1. LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Alexandre Dupuis Davide Marenduzzo Julia Yeomans FROM LIQUID CRYSTALS TO SUPERHYDROPHOBIC SUBSTRATES 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 Define a set of partial distribution functions, f i e i =lattice velocity vector i=1,…,8 (i=0 rest) in 2d i=1,…,14 (i=0 rest) in 3d Streaming with velocity e i Collision operator

5. The distributions f i are related to physical quantities via the constraints The equilibrium distribution function has to satisfy these constraints The constraints ensure that the NS equation is solved to second order mass and momentum conservation f i eq 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 UNIAXIAL PHASE BIAXIAL PHASE q 1 =q 2 =0 q 1 =-2q 2 =q(T) q 1 >q 2 -1/2q 1 (T) 3 deg. eig. 2 deg. eig. 3 non-deg. eig.

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

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

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

12. PERMEATION W. Helfrich, PRL 23 (1969) 372 helix direction flow direction x y z 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 Flow fixed boundaries free boundaries

15. Free Boundaries no back flow back flow

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

17. Fixed Boundaries no 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. 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) y z

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 surface term

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

24. Spreading on a heterogeneous substrate

25. Some experiments (by J.Léopoldès)

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

27. Impact near the centre of the lyophobic stripe

28. Impact near a lyophilic stripe

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

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

31. Base radius as a function of time

32. Characteristic spreading velocity A. Wagner and A. Briant

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

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

35. Substrate geometry eq =110 o

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. LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS 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)