Presentation:
- Video
- Slides (PDF)
- Slides (PPTX) with speaker notes
- GIF for the drive wall animation
This is a technical document containing lab notes and insights from building rod logic. Although an exploratory engineering work, its primary purpose is teaching the reader how to do things. How to reproduce the general design/analysis methods in similar projects.
This document is a computational and theoretical work, covering how to model and simulate structures. The content makes the design of large computer systems tractable. A related task, is then implementing the designed computer systems in experiment. Experimental validation is beyond the scope of the author's research and capabilities. However, it is hoped that this knowledge can benefit an experimental effort. Rod logic could be a more compact/efficient alternative to nanoelectronics, suitable for automating nanofactories during a bootstrapping effort.
The data and code are permissively licensed. This makes it an open alternative to Molecular Link Logic, which is patented by CBN Nano Technologies. In addition, rod logic is more compact and potentially more efficient. It does not require rotary mechanisms, making it more feasible to manufacture with primitive/limited mechanosynthesis capabilities. The only downside may be sliding friction, but the Landauer limit is a more pressing concern at proposed operation speeds. Note that with electronic transistors, the energy dissipated (~1 eV) is so large that the Landauer limit is not even considered.
Rod logic is made of diamond, crystalline group (IV) carbides, or crystalline elemental silicon. Although one could build related mechanisms out of DNA origami, that would likely have inferior efficiency. Potentially, exactly equal to modern "3 nm" transistors. It would be equivalent to 3D printing macroscale models of the mechanical machinery.
In the context of rod logic, "dopants" are lattice impurities inserted to improve the mechanical properties. This is different from dopants in semiconductors, which improve the electronic properties. For example, a phosphorus dopant would harm the mechanical properties of a rigid elemental silicon lattice. The unpaired valence electron would make the nanomachine more chemically reactive. A mechanical machine that has formed an unwanted chemical bond may not function at all.
The housing is made of cubic diamond with orthogonal (100) surfaces. The rods are made of lonsdaleite. The lattice constant of the rods misaligns with the housing, providing more flexibility to fine-tune the vdW potential. However, knobs must be placed irregularly. Their separations aren't evenly divisible by the cubic diamond lattice constant.
The structure was energy-minimized to allow (100) surfaces to render with the reconstructed geometry. The structure remained stable after 48 ps of MD simulation, with temperature from leftover vdW energy that the minimizer didn't eliminate (last image). In a previous simulation without the logic rods, the structure had fallen apart.
Here is a prototype of a manufacturable flywheel. It mixed three different materials into one covalent solid - diamond, moissanite, silicon. They span a wide range of lattice constants, causing immense lattice strain. This is why the part has such a small radius of curvature. In addition, the silicon protrusion had a few intentional carbon dopants (first image). It was originally warped and pointing to the right. The dopants created a counter-strain pulling it to the left. With some careful tuning, the net strain was approximately zero.
The flywheel was originally tested without protrusions, using rigid body dynamics. It was discovered that the flywheel remained intact at 10 GHz, but not 20 GHz or higher. The protrusions were added to hopefully increase the vdW binding energy. This change would keep the system intact at higher speeds. When the part was minimized in isolation, the protrusions were slightly skewed (second image). This shape was different than the energy minimum when interacting with the other parts. The simulation was switched to the slower molecular dynamics, to let the parts deform into the vdW minimum.
The revised simulation could have used rigid body dynamics, but the setup would be more complex.
The thermostat went as follows. Give each atom a velocity that matches 10 GHz overall rotation, but ensure nearby atoms have about the same velocity. The relative velocity between nearby atoms is the thermal component of the velocity. Therefore, the initial temperature is ~0 K. Let the system evolve over time, periodically resetting the thermal component to zero. The simulation followed this timeline:
initialize with 0 GHz bulk angular velocity
simulate descent into vdW minimum (2.4 ps)
erase thermal velocities
simulate descent into vdW minimum (0.4 ps)
erase thermal velocities
simulate descent into vdW minimum (0.4 ps)
erase thermal velocities
initialize with 10 GHz bulk angular velocity
simulate rotation (36.4 ps)
The third image was taken at the end of the simulation. It had rotated 36.4% of an entire revolution. Since the flywheel has threefold symmetry, a 1/3 rotation looks similar to the starting position.
The part was stable at 10 GHz, but still fell apart at 20 GHz. This shows that vdW attraction cannot hold the flywheel together at high speeds. One needs geometric constraints / overlap repulsion to force the parts together. The operation speed equated to ~250 m/s at the curved segment and ~500 m/s at the tip of the protrusion. These speeds are not satisfactory. The ideal flywheel would have most of its mass moving at ~1 km/s, but the same atom count as this prototype.
As part of an effort to reduce atom count, I considered replacing some bridgehead carbons with phosphorus. This would reduce the number of hydrogen atoms, which are a large portion of the total atom count. The surface area of these parts is quite large. Hydrogens cover the surface area. After some preliminary sketching based on MM4 bond lengths and vdW radii, I realized phosphorus wouldn't improve compactness much. However, nitrogen would be useful. Nitrogen wasn't currently supported by MM4, but the parameters could be added if necessary.
I predicted that N-terminating the logic rod could shrink each housing unit. The size reduction would be 1 lattice cell in two spatial directions. After putting the idea into atomic detail, I realized 0.5 lattice cells reduction was more realistic. I tested both the original C-H termination and the enhanced N-termination in GFN-FF, to check that they were roughly stable. The test system had 1400 atoms and took an entire minute to minimize. I then reduced the size to 800 atoms (~half). The system now took 1/4 as long to minimize because of GFN-FF's
The final test used GFN2-xTB to maximize accuracy and confidence that this was stable. GFN2-xTB has
| GFN2-xTB | GFN-FF | |
|---|---|---|
| Atoms | 196 | 832 |
| Bound State |  |
 |
| Unbound State |  |
 |
The N-terminated rod was kinetically stable inside the housing. I trust this result, as it was both predicted to be stable beforehand and corroborated by an accurate quantum mechanical method (GFN2-xTB). I also measured the energy difference between the rod being inside vs detached from the housing. If the latter is favored, the housing will reject the rod. The device will not function correctly because the rod is ejected from the housing.
Singlepoint calculation of whole system with low-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -162.041974221135 Eh ::
:: gradient norm 0.993214593789 Eh/a0 ::
Singlepoint calculation of inner region with low-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -37.608739606057 Eh ::
:: gradient norm 0.929836258917 Eh/a0 ::
Singlepoint calculation of inner region with high-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -339.999156565982 Eh ::
:: gradient norm 1.002894360532 Eh/a0 ::
:: HOMO-LUMO gap 5.359252848273 eV ::
-------------------------------------------------
| TOTAL ENERGY -464.432391181059 Eh |
| GRADIENT NORM 0.008500429453 Eh/α |
-------------------------------------------------
Bound state: -162.04 + 37.61 - 340.00 = -464.43 Ha
Singlepoint calculation of whole system with low-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -161.915062344301 Eh ::
:: gradient norm 0.998496036159 Eh/a0 ::
Singlepoint calculation of inner region with low-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -37.543272576790 Eh ::
:: gradient norm 0.930199287435 Eh/a0 ::
Singlepoint calculation of inner region with high-level method
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: SUMMARY ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::
:: total energy -339.990065593082 Eh ::
:: gradient norm 0.997945831572 Eh/a0 ::
:: HOMO-LUMO gap 6.417806487753 eV ::
-------------------------------------------------
| TOTAL ENERGY -464.361855360594 Eh |
| GRADIENT NORM 0.007629734036 Eh/α |
-------------------------------------------------
Unbound state: -161.92 + 37.54 - 339.99 = -464.36 Ha
The unbound state is higher in energy by 0.07 Hartree. However, the energies for the GFN2-xTB system are about the same in each case. The energy difference might be an artifact of GFN-FF's inaccuracy. To test this hypothesis, observe the energy from GFN2-xTB alone (~340 Ha). It increases by 0.009 Ha upon entering the unbound state. The energy difference is still tens of zeptojoules. This seems significant enough for the problem in context.
| Measurement | Atomic Units | SI Units |
|---|---|---|
| ONIOM | +0.071 Ha | +308 zJ |
| GFN2-xTB | +0.009 Ha | +37 zJ |
In conclusion, N termination has a high chance of succeeding in a real-world engineering effort. Whether I want to add parameters to simulate nitrogen, is another matter. It only decreases atom count by a marginal amount (single digit percent). There are more economical ways to get even greater size reductions. However, it could reduce sliding friction.
Prototype housing structure for the drive system. Dimensions 50 nm x 120 nm x 4 nm. 600,000 atoms, silicon carbide.
This experiment tested a mechanism for actuating logic rods without rotary motion. It uses van der Waals and Pauli repulsion forces to move rods between two possible states. SiC was used in the actuation mechanism because of its higher Hamaker constant. Rigid body dynamics was used and the system consisted of 35,000 atoms. A 64-picosecond simulation took 26.3 seconds to compute.
Some interesting things happened while simulating. First, the cutoff radius for vdW forces had to be changed. The default value is 1 nm to maximize simulation efficiency. However, that caused qualitative differences in behavior compared to larger cutoffs. 2 nm was used, providing a good tradeoff between compute cost and accuracy.
A FIRE energy minimizer was written from scratch, incorporating both FIRE 2.0 and ABC corrections. The minimizer had to be written because RBD lacks a built-in minimizer in MM4. MD is able to utilize L-BFGS from OpenMM. The parameters from the LAMMPS papers performed better than those in the INQ implementation. The overall source file was 235 lines, including data structure declarations and documentation comments.
NOTE: The code for the energy minimizer is qualitatively wrong. Rigid body dynamics requires 6x6 matrices for accurate motion. The code neglected some off-diagonal terms in this 6x6 matrix, treating it as two uncoupled 3x3 matrices.
Energy minimization is generally used to improve a scene prior to dynamical simulation. The next part of the code was the rigid body dynamics simulation. RBD simulators are often numerically unstable with a fixed timestep. The energy explodes quickly, sending parts into nonphysical overlapping positions and exorbitant velocities. One method to fix this is varying the timestep. Give a certain tolerance for energy drift. When the tolerance is exceeded, throw away the computations from the last timestep. Repeat that timestep with half the time interval. If it still fails, retry with an even shorter timestep.
A more convenient alternative is keeping the timestep fixed, but damping the kinetic energy. The damping emulates thermalization of kinetic energy from friction, which happens automatically in MD simulations. This experiment decreased the absolute value of each rod's momentum by 0.98x every 40 femtoseconds. It counteracted energy drift and kept the simulation numerically stable. However, the value of damping affects the simulated gate switching time. With 0.98x / 40 fs, the rods could switch in 23 ps. With 0.95x / 40 fs, the switching time had to rise to ~46 ps. Otherwise, the device would fail to function correctly.
Here is a rough animation of the clock cycle. Since we are doing reversible computing, there are two major phases. The forward phase (first three images; 0 - 23 ps) and the reverse phase (next two images; 23 - 46 ps). The final image (64 ps) shows what happens if the SiC drive wall moves beyond the designed range of motion. The drive wall's speed is 100 m/s during each phase.
To transmit signals in all 3 dimensions, a lonsdaleite rod must permit knobs on the +Y and +Z sides*. The +Y sides are decorated frequently in previous experiments. Y-oriented knobs always form six-membered rings.
*X is defined as the axis where the rod has the longest spatial dimension.
Z-oriented knobs pose an issue. They must contain 5-membered rings. This introduces unwanted warping in the +Z direction. To make the computer easier to manufacture (when inserting rods into shafts), or potentially for other reasons, one ought to avoid such warping.
Silicon Dopant Site (Front View):
Silicon Dopant Site (Top View):
One should be careful to not overshoot, and cause strain in the exact opposite direction to the initial warping. This would be the case if Ge was chosen as the dopant. It causes unacceptable strain in the -Z direction.
In the images below, a "carbon" dopant is equivalent to having no dopant at all. The intrinsic lattice is made of carbon atoms. The "carbon" dopant reveals the structure without proper doping to equalize strain.
Dopant Choices (Top View):
Dopant Choices (Side View):
| C | Si | Ge |
|---|---|---|
 |
 |
 |
When fixing strain in one dimension, it is easy to accidentally cause strain in a different dimension. One could even cause twisting along an axis that should be perfectly rectangular. There are many cases where all major modes can be suppressed simultaneously, to within acceptable design margins. Designing the dopants requires both energy minimization and molecular dynamics. MD can reveal ambient vibrational modes along a warping direction, which were missed by energy minimization.
Alternating Vertical Placement (Front View):
YouTube video: https://www.youtube.com/watch?v=xSOS9CYfYX8
