Abstract
The new generation of Lithography equipment of ASML uses rarefied gas flows in order to keep the mirrors from blackening. These flows cannot be modeled by conventional computational fluid dynamics based on the NavierStokes, NS, equation, because the continuum approach is no longer valid. The continuum approach is no longer valid, because of large non equilibrium due to the rarefaction. This is indicated by the Knudsen number, the ratio between the gas’ mean free path and the characteristic length scale, becoming larger than 1. Therefore, the Direct Simulation Monte Carlo, DSMC, method is used, which can be used up to very large Knudsen numbers. This method is, however, rather time consuming at small Knudsen numbers and thus impractical for quick design optimizations. Therefore, a new simulation tool is required. The simulation tool has to be able to model a flow within 10% of the DSMC result between Knudsen numbers 0.01 and 2 in a tenth of the time needed for a DSMC simulation.
The lattice Boltzmann method, LBM, is a promising alternative to DSMC simulations, which can be seen as a special discretization of the Boltzmann Transport Equation where the gas is allocated to a lattice and only a discrete set of velocities is allowed. The TU/e has developed an LBM implementation for ASML in a previous project. This project consists of validating this LBM implementation as well as investigating the capabilities of other LBM implementations.
The validation is performed by simulating both the flow in a 2D channel between two plates and in an axisymmetric channel between two disks. The LBM results of both are shown to be consistent with DSMC results and theory up to a Knudsen number of 0.06, which is much smaller than the desired 2. The reason for the deviation is attributed to the small number of discrete velocities implemented in this LBM, resulting in an LBM only able to model the slip flow. The results are reported in literature to improve when more discrete velocities are added to the LBM, resulting in a better approximation of the Boltzmann Transport Equation.
The capabilities of other LBM implementations are investigated by simulating similar 2D test cases with the LBM of Strathclyde, Frontier Lattices, Comsol and Exa. The simulation results are then used to determine the maximum Knudsen number at which the simulation results are still valid. None of the LBMs proved to be able to model a flow accurately enough up to a Knudsen number of 2. However, all of them were much faster than the required maximum time. The LBMs are ranked based on the highest Knudsen number and their capabilities. The Strathclyde LBM is the best and is shown to be valid up to a Knudsen number of 0.88. It should be noted that the Strathclyde LBM is specifically designed for this geometry and not able to model more complex geometries. The Frontier Lattices LBM is the runner up and valid up to a Knudsen number of 0.31. It has the ability to also import CAD drawings and simulate complex geometries. In addition, they claim to be able to simulate heat transfer and perform simulations in three dimensions. The Comsol LBM ends in the third place and is able to model the flow up to a Knudsen 0.36, however it is found to be quite computational intensive. The TU/e LBM ranks next with a maximum Knudsen number of 0.06 and Exa ranks last with a Knudsen number of 10−3. The results of all LBMs can be improved by increasing the number of discrete velocities.
Recommendations
The project focused on the gas flow and not on the heat transfer, which will also be needed in a tool for ASML. The Frontier Lattices LBM is the only LBM that claims to be able to perform accurately heat transfer, therefore it is recommended in a follow up project to check this together with their capabilities of 3D simulations. Alternatively, recommendations are provided to improve the TU/e LBM.
The lattice Boltzmann method, LBM, is a promising alternative to DSMC simulations, which can be seen as a special discretization of the Boltzmann Transport Equation where the gas is allocated to a lattice and only a discrete set of velocities is allowed. The TU/e has developed an LBM implementation for ASML in a previous project. This project consists of validating this LBM implementation as well as investigating the capabilities of other LBM implementations.
The validation is performed by simulating both the flow in a 2D channel between two plates and in an axisymmetric channel between two disks. The LBM results of both are shown to be consistent with DSMC results and theory up to a Knudsen number of 0.06, which is much smaller than the desired 2. The reason for the deviation is attributed to the small number of discrete velocities implemented in this LBM, resulting in an LBM only able to model the slip flow. The results are reported in literature to improve when more discrete velocities are added to the LBM, resulting in a better approximation of the Boltzmann Transport Equation.
The capabilities of other LBM implementations are investigated by simulating similar 2D test cases with the LBM of Strathclyde, Frontier Lattices, Comsol and Exa. The simulation results are then used to determine the maximum Knudsen number at which the simulation results are still valid. None of the LBMs proved to be able to model a flow accurately enough up to a Knudsen number of 2. However, all of them were much faster than the required maximum time. The LBMs are ranked based on the highest Knudsen number and their capabilities. The Strathclyde LBM is the best and is shown to be valid up to a Knudsen number of 0.88. It should be noted that the Strathclyde LBM is specifically designed for this geometry and not able to model more complex geometries. The Frontier Lattices LBM is the runner up and valid up to a Knudsen number of 0.31. It has the ability to also import CAD drawings and simulate complex geometries. In addition, they claim to be able to simulate heat transfer and perform simulations in three dimensions. The Comsol LBM ends in the third place and is able to model the flow up to a Knudsen 0.36, however it is found to be quite computational intensive. The TU/e LBM ranks next with a maximum Knudsen number of 0.06 and Exa ranks last with a Knudsen number of 10−3. The results of all LBMs can be improved by increasing the number of discrete velocities.
Recommendations
The project focused on the gas flow and not on the heat transfer, which will also be needed in a tool for ASML. The Frontier Lattices LBM is the only LBM that claims to be able to perform accurately heat transfer, therefore it is recommended in a follow up project to check this together with their capabilities of 3D simulations. Alternatively, recommendations are provided to improve the TU/e LBM.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution  
Supervisors/Advisors 

Award date  1 Jan 2013 
Place of Publication  Eindhoven 
Publisher  
Print ISBNs  9789044411942 
Publication status  Published  2013 