1. LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Julia Yeomans Rudolph Peierls Centre for Theoretical Physics University of Oxford

2. Binary fluid phase ordering and flow Wetting and spreading chemically patterned substrates superhydrophobic surfaces Liquid crystal rheology permeation in cholesterics Lattice Boltzmann simulations: discovering new physics

3. Binary fluids The free energy lattice Boltzmann model 1.The free energy and why it is a minimum in equilibrium 2.A model for the free energy: Landau theory 3.The bulk terms and the phase diagram 4.The chemical potential and pressure tensor 5.The equations of motion 6.The lattice Boltzmann algorithm 7.The interface 8.Phase ordering in a binary fluid

4. The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy A B

5. The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy A B

6. isothermal first law The free energy is a minimum in equilibrium constant T and V

7. n A is the number density of A n B is the number density of B The order parameter is The order parameter for a binary fluid

8. Models for the free energy n A is the number density of A n B is the number density of B The order parameter is

9. Cahn theory: a phenomenological equation for the evolution of the order parameter F

10. Landau theory bulk terms

11. Phase diagram

12. Gradient terms

13. Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

14. Getting from F to the pressure P and the chemical potential first law

15. Homogeneous system

17. Inhomogeneous system Minimise F with the constraint of constant N, Euler-Lagrange equations

18. The pressure tensor Need to construct a tensor which reduces to P in a homogeneous system has a divergence which vanishes in equilibrium

19. Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

20. The lattice Boltzmann algorithm Define two sets of partial distribution functions f i and g i Lattice velocity vectors e i, i=0,1…8 Evolution equations

21. Conditions on the equilibrium distribution functions Conservation of N A and N B and of momentum Pressure tensor Chemical potential Velocity

22. The equilibrium distribution function Selected coefficients

23. Interfaces and surface tension lines: analytic result points: numerical results

24. Interfaces and surface tension

25. N.B. factor of 2

26. surface tension lines: analytic result points: numerical results

27. Phase ordering in a binary fluid Alexander Wagner +JMY

28. Phase ordering in a binary fluid Diffusive ordering t -1 L -3 Hydrodynamic ordering t -1 L t -1 L -1 L -1

29. high viscosity: diffusive ordering

30. high viscosity: diffusive ordering

31. L(t) High viscosity: time dependence of different length scales

32. low viscosity: hydrodynamic ordering

33. low viscosity: hydrodynamic ordering

34. Low viscosity: time dependence of different length scales R(t)

35. There are two competing growth mechanisms when binary fluids order: hydrodynamics drives the domains circular the domains grow by diffusion

36. Wetting and Spreading 1.What is a contact angle? 2.The surface free energy 3.Spreading on chemically patterned surfaces 4.Mapping to reality 5.Superhydrophobic substrates

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

38. Surface terms in the free energy Minimising the free energy gives a boundary condition The wetting angle is related to h by

39. Variation of wetting angle with dimensionless surface field line:theory points:simulations

40. Spreading on a heterogeneous substrate

41. Some experiments (by J.Léopoldès)

42. 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.

43. 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

44. Base radius as a function of time

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

46. Superhydrophobic substrates Bico et al., Euro. Phys. Lett., 47, 220, 1999.

47. Two droplet states A collapsed droplet A suspended droplet * * * * He et al., Langmuir, 19, 4999, 2003

48. Substrate geometry  eq =110 o

49. Equilibrium droplets on superhydrophobic substrates On a homogeneous substrate,  eq =110 o Suspended,  ~160 o Collapsed,  ~140 o

50. Drops on tilted substrates

51. Droplet velocity

52. Dynamics of collapsed droplets

53. Drop dynamics on patterned substrates Lattice Boltzmann can give quantitative agreement with experiment Drop shapes very sensitive to surface patterning Superhydrophobic dynamics depends on the relative contact angles

54. Liquid crystals 1.What is a liquid crystal 2.Elastic constants and topological defects 3.The tensor order parameter 4.Free energy 5.Equations of motion 6.The lattice Boltzmann algorithm 7.Permeation in cholesteric liquid crystals

56. An ‘elastic liquid’

57. topological defects in a nematic liquid crystal

58. The order parameter is a tensor 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.

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

60. Equations of motion for the order parameter

61. The pressure tensor for a liquid crystal

62. The lattice Boltzmann algorithm Define two sets of partial distribution functions f i and g i Lattice velocity vectors e i, i=0,1…8 Evolution equations

63. Conditions on the additive terms in the evolution equations

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

65. 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

66. 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?

67. No Back Flow fixed boundaries free boundaries

68. Free Boundaries no back flow back flow

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

70. Fixed Boundaries no back flow back flow

71. 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 ?

72. 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

73. 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

74. Binary fluid phase ordering and hydrodynamics two times scales are important Wetting and spreading chemically patterned substrates final drop shape determined by its evolution superhydrophobic surfaces ?? Liquid crystal rheology permeation in cholesterics fixed boundaries – huge viscosity free boundaries – normal viscosity, but plug flow