Session PR24: Modeling Methods

Full Newton AMR solver with fast-slow model split and implicit multi-temperature chemistry

We present a multi-fluid, multi-temperature plasma solver with adaptive Cartesian mesh based on a full-Newton (non-linear, implicit) scheme for collisional low-temperature plasma. The implicit treatment of the coupled equations allows using large time steps, and the full-Newton method enables fast non-linear convergence at each time step, offering improved efficiency of fluid plasma simulations on dynamically adaptive grids [1]. We have enhanced our solver using a modular split of the fast (electron) and slow (ion) transport solvers to address the disparity of the electron and ion time scales. Significant speedup has been obtained, which scales well with increasing the number of ion species and becomes vital as the number of species grows in complex gas mixtures. The fast-slow split also facilitates using kinetic solvers for electrons to our hybrid framework. We have further enhanced our plasma solver with an implicit chemistry module enabling multi-temperature reaction rates for electron-induced reactions and reactions among heavy species. The implementation of the full-Newton method (FNM) with adaptive mesh refinement (AMR) utilizes state-of-the-art numerical CFD algorithms. The new solver enables solving challenging problems in a numerically efficient manner to study plasma self-organization in gas discharges and pattern formation during plasma-surface interactions.

 

[1] R.R. Arslanbekov and V.I. Kolobov, Implicit and coupled fluid plasma solver with adaptive Cartesian mesh and its applications to non-equilibrium gas discharges, Plasma Sources Sci. Technol. 30 (2021) 045013   

Modeling of Capacitively Coupled Plasmas Using Fluid Ion + Particle-in-Cell Electron Plasma Model

Low pressure (< 50 mTorr) capacitively coupled plasmas (CCP) are widely used for plasma processing in the semiconductor industry.  Kinetic effects play an important role in electron dynamics in these plasmas.  Fluid plasma models are often inadequate in capturing the behavior of these plasmas.  Full particle-in-cell (PIC) multi-dimensional models are more accurate but computationally very expensive and limited to simple chemistries.  A hybrid scheme for modeling low-pressure CCPs is described in this paper where electrons are modeled using PIC while ions and neutral species are treated as fluid.  Coupling between the fluid and PIC modules is done at every time step, where electron density from the PIC model is used during the solution of the Poisson equation.  The resulting model provides a good compromise between accuracy and computational speed, and it is suitable for industrial plasma system design applications.  One and 2-dimensional simulations of Ar and O₂ plasmas are discussed in this paper.  Single frequency, dual-frequency, and CCPs biased using non-sinusoidal waveforms are considered.  Plasma modeling results from this hybrid code are compared to corresponding results from fluid and full PIC simulations.  This comparison highlights the importance of using the appropriate boundary conditions and transport approximations for ions in fluid or hybrid plasma models.   

Surrogate Models for Low Temperature Plasma Simulations with Deep Learning

Advances in machine learning algorithms have recently generated considerable interest for constructing computationally efficient counterparts of complex dynamical systems such as low temperature plasmas. Accurate multiphysics/multiscale plasma simulations are often slow to execute for real world problems e.g. plasma reactors used for etching and deposition processes in semiconductor manufacturing, that limits their applicability for design and optimization purposes. A promising route to accelerate simulations by building fast and accurate surrogates with machine learning provides high computational gains for real-time prediction and multiscale simulations. In this work, we present a case study of development of a parametric emulation framework based on non-intrusive, data-driven methods using deep neural networks. We developed surrogate models for periodic steady-state radio frequency powered capacitively coupled plasmas where deep learning is used for parametric interpolation of reduced spatiotemporal modes and thus can be considered as equation-free approach for latent-space representation of plasmas. We assess the viability of such method to allow design parameter exploration and to enable new, previously unfeasible computational research.    

Investigation of non-ideal effects in wave-heated dense microplasmas including multiply charged ions and excited species using particle-in-cell Monte Carlo-collision modeling

In this work, we present a computational model for non-ideal plasma effects during the time evolution of a second-stage laser-heated discharge at high pressures.  The model extends a classical one-dimensional particle-in-cell Monte Carlo-collision (PIC-MCC) approach coupled with Maxwell’s equations for the laser-heating process of a xenon plasma.  Plasma non-ideality resulting from Coulomb coupling at high plasma densities is manifested as a depression in the effective ionization potential of atoms and the enhanced collision cross sections. We find that full ionization of the plasma is obtained on the picosecond time scale, starting from the skin layer and quickly expanding throughout the domain through an anomalous extension of the skin depth. More critically, we show that the inclusion of the non-ideal plasma effects results in more rapid ionization when compared to an ideal plasma, especially at higher pressures.  While the initial results in this work are obtained with a chemistry mechanism that includes singly charged ions only, the final work will consider multiply charged ions as well as excited species.  At higher pressures, higher charge states and excited states of xenon are expected to play an important role in the ionization process.   

Kinetic Monte Carlo simulations of plasma-chemistry

The interest in nanosecond pulsed discharges has been rising quickly due to their strong non-equilibrium properties. The typical plasma-chemistry models based on the low-anisotropy and quasi-stationary approximations to the Electron Energy Distribution Function may fail to describe these discharges correctly, due to the high reduced electric fields (E/N) and very short timescales involved. These limitations can be overcome with a self-consistent and unified formulation based on Kinetic Monte Carlo (KMC) techniques. A simultaneous KMC description of the electron and heavy-species kinetics would enable a rigorous inclusion of the time-dependent influence of different excited states in the electron kinetics and vice-versa, and would not be limited to a restricted range of E/N.

As a first step to achieve such unified formulation, a KMC algorithm to solve gas-phase chemistry was implemented and validated in the thermodynamic limit. However, the densities of the minority species have considerable fluctuations. Two novel variance-reduction techniques to achieve an efficient and accurate time-evolution of all the species, including the minority ones, are proposed and successfully applied.   

Development of gravitational N-body simulation algorithm for charged particle dynamics

Charged particles'  transport process is investigated employing a gravitational N-body numerical simulation algorithm. To circumvent severe computational restrictions at the close encounters of two charged particles, we employ analytical Kepler solutions which are smoothly connected to the numerical solutions. The connection is at the order ot Landau length. We also discuss how to incorporate the magnetic field perturbatively. Having alike charges in the system and thus Debye shielding, plasma N-body simulation is expected to be less demanding in constructing hierarchies compared to the gravitational N-body simulation whose central forces are all attractive. After examining the mechanism of the like-charged-particles' collisions and alike-charged-particles collsions, the statistical behavior of the system (particle's diffusion rate) is examined in comparison with the Bohm's analysis which is given by "D ~T/B".   

Integrating the Fokker-Planck approach to vibrational kinetics in a self-consistent N2 plasma chemistry model

The Fokker-Planck (FP) approach for the description of vibrational kinetics is coupled for the first time to a chemical kinetics model. Due to the importance of vibrational ladder climbing for the optimization of nitrogen fixation in a plasma reactor, nitrogen has been used as a test case for the validation of this method.

A self-consistent N2 plasma model is developed, including a complete description of chemical, vibrational and electron kinetics. This includes vibrational-translational, resonant and non-resonant vibrational energy exchanges, as well as electron-impact step-wise vibrational excitations.The complexity of the model has been gradually increased by adding vibrational-translational, resonant and non-resonant vibrational energy exchanges. Moreover, electron-impact step-wise vibrational excitations have been included, allowing self-consistent evolution and coupling of vibrational kinetics with electron kinetics. The model is coupled to the gas heating and vibrational energy equations, coupling with gas heating and vibrational energy has also been implemented, making the complete model suitable for investigating vibrational non-equilibrium conditions in a plasma reactor.

The time-resolved evolution and steady-state solution of the vibrational distribution function obtained through the FP approach have been benchmarked against results from the widely employed state-to-state method, showing excellent agreement and higher computational efficiency, while vibrational excitation and gas heating reproduce experimental results obtained at DIFFER.   

A new electron transport Monte Carlo code

We present a code for the Monte Carlo simulation of electron transport in an arbitrarily complex gas mixture. The code is designed so as to enable a relatively easy integration of an electron and heavy-particle Monte Carlo formulation [1]. Following the strategy of the LisbOn Kinetics Boltzmann solver (LoKI-B) [2], the simulation tool can address electron-neutral collisions with any target state (electronic, vibrational and rotational), characterized by any user-prescribed population, allowing to include the effects of superelastic collisions in a general manner. The influence of the thermal motion of the background molecules is considered, enabling to describe the swarm behavior at low reduced electric fields E/N. On output, the program provides: the electron energy distribution function and the electron velocity distribution function; flux and bulk swarm parameters; collision rates; power balance; and the spatiotemporal evolution of the electron swarm. The code will be released in the near future as part of the simulation tools provided by the N-PRiME group at Lisbon.

[1] T. C. Dias and V. Guerra, Plasma Sources Sci. Technol. 29 (2020) 115003

[2] A. Tejero-del-Caz et al, Plasma Sources Sci. Technol. 30 (2021) 065008