J.E. Sprittles (University of Oxford, U.K.) Y.D. Shikhmurzaev(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 50nm Channels 27mm Radius Tube 1 Million Orders of Magnitude! Millimetre scale Microfluidics Nanofluidics Emerging technologies Routine experimental measurement

Microdrop Impact Simulations ? 25  m water drop impacting at 5m/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 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.

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)

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 InterfacePlane of Symmetry

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

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

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

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

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

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 Coalescence simulation for 230cP liquid at t=0.01, 0.1, 1. Microdrop impact and spreading 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 Eggers et al 99 r=3.5t Not linear growth

Low Viscosity Drops ( )

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

Low Viscosity Drops: Benchmarking Paulsen et al 12

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

Hemispheres of water-glycerol mixture with:

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

Quantitative Comparison to Experiment 3.3mPas48mPas230mPas

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

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 Make a dry solid wet. Create a new/fresh liquid-solid interface. Class of flows with forming interfaces. Forming interface Formed interface Liquid-solidinterface Solid

Relevance of the Young Equation R σ 1e σ 3e - σ 2e 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) θeθe θdθd Static situationDynamic wetting σ1σ1 σ 3 - σ 2 R

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

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. Internal interface 180 o Infinite velocities as t->0 Interface instantaneously disappears

In the bulk (Navier Stokes): At contact lines: On free surfaces: Interface Formation Model θdθd e2e2 e1e1 n n f (r, t )=0 Interface Formation Modelling At the plane of symmery (internal interface):

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

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

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

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

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

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

For the lowest viscosity ( ) liquid:

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

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

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:

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.

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 ( ):

Capillary Rise: Models vs Experiments Interface formation & Lucas-Washburn ( ) vs experiments of Joos et al 90 Silicon oil of viscosity 12000cP for two capillary sizes (0.3mm and 0.7mm)

Lucas-Washburn vs Interface Formation Tube Radius = 0.36mm; Meniscus shape every 100secs Tube Radius = 0.74mm; Meniscus shape every 50secs After 100 secs LW IF After 50 secs LW IF

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

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

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