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1 Mesoscale modelling with the Lattice Boltzmann model Jonathan Chin <[email protected]> Lattice Boltzmann Method • Fluids represented by fictional “particles” with discrete velocities, occupying sites on a discrete lattice. • Particles described by real-valued distribution function • Density of single component given by (x) m f i (x) • Momentum of single component given by i u m f i ci i Time Evolution • Advection step: particles move to adjacent sites. • Collision step: distribution function relaxes to equilibrium value via BGK operator. fi ' fi 1 f i f i (eq) • The equilibrium distribution is a function only of density n and velocity uat a given site. fi (eq) ci ci u u nTi 1 2 2 cs 2 cs ci ci 2 cs • Weighting factors Ti chosen to ensure isotropic hydrodynamics. • Bounce-back boundaries give non-slip walls. • Velocity is perturbed to take account of collisions with other components, and forcing term. v u / F • Force term includes gravity and Shan-Chen immiscibility force. ˆ F (x ) g gravn z n (x ) n (x ci )ci • Immiscibility force proportional to single-component density gradient, repelling differing components. Spinodal decomposition • Two fluids may mix above some temperature TC • A mixture quenched below this temperature will separate into its component fluids: this process is called Spinodal Decomposition. • Simulation performed of demixing process using periodic boundary conditions. Phase separation images Timestep 0 500 1000 4000 8000 32000 2000 50000 Growth Regimes • Very early time: exponential growth in structure factor as interfaces form according to a Cahn-Hilliard model. • Early-time hydrodynamic regime dominated by viscosity. • Late-time inertial hydrodynamic regime: domains grow as two-thirds power of time. • Very late time turbulent mixing regime postulated but not seen. Early-time Cahn-Hilliard Growth • Circularly-averaged structure factor S(k) retains shape but grows exponentially in magnitude as domains form. • Although the model does not define any free energies, it produces results in agreement with free-energy models of phase separation. Domain Growth Laws Porod Law structure Power law growth Breakdown of scaling • Many theories assume that a phase-separating system contains a single length scale evolving in time. • Simulation shows regime with multiple length scales due to competition between different growth mechanisms. Surface tension measurement • “Laplace’s Law” states that the pressure drop across the interface of a bubble is inversely proportional to its radius. p R • Bubbles simulated with different coupling constants to measure surface tension.