Theoretical demonstration of a capacitive rotor for generation of alternating current from mechanical motion

Innovative concepts and materials are enabling energy harvesters for slower motion, particularly for personal wearables or portable small-scale applications, hence contributing to a future sustainable economy. Here we propose a principle for a capacitive rotor device and analyze its operation. This device is based on a rotor containing many capacitors in parallel. The rotation of the rotor causes periodic capacitance changes and, when connected to a reservoir-of-charge capacitor, induces alternating current. The properties of this device depend on the lubricating liquid situated between the capacitor’s electrodes, be it a highly polar liquid, organic electrolyte, or ionic liquid – we consider all these scenarios. An advantage of the capacitive rotor is its scalability. Such a lightweight device, weighing tens of grams, can be implemented in a shoe sole, generating a significant power output of the order of Watts. Scaled up, such systems can be used in portable wind or water turbines.

(1) The electric potential everywhere is then given by, The boundary conditions are: • Potential at = 0: • Continuity of potential at = : • Continuity of displacement field at = : • Potential at = + : In addition to the boundary conditions, the system is overall electroneutral, i.e., where, which gives the relation: By solving Supplementary Equations (6) - (9) and (12) for the parameter , we get: The capacitance per unit surface is given by:

Supplementary Note 2: Molecular Simulations
We performed long canonical (NVT) simulations, 60ns, of the liquids to compute dielectric constants, and NPT simulations to calculate bulk densities. We employed the v-rescale 1 thermostat to set the system temperature and the Berendsen barostat 2 for the constant pressure simulations. The slab simulations, as indicated in the main text, involved typically 40 ns trajectories. We employed the 3D correction 3 to compute the electrostatic interactions using the Ewald summation method, and the Lennard-Jones interactions were truncated at 1 nm using a spherical cut-off. The cross interactions were computed using geometric combination rules.
The system was surrounded by a large vacuum gap in the z direction, to prevent image-image interactions. The total length of the simulation in this direction, Lz, was 3Ls, at least. The simulations of the slab systems consisted of ~500 PC or ~1000 formamide molecules. The gold slabs consisted of 588 atoms. 196 atoms in the layer next to the PC liquid were assigned electrostatic charges to perform the capacitance computations. The position of the gold slabs was restrained to prevent drift and maintain the desired level of confinement. All the trajectories were integrated with the Leap Frog algorithm, a timestep of 2fs, and the code GROMACS 2020.
The initial geometry for PC was obtained from ATB following optimization with B3LYP/6-31G*, with the initial charges estimated using the ESP method of Merz-Kollman 4,5 Intermolecular and intramolecular parameters with parameters were taken initially from the GROMOS 54A7 forcefield. PC is a chiral molecule. All the simulations presented here were performed with the S-enantiomer. We performed control simulations using the R enantiomer obtaining the same results for density and dielectric constant, within the statistical uncertainty of our computations.
The initial forcefield for PC over-predicted the density of the liquid at 300 K, 1295.

Simulations of a capacitor with Room Temperature Ionic Liquids
The capacitances were obtained by analyzing trajectories spanning 16 ns, following the same procedure discussed above. The films were prepared to ensure the density in the middle of the film agreed with the bulk density of the model 8 , and they contained typically 2400 ion pairs, with a wall to wall distance of the order of 13 nm. We calculated the potential drop across the interfaces for surface charge in the range 0-3 μF cm 2 . We found that the voltage surface charge dependence follows a linear dependence, similar to what is observed in the molecular fluids or reported in previous simulations of ionic liquids 9 .
In Supplementary Figure 2, we show the electrostatic potential of the full system, including the substrate and RTIL contributions. Figure 2. The simulated electrostatic potential profile for ionic liquid, including the substrate and RTIL contributions.

Supplementary Note 3: Derivation of Drag Torque Acting on the Discs
Consider a disk of radius rotating about the z-axis with angular frequency . Situated parallel to that disk, at a distance ℎ, is an immobile disk of the same radius . For ≫ ℎ, and after taking advantage of the azimuthal symmetry, we get the Von-Karman differential equations on the velocity components r , φ , z in cylindrical coordinates: Where r , φ and z are functions of only, and = ( , ).
Taking a linear approximation and dropping all non-linear velocity terms: The first three differential equations, (20)-(22), present a coupling between r , z and . The fourth equation is uncoupled, and its solution is given by: Where and are constants to be determined by the boundary conditions of the problem.
Applying non-slip conditions for the rotating disk at = 0 and for the immobile disk is at = ℎ, we get: If we introduce a slipping length , Supplementary Equation (25) becomes, The shear stress is then given by: Consequently, the drag torque is given by, Using the relation between the charge on the capacitor and the voltage on its terminals we have, Where ( ) is the current in the circuit. Using Supplementary Equation (30) to eliminate fix : Dividing by (-) and identifying ( ) = , we get, The term in the brackets of Supplementary Equation (34) is just the equivalent inverse capacitance of the two capacitors connected in series. Since fix ≫ ( ), we can approximate the equivalent capacitance as that of the time changing capacitor alone. For such approximation we get, Which is independent of the reservoir capacitance. Let us first introduce the following rescaling parameters: Plugging these into Supplementary Equation (35) we get, This is a first order linear ODE with time-dependent coefficients. The general solution for it is given by, where: Initially, the current in the circuit is zero, so that the excess charge distributes between the two capacitors correspond to zero voltage between them. Therefore, initially, And so, At this point, to further continue the analysis, we need to model ( ). The capacitance ( ) changes continuously, on each period, from its minimal value, min , to its maximal value, max , and back to min . Therefore, let us model ( ) as:  The steady state current is given by: which is given by, (√ 1 − 1) tan ( 2 ) 1 + √ 1 tan 2 ( 2 ) ) ) ′ 0 (57) This derivative will result with two additive terms. The first's amplitude will still decay with , but the second term's amplitude will be independent of , as we shall show here. The stationary performance frequency of the rotor is determined by the power balance: where, in the main text, R is given by Equation (14), d is the drag torque given by Equation As seen in Supplementary Figure 5, the difference between the two curves becomes smaller as increases. The approximation made in the main text is within reason in the case of PC, and completely valid in the case of IL mixtures.