Homogenization and Optimization of Electrodes and Photovoltaic Cells with Periodic Nanoscopic Geometry

The work for this grant was performed at Purdue University from August 2011 to July 2014. During the first two years, the Academic Advisor was Professor James Longuski. During the final year, the Academic Advisor was Professor John Cushman. The goal of this project was to design and analyze a spacecraft propulsion system using spinning tethers and the electrodynamic force. Several previous researchers have considered spinning tethers and electrodynamic tethers. A notable spinning tether was the mechanical tether sling, shown in Figure 1, which consists of a tether spun up to high speeds using a solar-powered electric motor. This device has the advantage of high momentum transfer to the payload; however, it also has several disadvantages. The tether has a large moment of inertia, so to spin-up the tether, something else must be spun in the opposite direction to conserve angular momentum, adding complexity to the system. Additionally, when the tether boosts a payload, the orbit of the tether will decrease, again due to conservation of momentum. The mechanical tether sling needs some way to regain this orbital energy. Several designs were considered and final design required the tether sling to be mounted on an asteroid or small moon. Unfortunately, Earth has no such body, so the tether sling can only be used at places such at Mars. Electrodynamic tethers thrust against the Earth’s magnetic field and therefore have the rotational and angular momentum of the Earth as a source. Electrodyamic tethers are typically proposed as an onboard thrusting device, used only by a single spacecraft. A previous MSFC study considered a electrodynamic tether to first boost a payload to a higher orbit using its spin and then to reboost the tether using the electrodynamic capabilities. The MSFC tether rotated at a fixed rate. The electrodynamic tether sling, shown in Figure 2, combines a tether sling with an electrodynamic tether, combining the benefits of both. Like the mechanical tether sling, the electrodynamic tether sling uses a spinning tether to propel a spacecraft onto a more energetic trajectory. The electrodynamic force is then used to reboost the tether and reaquire orbital angular momentum. Additionally the electrodynamic force can be used to torque the tether and change its spin rate. Thus, the electrodynamic tether sling can recoup all of its orbital and rotational angular momentum for the magnetic field, eliminating the need for the asteroid or small moon; therefore, the electrodynamic tether sling can be used in Earth orbit. The electrodynamic tether sling operates in two modes: torqueing and thrusting. We developed a simple control law to pump rotational energy into the tether to increase the spin rate. The control law can be change the spin rate and even spin the tether up from zero angular velocity. A simulation of the tether spin-up is shown in Figure 3. We also developed a control law to increase the energy of the orbit. The magnetic field falls off rapidly with distance, so we needed a control law that would raise the apogee of the orbit while keeping the perigee in the region of (relatively) strong magnetic field. Figure 4 shows the evolution of the orbit radius with time using our elliptical spiral control law. After we developed the control laws and analyzed the dynamics, we needed to determine the cost-effectiveness of the electrodynamic tether sling. To this end, we compared the mass of the tether sling system to chemical rockets. These values, shown in Figure 5, determine the number of the launches need for the tether to break even. These values show that only 6-10 launches are required to break even. We then determined the time required for the tether to reset after each launch. Then, assuming that payloads are available as soon as the tether is reset, we can find the total time required for the tether to break even, as shown in Figure 6. We found that the tether can break even after only 5-6 years, after which the tether makes a profit for every launch. The goal of this project was to design and optimize carbon nanotube-based battery electrodes. Carbon nanotubes have radii on the order a few nanometers , while the overall scale of the electrode is the order of a few centimeters. This large range of length scales means that numerical simulations are not feasible; a grid having the size to cover the entire electrode with the grid density to accurately model the nanoscale physics would require an enormous number of grid points, making numerical simulations extremely expensive if not prohibitive. This separation of scales, however, means that techniques from porous media can be applied. For a porous medium, we model the system as possessing two distinct length scales, a large scale (the electrode) and a small scale (the nanotube). We assume that the object is periodic at the small scale. We are then able to separate the large-scale physics from the small-scale physics. The resulting macroscale equations ignore the microscale geometry; the microscale physics is absorbed into the physical constants at the macroscale. This porous media method allows us to separately numerically simulate the different length scales. We began by gathering the required equations. We modeled the electrode as a rectangular grid of vertically aligned carbon nanotubes. The spaces between the nanotubes (the “pores”) contained an electrolyte in which several ion species were dissolved. We ignored any electrolyte penetration into the nanotube. While most porous media are accurately described using three length scales, we included a third scale to capture the physics of the electric double layer at the surface of the nanotubes. The charge density in the electric double layer was described by the Poisson-Boltzmann Equation. At the micro-(pore)scale, the electrolyte flow was assumed to be Stokes flow, which is governed by a linearized version of the Navier-Stokes Equation. The Nernst-Planck Equations governed the charge concentration in the electrolyte. We invoked charge conservation to determine the electric potential of the electrolyte. After experiencing difficulty computing the potential in the electric double layer, we introduced Ohm’s Law and the electric potential in the nanotube. The project suffered numerous delays due to problems with the operation of the numerical solver. The problem was very sensitive to nonlinearities so we developed a bootstrap method to gradually introduce the nonlinearities. As stated above, we also experienced problems converging the numerical solution for the electric double layer. We had initially assumed that electric potential drop from the carbon nanotubes to the surrounding electrolyte would be equal to the total applied potential (assumed to be around 10V). We attempted to solve the electric double layer potential for with a boundary potential of 10V, the solution diverged. We attempted numerous fixes for this divergence. In the end, we realized we were overestimating the potential drop. We introduced Ohm’s Law inside the nanotubes, which would lower the potential difference between the surface of the nanotube and the electrolyte. While we were not able to obtain many results, particularly for the full electrode, we were able to solve the auxiliary problems. The nonzero components of the Cell Problem are shown in Figures 7-8. The nonzero components of the Darcy Problem are shown in Figures 9-12.