Smooth Particle Methods for Vapour Liquid Coexistence.

The vapour-liquid phase transition exhibits remarkable complexity and requires sophisticated mod- eling to explain in detail. The complex structure of a condensing quenched vapour suggests Lagrangian particle methods like smoothed particle hydrodynamics (SPH), smooth particle applied mechanics (SPAM) and dissipative particle dynamics (DPD) as natural numerical techniques. Lagrangian particle methods allow for arbitrarily complex flows, do not suffer from mesh tangling and avoid unstable modes associated with grid methods. Mesoscopic, coarse grained models of phase transitions are of current interest because of their potential applications to the fabrication of microstructures. In smooth particle methods the fluid is represented as a set of particles that carry the properties of the fluid and interact according to equations of motion that are derived in a clear and consistent way from the continuum equations and constitutive relations for the system. We see this as an advantage over lattice methods in which the relationship between the model dynamics and the continuum equations is less clear. We have developed a high performance smooth particle code for modeling small systems of the order of 1000s of particles. The code is built flexibly to allow for experimentation with the form of the smooth particle equa- tions and the various parameters. Lagrangian particle methods do bring their own set of numerical challenges, most of which have straightforward numerical corrections. Our code incorporates many of these well under- stood corrections including artificial viscosity and anti-clumping measures. Our technique is closely related to modern implementations of DPD with a key difference that we do not explicitly model thermal fluctuations. By solving the equations of motion for a mean-field van der Waals equation of state we are able to produce droplets, bubbles and planar gas-liquid interfaces under a variety of boundary conditions. Surface tension arises through the attractive part of the van der Waals equation of state and the van der Waals square gradient term. As scale decreases the capillary terms, which are driven by density gradients, become more important to the dynamics of the phase separating fluid. We do not need to use different particle types or different equations of state to represent the two phases. Instead the phase separation of the one component fluid arises naturally out of the van der Waals model. No explicit tracking of the vapour-liquid interface is required. To produce sensible results we need to ensure the cohesive pressure operates at a longer range than the repulsive pressure. We do not yet understand if this is required for consistency with mean-field theory, or is needed purely for numerical stability. Quenching a supercritical gas in a long periodic box we are able to realise the interconnected domains char- acteristic of spinodal decomposition, with indications that short time liquid-gas domain growth and long time hydrodynamic domain coalescence are correctly captured. Phase separating systems equilibrate to domains of coexisting vapour and liquid at the expected densities for the van der Waals model. Our method and code shows qualitative ability and quantitative promise. It is ready to be tested against real experimental data, under more complex boundary conditions and against more complicated phase diagrams involving more than one component. The smooth particle framework we have used can be naturally extended to handle concentration gradients, arbitrary boundary conditions, and other equations of state.