 

 

 

Table 3:  Force field bond stretch parameters

 

 

 

Table 4:  Force field angle bend parameters

 

 

 

 

Table 5:  Force field torsion parameters

 

 

 

Table 6:  Force field inversion parameters

 

 

 

Table 7:  Force field diagonal van der Waals parameters and combination rules