Interface width and bulk stability: Requirements for the simulation of deeply quenched liquid-gas systems

Abstract

Simulations of liquid-gas systems with extended interfaces are observed to fail to give accurate results for two reasons: the interface can get "stuck" on the lattice or a density overshoot develops around the interface. In the first case the bulk densities can take a range of values, dependent on the initial conditions. In the second case inaccurate bulk densities are found. In this communication we derive the minimum interface width required for the accurate simulation of liquid gas systems with a diffuse interface. We demonstrate this criterion for lattice Boltzmann simulations of a van der Waals gas. When combining this criterion with predictions for the bulk stability we can predict the parameter range that leads to stable and accurate simulation results. This allows us to identify parameter ranges leading to high density ratios of over 1000. This is despite the fact that lattice Boltzmann simulations of liquid-gas systems were believed to be restricted to modest density ratios of less than 20 [1].Application of lattice Boltzmann methods to the simulation of liquid-gas systems has been one of the early successful applications of lattice Boltzmann. Two very different algorithms were developed to do this: Swift et. al. [2,3,4,5] developed an algorithm based on implementing a pressure tensor and Shan et al. [6,7,8] developed an algorithm based on mimicking microscopic interactions. These algorithms have been succesfully applied to simulations of phase-separation [9], drop-collisions [1, 10], wetting dynamics and spreading [11], and the study of dynamic contact angles [12,13,14]. Only recently has it been shown that both algorithms perform very similarly when higher order corrections are taken into account [15].Previously there have been only heuristic analysis of what range of paramteres lead to accurate simulation results. In some parameter ranges not too far from the critical point too thin interefaces can lead to interfaces that stick to the lattice. This can lead to non-unique bulk densities for the liquid and gas phases that depend on the inital contitions. Further from the critical point density overshooting is observed at the interfaces. This also leads to incorrect bulk densities. In this communication we present a criterion that allows us to predict the range of acceptable values for the interface width. We will show that the accuracy of the algorithm rapidly deteriorates when this limit is exceeded. The second important contribution of this communication is a more general definition of the equation of state. Usual lattice Boltzmann methods recover the ideal gas equation of state with a pressure of p = ρ/3 when the gas is dilute. While relaxing this requirement has no effect on the interfacial properties it allows us to adjust the bulk-stability of the lattice Boltzmann method. Taken together this allows us to determine sets of paramters for which very deep quenches can be simulated. This is an important result since lattice Boltzmann methods were previously believed to be limited to density ranges of...

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