LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS

1. LATTICEBOLTZMANNSIMULATIONS OF COMPLEXFLUIDS FROM LIQUID CRYSTALS TO SUPERHYDROPHOBIC SUBSTRATES Alexandre Dupuis Davide Marenduzzo Julia Yeomans Rudolph Peierls Centre for Theoretical Physics University of Oxford

2. molecular dynamics stochastic rotation model dissipative particle dynamics

3. lattice Boltzmann computational fluid dynamics experiment simulation

4. The lattice Boltzmann algorithm ei=lattice velocity vector i=1,…,8 (i=0 rest) in 2d i=1,…,14 (i=0 rest) in 3d Define a set of partial distribution functions, fi Collision operator Streaming with velocity ei

5. The distributions fi are related to physical quantities via the constraints The equilibrium distribution function has to satisfy these constraints mass and momentum conservation The constraints ensure that the NS equation is solved to second order fieq can be developed as a polynomial expansion in the velocity The coefficients of the expansion are found via the constraints

6. Permeation in cholesteric liquid crystals Davide Marenduzzo, Enzo Orlandini Wetting and Spreading on Patterned Substrates Alexandre Dupuis

7. Liquid crystals are fluids made up of long thin molecules orientation of the long axis = director configuration n 1) NEMATICS Long axes (on average) aligned n homogeneous 2) CHOLESTERICS Natural twist (on average) of axes n helicoidal Direction of the cholesteric helix

8. The director field model considers the local orientation but not the local degree of ordering This is done by introducing a tensor order parameter, Q ISOTROPIC PHASE q1=q2=0 3 deg. eig. q1=-2q2=q(T) UNIAXIAL PHASE 2 deg. eig. q1>q2-1/2q1(T) 3 non-deg. eig. BIAXIAL PHASE

9. Free energy for Q tensor theory bulk (NI transition) distortion surfaceterm

10. Beris-Edwards equations of liquid crystal hydrodynamics 1. Continuity equation 2. Order parameter evolution couplingbetween director rotation & flow molecular field ~ -dF/dQ 3. Navier-Stokes equation pressure tensor: gives back-flow (depends on Q)

11. A rheological puzzle in cholesteric LC Cholesteric viscosity versus temperature from experiments Porter, Barrall, Johnson, J. Chem Phys. 45 (1966) 1452

12. z y x PERMEATION W. Helfrich, PRL 23 (1969) 372 helix direction flow direction Helfrich: Energy from pressure gradient balances dissipation from director rotation Poiseuille flow replaced by plug flow Viscosity increased by a factor

13. BUT What happens to the no-slip boundary conditions? Must the director field be pinned at the boundaries to obtain a permeative flow? Do distortions in the director field, induced by the flow, alter the permeation? Does permeation persist beyond the regime of low forcing? How does the channel width affect the flow? What happens if the flow is perpendicular to the helical axis?

14. No Back Flowfixed boundaries free boundaries

15. Free Boundariesno back flow back flow

16. These effects become larger as the system size is increased

17. Fixed Boundariesno back flow back flow

18. Summary of numerics for slow forcing • With fixed boundary conditions the viscosity increases by ~ 2 orders of magnitude due to back-flow • This is NOT true for free boundary conditions: in this case one has a plug-like flow and a low (nematic-like) viscosity • Up to which values of the forcing does permeation persist? What kind of flow supplants it ?

19. z y Above a velocity threshold ~5 m/s fixed BC, 0.05-0.1 mm/s free BC chevrons are no longer stable, and one has a doubly twisted texture (flow-induced along z + natural along y)

20. Permeation in cholesteric liquid crystals • With fixed boundary conditions the viscosity increases by ~ 2 orders of magnitude due to back-flow • This is NOT true for free boundary conditions: in this case one has a plug-like flow and a low (nematic-like) viscosity • Up to which values of the forcing does permeation persist? What kind of flow supplants it ? • Double twisted structure reminiscent of the blue phase

21. Lattice Boltzmann simulations of spreading drops:chemically and topologically patterned substrates

22. Free energy for droplets bulk term interface term surfaceterm

23. Wetting boundary conditions Surface free energy Boundary condition for a planar substrate An appropriate choice of the free energy leads to

24. Spreading on a heterogeneous substrate

25. Some experiments (by J.Léopoldès)

26. LB simulations on substrate 4 • Two final (meta-)stable state observed depending on the point of impact. • Dynamics of the drop formation traced. • Quantitative agreement with experiment. Simulation vs experiments Evolution of the contact line

27. Impact near the centre of the lyophobic stripe

28. Impact near a lyophilic stripe

29. LB simulations on substrate 4 • Two final (meta-)stable state observed depending on the point of impact. • Dynamics of the drop formation traced. • Quantitative agreement with experiment. Simulation vs experiments Evolution of the contact line

30. Effect of the jetting velocity Same point of impact in both simulations With an impact velocity t=0 t=10000 t=20000 t=100000 With no impact velocity

31. Base radius as a function of time

32. Characteristic spreading velocityA. Wagner and A. Briant

33. Superhydrophobic substrates Öner et al., Langmuir, 16, 7777, 2000. Bico et al., Euro. Phys. Lett., 47, 220, 1999.

34. Two experimental droplets He et al., Langmuir, 19, 4999, 2003.

35. Substrate geometry qeq=110o

36. A suspended superhydrophobic droplet

37. A collapsed superhydrophobic droplet

38. Drops on tilted substrates

39. A suspended drop on a tilted substrate

40. Droplet velocity

41. Water capture by a beetle

42. LATTICEBOLTZMANNSIMULATIONS OF COMPLEXFLUIDS • Permeation in cholesteric liquid crystals • Plug flow and high viscosity for fixed boundaries • Plug flow and normal viscosity for free boundaries • Dynamic blue phases at higher forcing • Drop dynamics on patterned substrates • Lattice Boltzmann can give quantitative agreement with experiment • Drop shapes very sensitive to surface patterning • Superhydrophobic dynamics depends on interaction of contact line and substrate

43. Some experiments (by J.Léopoldès)