Session F07: Rarefied Flows

Data-Driven Approach for the Rosenbluth Fokker-Planck Equation

In this study, we demonstrate a data-driven technique to efficiently obtain an accurate approximation of the transport coefficients in the Rosenbluth-Fokker-Planck equation. Our approach is based on an end-to-end mapping between the statistical moments of plasma particles and the corresponding transport coefficients. The mapping is accomplished by training the finely tuned multilayer perceptron (MLP) architecture. The training data is created by introducing the accurately approximated probability density function (PDF) using the entropic closure, and the PDF bridges the moments and the coefficients by solving the Poisson equation of the Rosenbluth potential. This approach ensures accurate consideration of all nonlinearities in the system, hence the quality of the training data is guaranteed. We integrated the proposed model in a stochastic particle Fokker-Planck method. By adopting the particle method,  we exploit benefits in the computational costs and are able to fix the imbalance of the collision symmetries induced by the discretization in the computational domain. The proposed scheme is validated for a highly nonlinear system and compared to the Direct Simulation Monte Carlo (DSMC) method. The results show that our approach successfully reproduces the correct anisotropic relaxation behavior and improves the computational cost compared to not only the direct evaluation of the transport coefficients but also the DSMC method.   

Improved Particle Time Integration for Solving the Vlasov-Fokker-Planck Equation

In this study, we extend the previous accurate particle time integration scheme of the Vlasov-Fokker-Planck equation to accommodate spatially inhomogeneous electromagnetic fields. The new approach stems from an improvement of time-symmetry by introducing an iterative mid-step correction. Further, we adopted an additive correction to the electric field to assure consistency with the negative gradient of the electric potentials in the direction of the particle displacement. Therefore,  the proposed approach ensures strict conservation of the Hamiltonian structure and achieves accurate time integration of the particle position and velocity. In addition, a linear drift and a constant diffusion coefficient are considered to deal with Coulomb collisions. The validation of the scheme is performed first for the 2D dynamics of a charged particle. Our method shows that an increase of mid-step evaluations improves time symmetry. Moreover, additive correction guarantees conservation of invariants, as total energy and angular momentum. Therefore, time step sizes beyond the gyro-frequency limits are realizable. In addition, collisional test cases are considered and the results are compared to those obtained with Direct Simulation Monte Carlo (DSMC). It can be observed that our method still is accurate for large time steps beyond the gyro- and collision-frequency limits.   

Numerical study of internal flow behavior at low vacuum in semiconductor industry

In the semiconductor industry, internal flow, especially occurring in a vacuum, is of interest because it is important to estimate particle behavior by fluid flow. resulting in defects that significantly affect the loss of mass production. However, due to the lack of experimental data on vacuum flow properties, it is difficult to accurately predict or analyze flow behavior, and thus we rely on the flow simulation by assuming the flow as a continuum. In the present study, we have conducted flow simulation using commercial package ANSYS, despite being used due to the low computational cost requirements of industrial processes, to estimate particle behavior caused by vacuum flow inside the semiconductor equipment such as extreme ultraviolet (EUV) machinery and gas transport pipeline. The continuum Navier-Stokes equations together with particle dynamics were used to predict particle and flow behaviors and the results are in quite good agreement with inspection results or equipment tendency.   

Experimental investigation of a supersonic jet in near vacuum

Most semiconductor processes are carried out in vacuum conditions due to purity requirements, deposition efficiency, and plasma generation. Although the overall conditions are near vacuum, air jets are utilized for various processes within vacuum process chambers. These jets expand rapidly and propagate at a very fast speed when they enter the vacuum environment, and the detailed flow physics of this process are not well understood. Due to the difficulty of measuring such flows in a vacuum environment, previous work has relied mostly on computational simulations, and there is a strong need for experimental validation. This study quantitatively visualized the flow structure of a round supersonic free jet at various vacuum conditions, utilizing particle image velocimetry (PIV) with ultra-fine tracer particles tens of nanometers in size. In addition, the shock structure was imaged via shadowgraphy. These results will provide a baseline for further studies of jets in near vacuum environments.   

On the drag and lift coefficients of ellipsoidal particles under rarefied flow conditions

The capability to simulate a two-way coupled interaction between a rarefied gas and an arbitrary-shaped colloidal particle is important for many practical applications, such as aerospace engineering, lung drug deliver and semiconductor manufacturing. By means of numerical simulations based on the Direct Simulation Monte Carlo (DSMC) method, we investigate the influence of the orientation of the particle and rarefaction on the drag and lift coefficients, in the case of prolate and oblate ellipsoidal particles immersed in a uniform ambient flow. This is done by modeling the solid particles using a cut-cell algorithm embedded within our DSMC solver. In this approach, the surface of the particle is described by its analytical expression and the microscopic gas-solid interactions are computed exactly using a ray-tracing technique. The measured drag and lift coefficients are used to extend the correlations available in the continuum regime to the rarefied regime, focusing on the transitional  and free-molecular regimes. The functional forms for the correlations for the ellipsoidal particles are chosen as a generalization from the spherical case. We show that the fits over the data from numerical simulations can be extended to regimes outside the simulated range of Knudsen by testing the obtained predictive model on values of Knudsen that where not included in the fitting process, allowing to achieve an higher precision when compared with existing predictive models from literature. Finally, we underline the importance of this work in providing new correlations for non-spherical particles that can be used for point-particle Euler-Lagrangian simulations to address the problem of contamination from finite-size particles in high-tech mechanical systems.   

Not Participating

The thermophoretic force may be an effective mechanism of preventing deposition of micron-sized particles in high-tech micromechanical systems under low pressure conditions. This force occurs when a particle is placed within a temperature gradient and its direction is opposite to this temperature gradient. For small particles at low pressure this force can be an active form of transport. We used the Direct Simulation Monte Carlo (DSMC) method to obtain the thermophoretic force on a micron-sized particle. Varying the size of the particle and the gas pressure we evaluated the force in the whole transition regime (Kn based on the particle radius=0.1-20). The particle has a finite size and is thus physically present inside the simulation domain. Gas molecules collide with the particle. The gas molecules impact location on the particle surface is determined by means of a raytracing method in combination with an analytical representation of the particle shape. The force exerted by the gas on the particle surface is evaluated via the momentum exchange between the gas molecules and the particle. We compare the results from simulations to the thermophoretic force values in the free molecular and continuum regime for which analytical expressions are available.