Mesoscale modelling with the Lattice Boltzmann model

1. Mesoscale modelling with the Lattice Boltzmann model Jonathan Chin <j.chin@qmul.ac.uk>

2. 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 • Momentum of single component given by

3. Time Evolution • Advection step: particles move to adjacent sites. • Collision step: distribution function relaxes to equilibrium value via BGK operator.

4. The equilibrium distribution is a function only of density and velocity at a given site. • Weighting factors chosen to ensure isotropic hydrodynamics. • Bounce-back boundaries give non-slip walls.

5. Velocity is perturbed to take account of collisions with other components, and forcing term. • Force term includes gravity and Shan-Chen immiscibility force. • Immiscibility force proportional to single-component density gradient, repelling differing components.

6. Spinodal decomposition • Two fluids may mix above some temperature • 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.

7. Phase separation images Timestep 0 500 1000 2000 4000 8000 32000 50000

8. 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.

9. 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.

10. Domain Growth Laws Porod Law structure Power law growth

11. 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.

12. Surface tension measurement • “Laplace’s Law” states that the pressure drop across the interface of a bubble is inversely proportional to its radius. • Bubbles simulated with different coupling constants to measure surface tension.