Coalescence of Liquid Drops: Different Models vs Experiments

1. Coalescence of Liquid Drops: Different Models vs Experiments J.E. Sprittles (University of Oxford, U.K.) Y.D. Shikhmurzaev (University of Birmingham, U.K.) Workshop on the Micromechanics of Wetting & Coalescence

2. Microfluidic Technologies • Often the key elements are the interaction of: • Drops with a solid - Dynamic Wetting • Drops with other drops - Coalescence

3. Dynamic Wetting Phenomena Emerging technologies Routine experimental measurement 1 Million Orders of Magnitude! 50nm Channels 27mm Radius Tube Millimetre scale Microfluidics Nanofluidics

4. Microdrop Impact Simulations ? 25mm water drop impacting at 5m/s. Experiments: Dong et al 06

5. Coalescence of Liquid Drops Hemispheres easier to control experimentally Thoroddsen et al 2005 Ultra high-speed imaging Paulsen et al 2011 Sub-optical electrical (allowing microfluidic measurements) Thoroddsen et al 2005

6. A Typical Experiment • 230cP water-glycerol mixture: • Length scale is chosen to be the radius of drop • Time scale is set from so that Electrical: Paulsen et al, 2011. Optical:Thoroddsen et al, 2005.

7. Coalescence of Liquid Drops:Conventional Modelling

8. Coalescence • Frenkel 45 • Solution for 2D viscous drops using conformal mapping • Hopper 84,90,93 & Richardson 92 • Scaling laws for viscous-dominated flow • Eggers et al 99 (shows equivalence of 2D and 3D) • Scaling laws for inertia-dominated flow • Duchemin et al 03 (toroidal bubbles, Oguz & Prosperetti 89)

9. Problem Formulation • Two identical drops coalesce in a dynamically passive inviscid gas in zero-gravity. • Conventional model has: • A smooth free surface • An impermeable zero tangential-stress plane of symmetry • Analogous to wetting a geometric surface with: • The equilibrium angle is ninety degrees • Infinite ‘slip length’.

10. Problem Formulation Bulk Free Surface Liquid-Solid Interface Plane of Symmetry

11. Conventional Model’s Characteristics Initial cusp is instantaneously smoothed • Bridge radius: • Undisturbed free surface: • Longitudinal radius of curvature:

12. Conventional Model’s Characteristics • Surface tension driving force when resisted by viscous forces gives (Eggers et al 99):

13. Assumed valid while after which (Eggers et al 99):

14. Traditional Use of Scaling Laws Test scaling laws by fitting to experiments No guarantee this is the solution to the conventional model

15. Computational Works • Problem demands resolution over at least 9 orders of magnitude. • The result been the study of simplified problems: • The local problem – often using the boundary integral method for Stokes flow (e.g. Eggers et al 99) or inviscid flow. • The global problem - bypassing the details of the initial stages • Our aim is to resolve all scales so that we can: • Directly compare models’ predictions to experiments • Validate proposed scaling laws

16. A Finite Element Based Computational Framework JES & YDS 2011, Viscous Flows in Domains with Corners, CMAME JES & YDS 2012, Finite Element Framework for Simulating Dynamic Wetting Flows, Int. J. Num. Meth Fluids. JES & YDS, 2012, The Dynamics of Liquid Drops and their Interaction with Surfaces of Varying Wettabilities, Phy. Fluids. JES & YDS, 2013, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, J. Comp. Phy.

17. Resolving Multiscale Phenomena

18. Arbitrary Lagrangian Eulerian Mesh Based on the ‘spine method’ of Scriven and co-workers Coalescence simulation for 230cP liquid at t=0.01, 0.1, 1. Microdrop impact and spreading simulation.

19. Benchmark Simulations • ‘Benchmark’ code against simulations in Paulsen et al 12 for identical spheres coalescing in zero-gravity with • Radius • Density • Surface tension • Viscosities • Giving two limits of Re to investigate: • Hence establish validity of scaling laws for the conventional model

20. High Viscosity Drops ( )

21. High Viscosity Drops: Benchmarking • Influence of minimum radius lasts for time Paulsen et al 12

22. High Viscosity Drops: Scaling Laws r=3.5t Not linear growth Eggers et al 99

23. Low Viscosity Drops ( )

24. Low Viscosity Drops: Toroidal Bubbles Toroidal bubble Increasing time As predicted in Oguz & Prosperetti 89 and Duchemin et al 03

25. Low Viscosity Drops: Benchmarking Paulsen et al 12

26. Low Viscosity Drops: Benchmarking Crossover at Actually nearer Duchemin et al 03 Eggers et al 99

27. Comparison to Experiments Hemispheres of water-glycerol mixture with:

28. Qualitative Comparison to Experiment Experimental images courtesy of Dr J.D. Paulsen Coalescence of 2mm radius water drops. Simulation assumes symmetry about z=0

29. Quantitative Comparison to Experiment 3.3mPas 48mPas 230mPas

30. Conventional Modelling: Key Points Accuracy of simulations is confirmed Scaling laws approximate conventional model well Conventional model doesn’t describe experiments

31. Coalescence & Dynamic Wetting:Processes of Interface Formation/Disappearance YDS 1993, The moving contact line on a smooth solid surface, Int. J. Mult. Flow YDS 2007, Capillary flows with forming interfaces, Chapman & Hall.

32. Interface Formation in Dynamic Wetting Liquid-solid interface Solid Forming interface Formed interface • Make a dry solid wet. • Create a new/fresh liquid-solid interface. • Class of flows with forming interfaces.

33. Relevance of the Young Equation Static situation Dynamic wetting σ1e σ1 θe θd σ3 - σ2 σ3e - σ2e R R Dynamic contact angle results from dynamic surface tensions. Theangle is now determined by the flow field. Slip created by surface tension gradients (Marangoni effect)

34. Dynamic Wetting • Conventional models: contact angle changes in zero time. • Interface formation: new liquid-solid interface is out of equilibrium and determines angle. o 180 Liquid-solid interface forms instantaneously Free surface pressed into solid Free surface pressed into solid Liquid-solid interface takes a time to form

35. Coalescence • Standard models: cusp becomes “rounded” in zero time. • IFM: cusp is rounded in finite time during which surface tension forces act from the newly formed interface. o 180 Infinite velocities as t->0 Interface instantaneously disappears Internal interface

36. f (r, t )=0 e1 n n θd e2 Interface Formation Modelling In the bulk (Navier Stokes): Interface Formation Model At the plane of symmery (internal interface): On free surfaces: At contact lines:

37. Coalescence: Models vs Experiments 230mPas Conventional Interface Formation Parameters from Blake & Shikhmurzaev 02 apart from

38. Coalescence: Free surface profiles Time: 0 < t < 0.1 Conventional theory Interface formation theory Water- glycerol mixture of 230cP

39. Disappearance of the Internal Interface s is the distance from the contact line.

40. Free Surface Evolution s is the distance from the contact line.

41. Coalescence: Models vs Experiments 48mPas Wider gap Conventional Interface Formation Parameters from Blake & Shikhmurzaev 02 apart from

42. Coalescence: Models vs Experiments 3.3mPas Widening gap Conventional Interface Formation Parameters from Blake & Shikhmurzaev 02

43. Influence of a Viscous Gas on the Conventional Model’s Predictions For the lowest viscosity ( ) liquid:

44. Influence of a Viscous Gas • Toroidal bubble formation suppressed by viscous gas which forms a pocket in front of the bridge Eggers et al, 99: gas forms a pocket of radius

45. Influence of a Viscous Gas 3.3mPas Conventional Black: inviscid passive gas Blue: viscous gas Interface Formation Eggers et al, 99

46. Outstanding Questions How does the viscous gas effect the interface formation dynamics? Can a non-smooth free surface be observed optically? Can the electrical method be used in wetting experiments? How do the dynamics scale with drop size? Are singularities in the conventional model the cause of mesh-dependency in computation of flows with topological changes (Hysing et al 09)?

47. Funding • Funding • This presentation is based on work supported by:

48. Thanks

49. Early-Time Free Surface Shapes Eddi, Winkels & Snoeijer (preprint) How large is the initial contact?

50. Initial Positions Conventional model takes Hopper’s solution: for and chosen so that . IFM is simply a truncated sphere: Notably, as we tend to the shape