**Computational Optimization of Emitter Density in a Planar Electrospray Array**

**Chase Coffman** (SM Student), Paulo Lozano (Associate Professor)

### Collaborators:

**Objective of the Research **

Electrospray thrusters are of significant interest as in-space propulsion platforms for their high degree of throttlability, efficiency, and favorable thrust-to-mass ratio. One of the primary limitations in these devices is the inherently small thrust attainable from a single ion emission source, on the order of 0.1 μN - 1.0 μN, forcing designers to multiplex a large number of sources to meet small spacecraft requirements [1]. Recent research has targeted several multiplexing techniques, but these efforts have largely been proof-of-concept, with little focus in the way of design space mapping or performance optimization. The objective of the current work is to gain a more fundamental understanding of the packing limitations in the multiplexing approach, and how it applies specifically to microfabricated, porous metal electrospray emitters.

**Approach / Tools **

Low-order models of the perforated extractor electrode structural mechanics and fluidic transport processes are being developed and coupled with a computational, finite-element approach to the electro-quasi-statics (EQS) problem governing electrospray operation. A typical multiplexed electrospray consists of a perforated extracting electrode atop a planar array of emission sources. The difficulty in modeling such a device lies primarily in the geometric complexity of the electrical system, making the solution of Poisson's equation in the inter-electrode space rigorous. For emitter and extractor configurations of practical interest, a closed-form analytical solution does not exist. Consequently, a computational approach has been adopted whereby an FEA technique is used to compute the electric field using COMSOL Multiphysics®. To mitigate computational cost in the FEA simulations, the free charge density within the electrospray is considered negligible as a consequence of the relatively large emitter pitches and low per emitter current throughput. This assumption calls only for a solution to the homogeneous form of Poisson's equation, the Laplacian, and obviates the need for a time-consuming iterative computation of space charge.

The extracting electrode is characterized by a large number of perforations, each corresponding to a particular emitter, making it likewise expensive to model in a computational sense. Rather than do so, it has been described as a thin flat plate of an effective elastic constant. This description facilitates a wholly analytical approach to the bending solution for the plate and reduces overall computation time by a significant factor. Approximations for the bending of plates of various edge constraints are readily available in the literature, as are effective elastic constants [2, 3].

Results from the COMSOL Multiphysics® simulations are actively coupled with the structural model to determine deformation characteristics of the extracting electrode and how they impact the underlying electric field. A gradient-based optimization routine has been developed to determine the maximum supportable emitter density for a given array configuration, subject to a user-defined deformation threshold imposed on the extractor. Models being developed for the fluidic transport processes will be coupled to these results in the future to assess whether or not high-degrees of emitter multiplexing have an adverse impact on propellant delivery to individual emission sources.

**Results**

The optimization yields information regarding emitter density, extracting potential, and per emitter current throughput. These parameters can be used to compute relevant thruster metrics such as the specific impulse, thrust-to-power ratio, and thrust density. Preliminary results indicate that the maximum structurally supportable emitter density is a strong function of both the emitter geometry itself and the inter-electrode spacing in the device, or the distance between the emitter tips and extracting electrode. Furthermore, these results have shown that sharper emission sources significantly reduce the required extracting potential for emission onset and the electric pressure loading on the extracting electrode, resulting in appreciably higher degrees of allowable multiplexing. Minimizing the inter-electrode distance produces similar results in the same fashion. In the most desirable cases, these results suggest that highly multiplexed electrosprays are capable of reaching and exceeding the thrust densities of modern, high-power Hall thrusters while conserving their mass and volume advantages. Figure 1, below, delineates a family of representative emitter density curves as functions of the electrospray geometric parameters.

Figure 1: Maximum emitter density as a function of the electrode spacing. Red, blue, and black lines signify emitter tips of radii 5 μm, 10 μm, and 15 μm, respectively. Triangles, circles, and diamonds signify arrays of area 50 mm², 100 mm², and 200 mm², respectively. |

**References**

[1] Courtney, D., et al., "On the Validation of Porous Nickel as Substrate Material for Electrospray Ion Propulsion," 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Nashville, TN, AIAA 2010-7020, July 2010.

[2] Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, New York: McGraw-Hill, 1959.

[3] Cepkauskas, M.M., Yang, J., "Equivalent Properties for Perforated Plates - An Analytical Approach," 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18), Beijing, China, SMiRT 18-F06-1, Aug. 2005.