Coalescence of Liquid Drops Different Models vs Experiments

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

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

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

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

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

A Typical Experiment 230 c. P 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.

Coalescence of Liquid Drops: Conventional Modelling

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

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’.

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

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

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

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

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

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

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.

Resolving Multiscale Phenomena

Arbitrary Lagrangian Eulerian Mesh Based on the ‘spine method’ of Scriven and co-workers Microdrop impact Coalescence simulation and spreading for 230 c. P liquid at t=0. 01, 0. 1, 1. simulation.

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

High Viscosity Drops ( )

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

High Viscosity Drops: Scaling Laws r=3. 5 t Not linear growth Eggers et al 99

Low Viscosity Drops ( )

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

Low Viscosity Drops: Benchmarking Paulsen et al 12

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

Comparison to Experiments Hemispheres of water-glycerol mixture with:

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

Quantitative Comparison to Experiment 230 m. Pas 48 m. Pas 3. 3 m. Pas

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

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.

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.

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

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

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

Interface Formation Modelling f (r, t )=0 In the bulk (Navier Stokes): e 1 θd e 2 n n Interface Formation Model On free surfaces: At the plane of symmery (internal interface): At contact lines:

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

Coalescence: Free surface profiles Time: 0 < t < 0. 1 Conventional theory Interface formation theory Waterglycerol mixture of 230 c. P

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

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

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

Coalescence: Models vs Experiments 3. 3 m. Pas Conventional Widening gap Interface Formation Parameters from Blake & Shikhmurzaev 02

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

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

Influence of a Viscous Gas 3. 3 m. Pas Conventional Black: inviscid passive gas Blue: viscous gas Interface Formation Eggers et al, 99

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)?

Funding This presentation is based on work supported by:

Thanks

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

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 .

Influence of Gravity On the predictions of the conventional model.

Simulations vs Asymptotics

Benchmark Simulations Consider a steady meniscus propagating through a capillary. To validate the asymptotics for ): take (with

Profiles of Interface Formation Profiles along the free surface for:

Profiles of Interface Formation Profiles along the liquid-solid interface for:

Value of the Dynamic Contact Angle For we obtain compared to an asymptotic value of (Shikhmurzaev 07): Outside region of applicability of asymptotics ( ):

Simulations vs Experiments

Capillary Rise: Models vs Experiments Interface formation & Lucas-Washburn ( experiments of Joos et al 90 ) vs Silicon oil of viscosity 12000 c. P for two capillary sizes (0. 3 mm and 0. 7 mm)

Lucas-Washburn vs Interface Formation After 50 secs After 100 secs LW IF Tube Radius = 0. 36 mm; Meniscus shape every 100 secs LW IF Tube Radius = 0. 74 mm; Meniscus shape every 50 secs

Comparison to Experiment Meniscus height h, in cm, as a function of time t, in seconds. Washburn Full Simulation JES & YDS 2013, J. Comp. Phy. Full Simulation

Microdrop Impact and Spreading

Microdrop Impact 25 micron water drop impacting at 5 m/s on left: wettable substrate right: nonwettable substrate

Microdrop Impact Pressure Scale Velocity Scale 25 mm water drop impacting at 5 m/s.