Liquid Droplet Dynamics: Variations on a Theme

1. Liquid Droplet Dynamics: Variations on a Theme • Collaborators: • S.H. Davis, Northwestern University • M.G. Worster, University of Cambridge • M.G. Forest, University of North Carolina • R. Superfine, University of North Carolina • W.W. Schultz, University of Michigan • J. Siddique, George Mason University • E. Barreto, George Mason University • B. Gluckman, George Mason University/Penn. State University Daniel M. Anderson Department of Mathematical Sciences George Mason University Supported by NASA (Microgravity Science), 3M Corporation and NSF (Applied Mathematics – DMS-0306996)

2. Free-Boundary Problems in Fluid Dynamics • the location of the free surface is part of the solution • - surface waves in oceans, lakes wind-driven waves

3. Free-Boundary Problems in Fluid Dynamics • the location of the free surface is part of the solution • - surface waves in oceans, lakes canine-driven waves wind-driven waves

4. Free-Boundary Problems in Fluid Dynamics Fluids Spreading on Solids • free surface with moving contact lines – LARGE SCALE: • floods, lava flows (gravity)

5. Free-Boundary Problems in Fluid Dynamics The Great Molasses Flood – Boston, MA 1919 • From The Boston Globe, May 28, 1996 “About 2 million gallons of raw molasses burst from a storage tank at the corner of Foster and Commercial streets about noon on January 15, 1919. The black wave of the sticky substance was so powerful that it knocked buildings off their foundations and killed 21 people. Newspapers described the cleanup effort as nightmarish …”

6. Free-Boundary Problems in Fluid Dynamics Fluids Spreading on Solids • free surface with moving contact lines – SMALL SCALE: • micro-fluidics, nano-fluidics (surface tension) 1mm

7. Outline of Talk: • Isothermal Spreading Droplet (‘Plain vanilla’) • [Greenspan, 1978] • Non-Isothermally Spreading Droplet • [Ehrhard & Davis, 1991] • Migrating Droplet • [Smith, 1995] • Evaporating Droplet • [Anderson & Davis, 1995] • Freezing Droplet • [Anderson, Worster, Davis, Schultz, 1996, 2000] • Melting Droplet • [Anderson, Forest & Superfine, 2001] • Imbibing Droplet, Rigid Porous Substrate • [Hocking & Davis, 2000] • Imbibing Droplet, Deformable Porous Substrate • [Anderson, 2005] • Vibrating Droplet • [Vukasinovic, Smith, Glezer, James, 2003, 2004]

8. Spreading Droplet Isothermal

9. Isothermal System Solid Boundary Anatomy of a Spreading Droplet

10. Spreading Droplet: ‘Full’ Problem • In the liquid: • - Navier-Stokes Equations • Free-Surface Conditions: • - Normal and tangential stress balances • - Mass balance (kinematic condition) • Conditions at the solid boundary: • - velocity normal to interface is zero • - ‘slip’ allowed in tangential velocity • Contact-line conditions: • - ‘contact’ (droplet height is zero) • - condition on contact angle air liquid solid substrate GOAL: Identify a physical regime that corresponds to experiments and allows isolation of important physical effects. Reduce mathematical model accordingly.

11. Thin Film Equations: Original Form

12. Thin Film Equations: Rescaled-Dimensionless Form

13. Thin Film Equations: Lubrication Theory Limit

14. Isothermal Spreading Droplet: Lubrication Theory [Greenspan, 1978; Ehrhard & Davis, 1991, 1993; Haley & Miksis, 1991] • Slow flow (Re << 1) and slender geometry, zero gravity • Full problem reduces to an evolution equation for the interface shape Capillary number • symmetry conditions at • contact line conditions: at where [Dussan V. 1979; Ehrhard & Davis, 1991, 1993]

15. Isothermal Droplet Spreading • large surface tension and • Analytical formula for • interface shape and • contact line position Droplet Evolution

16. Spreading Droplet Non-Isothermal

17. Non-Isothermal System Hot (or Cold) Solid Boundary Anatomy of a Non-isothermally-Spreading Droplet [Ehrhard & Davis, 1991]

18. Non-Isothermal Spreading Droplet: Lubrication Theory [Ehrhard & Davis, 1991, 1993] • Slow flow, slender geometry, zero gravity, temperature-dependent surface tension • Full problem reduces to an evolution equation for the interface shape Marangoni effects (surface tension gradients) capillarity (surface tension) unsteady term Marangoni number Biot number (interface heat transfer) • quasi-steady temperature: • contact line conditions: at contact line

19. Non-Isothermal Spreading Droplet: Results [Ehrhard & Davis, 1991, 1993] • Thermocapillary forces on interface (Marangoni effects) • drive a flow from warmer regions to colder regions • (surface tension decreases with temperature). • Spreading is enhanced when substrate is cooled. • Spreading is retarded when substrate is heated. • Experiments using paraffin oil and silicone oil spreading • on glass confirm these predictions.

20. Migrating Droplet Non-Isothermal

21. Non-Isothermal System Hot Cold Anatomy of a Migrating Droplet [Smith, 1995]

22. Migrating Droplet: Lubrication Theory [Smith 1995] • Slow flow, slender geometry, zero gravity, temp.-dep surface tension • Imposed temperature variation along solid boundary • Full problem reduces to an evolution equation for the interface shape Marangoni effects (surface tension gradients) capillarity (surface tension) unsteady term • contact line conditions: at left and right contact lines NOT SYMMETRIC!

23. Migrating Droplet: Results [Smith, 1995] • Droplet placed on a non-uniformly heated substrate • migrates towards colder temperature region (for • sufficiently large temperature gradients). • Steady-state solutions include motionless drops and drops • moving at constant speed (towards cooler regions). • Thermocapillary-driven fluid flow in the drop distorts • the free surface, modifies the apparent contact angle • which in turn modifies contact line speed.

24. Migrating Droplet: Results [Smith, 1995] COLD HOT [video compliments of Marc Smith, 2006]

25. Evaporating Droplet

26. Non-isothermal System Heated Boundary Anatomy of an Evaporating Droplet

27. EvaporatingDroplet • (Anderson & Davis, 1994; Hocking 1995). • Lubrication theory leads to an evolution equation Marangoni effects (surface tension gradients) evaporation (mass loss) vapor recoil capillarity (surface tension) Evaporation number Marangoni number Scaled density ratio Nonequilibrium param. Slip coefficient Capillary number

28. EvaporatingDroplet • [Anderson & Davis, 1994; Hocking 1995]. • Lubrication theory leads to an evolution equation boundary conditions contact line condition symmetry at at at liquid volume is not constant in time (droplet vanishes in finite time)

29. EvaporatingDroplet • Small capillary number (large surface tension) [Anderson & Davis, 1994]. where contact line condition global mass balance plus initial conditions • Competition between spreading and evaporation • EVAPORATION EVENTUALLY WINS!

30. Evaporating Droplet • strong evaporation, • weak spreading • contact line position recedes • monotonically • contact angle increases initially • and remains relatively constant

31. Evaporating Droplet • weak evaporation, • strong spreading • contact line position advances • initially • contact angle decreases • monotonically and has a nearly • constant intermediate region

32. Evaporating Droplet: Results [Anderson & Davis, 1995] • Evaporative effects are strongest near the contact-line • region due to largest thermal gradients there. • Effects that increase the contact angle retard evaporation • - thermocapillarity: flow directed toward the colder • droplet center • - vapor recoil: nonuniform pressure (strongest at contact • line) tends to contract the droplet • Effects that decrease the contact angle promote evaporation • - contact line spreading

33. Freezing Droplet

34. Freezing Droplet • This problem is motivated by the need to understand crystal growth problems and ‘containerless’ processing systems such as Czochralski growth, float-zone processing or surface melting. • The common feature in these systems is the presence of a ‘tri-junction’ – where a liquid, its solid and a vapor phase meet – at which phase transformation occurs. • Simple Model Problem: • WHAT HAPPENS WHEN WE FREEZE A LIQUID • DROPLET FROM BELOW ON A COLD SUBSTRATE?

35. Experimental Investigation (Water/Ice) [Anderson, Worster & Davis (1996)] Initial, motionless, water droplet at room temperature

36. Experimental Investigation (Water/Ice) [Anderson, Worster & Davis (1996)] Initial, motionless, water droplet at room temperature Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate) Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate) ?

37. Experimental Investigation (Water/Ice) [Anderson, Worster & Davis (1996)] Initial, motionless, water droplet at room temperature Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate) Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate)

38. Non-isothermal System Cold Boundary Anatomy of a Freezing Droplet

39. Latent heat Thermal diffusivity Specific heat Constant contact angle ‘Fixed’ contact line Nonzero growth angle FreezingDroplet • surface tension dominated liquid shape [Anderson, Worster & Davis, 1996]. Mass balance Capillarity and gravity relate Assume the solid – liquid interface is planar (1D heat conduction from cold boundary of temperature ; isothermal liquid at temperature ) Tri-junction condition (3 models)

40. Freezing Droplet: Constant Contact Angle Model • Contact angle in liquid is constant Droplet Evolution • Solidified Shape = Cone! • no inflexion points • solid shape is independent of growth rate

41. Freezing Droplet: Experimental Evidence • Solidified silicon in crucible of e-beam evaporation system (Phil Adams, LSU, 2005)

42. Freezing Droplet: Fixed Contact Line Model • The tri-junction moves tangent to the liquid – vapor interface; the liqiud contact angle is free to vary Solidified Shapes concave down (zero slope) concave down (nonzero slope) concave up (nonzero slope) • no inflexion points • water/ice predicted to have zero slope at top • solid shape is independent of growth rate

43. Freezing Droplet: Nonzero Growth Angle Model [Satunkin et al. (1980), Sanz (1986), Sanz et al. (1987)] • The tri-junction moves at a fixed growth angle to the liquid – vapor interface (angle through vapor phase is ) Solidified Shapes concave down (nonzero slope) concave up (nonzero slope) • no inflexion point • all materials with nonzero growth angle have pointed top • solid shape is independent of growth rate

44. Freezing Droplet: Nonzero Growth Angle simulation Experiment: ice

45. FreezingDroplet • A two-dimensional model for the thermal field in the solid was obtained by a boundary integral method [Schultz, Worster & Anderson, 2000].

46. FreezingDroplet [Schultz, Worster & Anderson, 2000] Results: • both peaks and dimples • can form at the top of • the drop (depending • on the growth angle • and density ratio) • inflexion points are also • possible

47. Melting Droplet

48. Melting Droplet: • Motivated by experiments on polystyrene spheres (1mm radius) • by D. Glick [UNC Physics Ph.D. 1998 – with R. Superfine] Glick Contact Angle Data • thermal diffusion time • ~ 10 – 25 seconds • data collapse if time is • scaled with 138C 99C = viscosity (varies by 3 orders of magnitude in experiment) = surface tension (varies by ~ 10%) = ad hoc length scale, increases with temperature

49. Non-isothermalSystem solid liquid Hot Boundary Anatomy of a Melting Droplet

50. Melting Droplet: Model [Anderson, Forest & Superfine, 2001] • initially spherical solid • no gravity • surface tension dominates – quasi-steady liquid vapor interface • solid-liquid interface assumed planar Liquid Shape – Spherical Nine Unknown Functions of Time angles lengths volumes and pressure