1. Drop Impact and Spreading on Surfaces of Variable Wettability J.E Sprittles Y.D. Shikhmurzaev Bonn 2007

2. Swansea 2007 Motivation Drop impact and spreading occurs in many industrial processes. 100 million inkjet printers sold yearly. NEW: Inkjet printing of electronic circuits. Why study the ‘old problem’ of drops spreading on surfaces?

3. Swansea 2007 Worthington 1876 – First Experiments

4. Swansea 2007 Worthington’s Sketches Millimetre sized drops of milk on smoked glass.

5. Swansea 2007 Modern Day Experiments (mm drops of water) Courtesy of Romain Rioboo

6. Swansea 2007 Xu et al 03 Drops don’t splash at the top of Everest!

7. Swansea 2007 Renardy 03 et al - Pyramidal Drops Impact of oscillating water drops on super hydrophobic substrates 1 23 4

8. Swansea 2007 The Simplest Problem How does a drops behaviour depend on: fluid properties, drop speed, drop size,.. etc? Spread Factor Apex Height Contact Angle a

9. Swansea 2007 The Contact Angle In equilibrium the angle defines the wettability of a solid-liquid combination. How should we describe it in a dynamic situation? More Wettable (Hydrophilic) Less Wettable (Hydrophobic) Solid 1 Solid 2

10. Swansea 2007 Modelling of Drop Impact and Spreading Phenomena The Moving Contact Line Problem Conventional Approaches and their Drawbacks The Shikhmurzaev Model

11. Swansea 2007 The Moving Contact Line Problem Liquid Inviscid Gas Contact line Contact angle Solid

12. Swansea 2007 The Moving Contact Line Problem No-Slip Impermeability Kinematic condition Dynamic condition Navier-Stokes Continuity Contact angle prescribed No solution!!!

13. Swansea 2007 The Conventional Approach These are treated separately by: 1)Modifying the no-slip condition near the Contact Line (CL) to allow slip, e.g. 2)Prescribing the Contact Angle as a function of various parameters, e.g. One must: 1)Allow a solution to be obtained. 2)Describe the macroscopic contact angle.

14. Swansea 2007 Experiments Show This Is Wrong… Can one describe the contact angle as a function of the parameters? “There is no universal expression to relate contact angle with contact line speed”. (Bayer and Megaridis 06) “There is no general correlation of the dynamic contact angle as a function of surface characteristics, droplet fluid and diameter and impact velocity.” (Sikalo et al 02)

15. Swansea 2007 As in Curtain Coating Used to industrially coat materials. Conventional models: Fixed substrate speed => Unique contact angle

16. Swansea 2007 ‘Hydrodynamic Assist of Dynamic Wetting’ The contact angle depends on the flow field. See: Blake et al 1994, Blake et al 1999, Clarke et al 2006

17. Swansea 2007 Angle Also Dependent On The Geometry: Flow Through a Channel The contact angle is dependent on d and U. ( Ngan & Dussan 82) U U d Conclusion: Angle is determined by the flow field

18. Swansea 2007 The Shikhmuraev Model’s Predictions Unlike conventional models: The contact angle is determined by the flow field. No stagnation region at the contact line. No infinite pressure at the contact line => Numerics easier

19. Swansea 2007 Shikhmurzaev Model What is it? Generalisation of the classical boundary conditions. Considers the interface as a thermodynamic system with mass, momentum and energy exchange with the bulk. Used to relieve paradoxes in modelling of capillary flows such as …..

20. Swansea 2007 Some Previous Applications

21. Swansea 2007 The Shikhmurzaev Model Qualitatively (Flow near the contact line) Solid Gas Width of interfacial layer Liquid

22. Swansea 2007 Shikhmurzaev Model Solid-liquid and liquid-gas interfaces have an asymmetry of forces acting on them. In the continuum approximation the dynamics of the interfacial layer should be applied at a surface. Surface properties survive even when the interface's thickness is considered negligible. Surface tension Surface density Surface velocity

23. Swansea 2007 Shikhmurzaev Model On liquid-solid interfaces: On free surfaces:At contact lines: θdθd e2e2 e1e1 n n f (r, t )=0

24. Swansea 2007 What if (the far field) ? On liquid-solid interfaces: On free surfaces:At contact lines:

25. Swansea 2007 Summary Classical Fluid Mechanics => No Solution Conventional Methods Are Fundamentally Flawed The Shikhmurzaev Model Should Be Investigated

26. Swansea 2007 Our Approach Bulk: Incompressible Navier-Stokes equations Boundary: Conventional Model (for a start!) Use Finite Element Method. Assume axisymmetric motion (unlike below!).

27. Swansea 2007 Numerical Approach Use the finite element method: Velocity and Free Surface quadratic Pressure Linear The ‘Spine Method’ is used to represent the free surface ~2000 elements Second order time integration

28. Swansea 2007 The Spine Method (Scriven and co-workers) The Spine Nodes fixed on solid. Nodes define free surface.

29. Swansea 2007 Code Validation Consider large deformation oscillations of viscous liquid drops. Compare with results from previous investigations, Basaran 91 and Meradji 01. Microgravity Experiment Compare aspect ratio of drop as a function of time. Starting position is

30. Swansea 2007 Second Harmonic – Large Deformation For Re=100, f2 = 0.9

31. Swansea 2007 Second Harmonic – Large Deformation (cont) Aspect ratio of the drop as a function of time. A damped wave.

32. Swansea 2007 Fourth Harmonic – Large Deformation For Re=100, f4 = 0.9

33. Swansea 2007 Drop Impact on a Hydrophilic (Wettable) Substrate Re=100, We=10, β = 100,.

34. Swansea 2007 The Experiment – Water on Glass Courtesy of Dr A. Clarke (Kodak)

35. Swansea 2007 Drop Impact on a Hydrophobic (non-wettable) Substrate Re=100, We=10, β = 100,.

36. Swansea 2007 The Experiment – Water on Hydrophobe Courtesy of Dr A. Clarke (Kodak)

37. Swansea 2007 High Speed Impact Radius = 25  m, Impact Speed = 12.2 m/s Re=345, We=51, β = 100,.

38. Swansea 2007 Non-Spherical Drops on Hydrophobic Substrates Radius = 1.75mm, Impact Speed = 0.4 m/s, Re=1435, We=8,.

39. Swansea 2007 Impact + Spreading of Non-Spherical Drops on Hydrophobic Substrates

40. Swansea 2007 Impact + Spreading of Non-Spherical Drops on Hydrophobic Substrates The Pyramid!

41. Swansea 2007 Impact + Spreading of Non-Spherical Drops on Hydrophobic Substrates Experiment shows pinch off of drops from the apex

42. Swansea 2007 Impact + Spreading of Non-Spherical Drops on Hydrophobic Substrates As in experiments, drop becomes toroidal

43. Swansea 2007 Current Work Quantitatively compare results against experiment. Incorporate the Shikhmurzaev model. Consider variations in wettability ….

44. Swansea 2007 How to Incorporate Variations in Wettability? Technologically, why are flows over patterned surfaces important? What are the issues with modelling such flows? How will a single change in wettability affect a flow? How about intermittent changes?

45. Swansea 2007 Using Patterned Surfaces Manipulate free surface flows using unbalanced surface tension forces.

46. Swansea 2007 Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces Pattern a surface with areas of differing wettability. ‘Corrects’ deposition.

47. Swansea 2007 Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces Pattern a surface to ‘correct’ deposition. Courtesy of Professor Roisman

48. Swansea 2007 The Problem What if there is no free surface? Do variations in the wettability affect an adjacent flow? Solid 1 Solid 2 What happens in this region? Shear flow in the far field

49. Swansea 2007 Molecular Dynamics Simulations Courtesy of Professor N.V. Priezjev More wettable Compressed More wettable Compressed Less wettable Rarefied Less wettable Rarefied

50. Swansea 2007 Hydrodynamic Modelling: Defining Wettability Defining wettability The Young equation: The contact line Solid 1

51. Swansea 2007 Hydrodynamic Modelling: Which Model? No-Slip No effect Slip Models (e.g. Navier Slip) There is no theta! A Problem.. We have no tools!

52. Swansea 2007 Qualitative Picture Bulk Fluid particles are driven into areas of differing wettability. Surface properties take a finite time to relax to their new equilibrium state. What happens when flow drives fluid particles along the interface? Mass, momentum and energy exchange between surface and bulk. The process of interface formation. Solid 1 Solid 2 Consider region of interest. Finite thicknes: For Visualisation Only Finite thicknes: For Visualisation Only

53. Swansea 2007 Interface Formation Equations – Hydrodynamic of Interfaces Surface density is related to surface tension: Equilibrium surface density defines wettability: Surface possesses integral properties such as a surface tension, ; surface velocity, and surface density,. Equation of State Input of Wettability

54. Swansea 2007 The Shikhmurzaev Model: Constant Wettability Bulk Interfacial Layer: For Visualisation Only. In the continuum limit.. Interfacial Layer: For Visualisation Only. In the continuum limit.. If then we have Navier Slip

55. Swansea 2007 Solid-Liquid Boundary Conditions – Shikhmurzaev Equations Bulk Tangential velocity Surface velocity Solid facing side of interface: No-slip Layer is for VISUALISATION only. In the continuum limit…

56. Swansea 2007 Solid-Liquid Boundary Conditions – Shikhmurzaev Equations Bulk Continuity of surface mass Normal velocity Solid facing side of interface: Impermeability Layer is for VISUALISATION only. In the continuum limit…

57. Swansea 2007 Problem Formulation 2D, steady flow of an incompressible, viscous, Newtonian fluid over a stationary flat solid surface (y=0), driven by a shear in the far field. Bulk –Navier Stokes equations: Boundary Conditions –Shear flow in the far field, which, using gives:

58. Swansea 2007 Results - Streamlines Consider solid 1 (x 0). Coupled, nonlinear PDEs were solved using the finite element method.

59. Swansea 2007 Results – Different Solid Combinations Consider different solid combinations.

60. Swansea 2007 Results – Size of The Effect Consider the normal flux out of the interface, per unit time, J. We find: The constant of proportionality is dependent on the fluid and the magnitude of the shear applied.

61. Swansea 2007 Results - The Generators of Slip Variations in slip are mainly caused by variations in surface tension. 1) Deviation of shear stress on the interface from equilibrium. 2) Surface tension gradients. 1) Deviation of shear stress on the interface from equilibrium. 2) Surface tension gradients.

62. Swansea 2007 Periodically Patterned Surface Consider Solid 1 More Wettable. Consider a=1 -> Strips Have Equal Width.

63. Swansea 2007 Results - Streamlines Solid 2 less wettable Qualitative agreement

64. Swansea 2007 Results – Velocity Profiles Tangential (slip) velocity varies around its equilibrium value of u=9.8. Fluxes are both in and out of the interfacial layer. Overall mass is conserved.

65. Swansea 2007 Further Work  Compare results with molecular dynamics simulations.  Devise experiments to test predictions.  Fully investigate periodic case. Single transition investigation is in: Sprittles & Shikhmurzaev, Phys. Rev. E 76, 021602 (2007).

66. Swansea 2007 Thanks!

67. Swansea 2007 Numerical Analysis of Formula for J Shapes are numerical results. Lines represent predicted flux Shapes are numerical results. Lines represent predicted flux

68. Swansea 2007 Interface Formation Equations + Input of Wettability Transition in wettability centred at x=y=0. Input of wettability

69. Swansea 2007 Surface Equation of State Break-upDynamic wetting

70. Swansea 2007 Deviation of The Actual Contact Angle => Non Equilibrium Surface Tensions Left:Curtain Coating Experiments (+) vs Theory (lines) Blake et al 1999 Wilson et al 2006 Right:Molecular Dynamics Koplik et al 1989

71. Swansea 2007 Comparison of Theory With Experiment Perfect wetting (Hoffman 1975; Ström et al. 1990; Fermigier & Jenffer 1991) Partial wetting (□: Hoffman 1975;  : Burley & Kennedy 1976; ,,  : Ström et al. 1990) It has been shown that the theory is in good agreement with all experimental data published in the literature.

72. Swansea 2007 Mechanism of Relaxation Comparison of the theory with experiments on fluids with different viscosity (1.5-672 cP) confirms that the mechanism of the interface formation is diffusive in nature (J. Coll. Interface Sci. 253,196 (2002)). Estimates for parameters of the model have been obtained, in particular, showing that for water-glycerol mixtures one has: where