# Research

Plasma Dynamics Modeling Laboratory (PDML) focuses on development of computational and theoretical models to investigate weakly-ionized gas, collisionless plasmas, and plasma discharges.

### Physics of Low-Temperature Magnetized Plasmas

One of the most challenging problems in the community is related to electron transport, which determines the efficiency, operating mode (oscillatory/stable), and thrust level. It is known that the electron mobility obtained from experimental results does not agree with classical theory, which is attributed to "anomalous" electron transport. There are signatures that the anomalous electron transport is attributed to high frequency oscillations, turbulence, plasma waves, and etc. In PDML, we are developing both kinetic and fluid codes to investigate the electron transport mechanism and its effect on the thruster performance.

Fig. 1. Multiscale, multiphysics in low-temperature magnetized plasmas, such as in magnetron discharges and Hall effect thrusters |

### Physics-Based Plasma Models

#### Particle-Based and Grid-Based Approaches

In gas kinetics, the velocity distribution functions (VDFs) of the gas species play an important role in the overall gas dynamics and chemical reactions. When the flow is collisional, the VDFs relax to Maxwellian distribution, for which fluid description (conservation laws) is valid. On the other hand, when the flow is rarefied or collisionless, the VDFs can be any non-Maxwellian distribution and kinetic approach must be used. The gas dynamics is further coupled with the electromagnetic forces, which further make the system more nonlinear and complex. The most commonly used kinetic simulation technique is particle based methods, such as particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC). Such particle-based kinetic methods can model non-Maxwellian nature of the gas and plasma flows, but the statistical noise due to the use of such discrete particles may lead to unphysical oscillations and instabilities. We have been developing a new kinetic approach, which we call the direct kinetic (DK) method, in which the kinetic equations such as the Boltzmann and Vlasov equations are solved directly on discrete phase space. As the kinetic equations are hyperbolic partial differential equations (advection type), the numerical methods and algorithms developed in the computational fluid dynamics (CFD) community can be employed. The advantage of using a DK method is that the statistical noise is essentially absent, making the method applicable to investigate oscillations and instabilities.

#### Fluid Moment Models

Fluid models are constructed by taking the moments of the first-principles gas kinetic equations. The key challenge for the fluid models is the closure problem, which essentially is dependent on how non-Maxwellian the gas and plasma particles are. In the low-temperature plasma (LTP) community, drift-diffusion (DD) approximation that neglects the electron inertia term and the unsteady term (which are valid when the electron bulk velocity is much smaller than the electron thermal speed, cf. low-Mach number approximation, and the electron time scales are much faster than the other time scale) is often used. However, our group has shown that the DD approximation may not be valid when the charged particles are magnetized. The fluid moment models that retain the inertia and unsteady terms can capture some of the physical processes in low-temperature magnetized plasmas. We have been developing full fluid moment (FFM) models for Hall effect thrusters, rotating spokes, and capacitively coupled plasma (CCP) sources.

Fig. 2. Plasma simulation methods: (left) fluid approach, (middle) particle method, (right) grid-based direct kinetic (DK) method |

### Plasma Instabilities, Oscillations, and Turbulence

Due to the nonlinearity of the system, plasmas can experience a variety of instabilities and oscillations. We have used both particle and grid based kinetic methods to investigate the instabilities and oscillations. Due to the discrete velocity space, DK method has been used for nonlinear problems in thermal plasmas, including trapped particle instabilities and ladder climbing of plasma waves using external chirped field. On the other hand, PIC simulations are useful for beam-plasma interactions such as the instabilities induced by neutralized ion beam. Additionally, ionization oscillations in Hall thruster discharge plasmas are investigated using a hybrid-DK method. Utilization of different numerical modeling techniques (fluid and kinetic) is important to gain understanding of such oscillation phenomena.

Fig. 3. Fundamental plasma physics: (left) trapped particle dynamics in nonlinear plasma waves, (right) two-stream instability by the neutralized ion beam propagating in a quasi-neutral plasma |

### Real-Time State Estimation of Plasma Dynamics

Coupled physical and chemical phenomena, uncertain or approximated representations of nonlinear processes, and the sheer complexity of plasma physics have been the key limitation of a truly predictive computational model. Yet, the inability to measure detailed, dynamic, and multiscale plasma behavior creates the need for a predictive model to continue advancing technologies. Recent advances in computational capability have paved the way for data-driven modeling, or data-model fusion, where experimental data is used to supplement and enhance existing computational models. This can be performed to improve the results of lower-dimensional models to save on computational costs, or it can be used to study uncertain physics within a given model. Common techniques include neural networks, reduced-order modeling, dynamic mode decomposition, and phase space representations. In particular, the field of state estimation enables real-time applications, allowing for the simulation to run while data is continually being collected as no training of the model is required. This is a beneficial capability for studying real-time effects such as on-orbit maneuvers for satellites. Our current research focuses on using the suite of Kalman filters to estimate the time histories of unknown, dynamic quantities in plasma physics applications ranging from industry plasmas to space propulsion.

Fig. 4. Data-driven model: Online vs Offline approaches |

### Plasma-Material Interactions

For any plasma applications, plasma-material interaction plays an important role in controlling the plasma. Typically, electrons are much faster than ions due to the mass difference (more than 1800 times smaller). This results in the plasma-immersed materials to be negatively biased and formation of a potential drop near the material, which is called the plasma sheath. The potential drop accelerates the ions, which may lead to enhanced sputtering of the material, and decrease the electron heat flux to the wall. In the presence of plasmas, it is known that electrons can be emitted from the materials due to the ion and/or electron bombardment. The emitted electrons, either by thermionic or secondary electron emission, reduces the potential drop, which can in turn affect how the plasma is confined. As the distribution functions are non-Maxwellian and the plasma is non-neutral in the sheath region, DK simulation is useful to model the plasma sheath.

Fig. 5. Plasma sheaths that form around plasma-immersed materials |

### Atmospheric-Pressure Plasma Discharge

The local thermodynamic equilibrium (LTE) condition is widely used in many fluid and plasma applications. The LTE condition can be achieved when the collision frequency is large so that the electron temperature becomes equal to various other temperatures, such as the translational, rotational, vibrational, and electronic excitation temperatures. We have developed a multispecies model that accounts for chemical, velocity, and thermal nonequilibrium between species. The boundary conditions due to the sheath formation and surface reactions in reactive plasma flows are also implemented. Our preliminary results suggest that the radiation transport in the discharge and at the electrode surface is critical to the plasma discharge. We are currently collaborating with Dr Alexandros Gerakis (Luxembourg Institute of Science and Technology) to investigate the LTE condition using their innovative laser diagnostic tools.

Fig. 6. Modeling of an ablating arc discharge using graphite electrodes |

### Low-Temperature Plasma Processing Devices

PDML has been developing computational models of low-pressure, low-temperature plasma sources, such as capacitively coupled plasma (CCP) and inductively coupled plasma (ICP) sources. These research activities are performed in conjunction with our industrial partners in semiconductor processing.

Fig. 7. Low-pressure capacitively coupled plasma (CCP) model using particle-in-cell (PIC) Monte Carlo collision (MCC) approach

### Particle-Light Coupling

The interaction between light (photon) beam and particles plays an important role in many applications, such as laser diagnostics and optical tweezers. We have been working on theoretical and computational modeling of the nonlinear coupling between particles and light to study the feasibility of beam-powered spacecraft propulsion (cf. Breakthrough Starshot project). The refractive index of the gaseous media is affected by the atomic/molecular polarizability and light frequency, which can contribute to focusing or diffracting the light beam. Simultaneously, the particles may experience optical dipole force and radiation pressure force, leading to trapping or detrapping of the particles. The two mechanisms combined can result in a mutually guided particle-light beam.

Fig. 8. Modeling of light-particle coupled beam propagation using DSMC for particle dynamics with Helmholtz equation for light propagation