J.E. Sprittles (University of Oxford, U.K.)Y.D. Shikhmurzaev (University of Birmingham, U.K.)

Workshop on the Micromechanics of Wetting & Coalescence

Microfluidic TechnologiesOften the key elements are the interaction of:

Drops with a solid - Dynamic WettingDrops with other drops - Coalescence

Dynamic Wetting Phenomena

50nm Channels27mm Radius Tube

1 Million Orders of Magnitude!

Millimetrescale

Microfluidics Nanofluidics

Emerging technologies

Routine experimental measurement

Microdrop Impact Simulations

?

25m water drop impacting at 5m/s. Experiments: Dong et al 06

Coalescence of Liquid DropsHemispheres easier to control

experimentally

Thoroddsen et al 2005Ultra high-speed imaging

Paulsen et al 2011Sub-optical electrical (allowing microfluidic measurements)

r

Thoroddsen et al 2005

A Typical Experiment230cP water-glycerol mixture:

Length scale is chosen to be the radius of dropTime scale is set from so that

Electrical: Paulsen et al, 2011. Optical:Thoroddsen et al, 2005.

(mm)L R O

/U / (ms)T R O

Mathematics

Consider a new approach - use the ifm derived in 1993 by yds.1) Briefly describe the modelShow you how in this framework2)we see how the additional physics naturally resolved two issues of no-solution and dynamic angle without ad-hoc assumptions3) Show limits in which analytic progress is possible. before moving onto full problem

CoalescenceFrenkel 45

Solution for 2D viscous drops using conformal mapping Hopper 84,90,93 & Richardson 92

Scaling laws for viscous-dominated flowEggers et al 99 (shows equivalence of 2D and 3D)

Scaling laws for inertia-dominated flowDuchemin et al 03 (toroidal bubbles, Oguz & Prosperetti 89)

Problem FormulationTwo identical drops coalesce in a dynamically

passive inviscid gas in zero-gravity.

Conventional model has:A smooth free surfaceAn impermeable zero tangential-stress

plane of symmetry

Analogous to wetting a geometric surface with:The equilibrium angle is ninety degreesInfinite ‘slip length’.

90d

d e

Problem Formulation

n P (I nn) = 0

u n = 0

u 0, n P (I nn) = 0, n P n = nf

ft

n P (I nn) = u (I nn)

u n = 0

Bulk

Free Surface

Liquid-Solid Interface Plane of Symmetry

uu 0, u u P, P = - I + u + u

Tp

t

d e 90d

( )r t

( )h t

( )d t

2( )h O r

3( )d O r

Bridge radius:

Undisturbed free surface:

Longitudinal radius of curvature:

( )r t

Conventional Model’s Characteristics

Initial cusp is instantaneously smoothed

dim dim dimln ,visc

r t t RC T

R T T

( )r t

( )h t

( )d t

ln ,viscr C t t

Surface tension driving force when resisted by viscous forces gives (Eggers et al 99):

Conventional Model’s Characteristics

/ ( )d t

0dr

u as tdt

( )r t

( )h t

( )d t

1/2 1/23dim dim ,inert inert

inert

r t RC T

R T

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

2Re / 1r r

Test scaling laws by fitting to experimentsNo guarantee this is the solution to the conventional model

Traditional Use of Scaling Laws

ln

1visc

visc

r C t t

C

ln

0.1visc

visc

r C t t

C

ln

0.2visc

visc

r C t t

C

Computational WorksProblem 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 experimentsValidate proposed scaling laws

JES & YDS 2011, Viscous Flows in Domains with Corners, CMAMEJES & 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 MeshBased 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 withRadiusDensitySurface tensionViscosities

Giving two limits of Re to investigate:

Hence establish validity of scaling laws for the conventional model

3

1

1mm

970kg m

20mN m

1mPa s & 58000mPa s

R

4 6Re 1.9 10 & 5.8 10

High Viscosity Drops ( )6Re 5.8 10

High Viscosity Drops: BenchmarkingInfluence of minimum radius lasts for time mint O r

Paulsen et al 12

High Viscosity Drops: Scaling Laws

Eggers et al 99

r=3.5t

ln

0.2visc

visc

r C t t

C

Not linear growth

Low Viscosity Drops ( ) 4Re 1.9 10

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: BenchmarkingCrossover atActually nearer

4Re Re 1 10r r r 2 2Re Re 1 10h r r

3

1

4

2mm

1200kg m

65mN m

3.3mPa s , 48mPa s & 230mPa s

Re = 1.4 10 , 68 & 2.9

R

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 PointsAccuracy 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.

Forminginterface

Formed interface

Liquid-solidLiquid-solidinterfaceinterface

SolidSolid

Relevance of the Young Equation

U

1 3 2cose e e e 1 3 2cos d

R

σ1e

σ3e - σ2e

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)

θe θd

Static situation Dynamic wetting

σ1

σ3 - σ2

R

Free surface pressed into solid

Dynamic WettingConventional 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

lgd

ls

180o

Liquid-solid interface forms instantaneously

e

Free surface pressed into solid

CoalescenceStandard 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.

lglg

Internal interface

d

180o

Infinite velocities as t->0

ll

Interface instantaneously disappears

2u 1u 0, u u up

t

s s1 1 1 2 2 2

1 3 2

v e v e 0

cos

s s

d

s1

*1

*1

s 1 11

s 1 111 1

1 1|| ||

v 0

n [( u) ( u) ] n n

n [( u) ( u) ] (I nn) 0

(u v ) n

( v )

(1 4 ) 4 (v u )

s se

s sss e

s

ff

t

p

t

In the bulk (Navier Stokes):

At contact lines:

On free surfaces:

Interface Formation Model

θd

e2

e1

n

nf (r, t )=0

Interface Formation Modelling

s1

*1

*1

v 0

n [( u) ( u) ] n n

n [( u) ( u) ] (I nn) 0

ff

t

p

Kinematic equation :

Normal stress balance :

Tangential stress balance :

*2

s2

n [ u ( u) ] (I nn) 0

v n 0

Balance of tangential stress :

Normal velocity :

*2

s 2 22

s 2 222 2

2|| || 2

21,2 1,2 1,2

n [ u ( u) ] (I nn) 0

(u v ) n

( v )

4 v u 1 4

( )

s se

s sss e

s

s s

t

A

a b

At the plane of symmery (internal interface):

1 1 2 2

As 0 :

,

90

(conventional model)

e e

d e

T

As 0 :T

Coalescence: Models vs Experiments

Interface Formation

Conventional

Parameters from Blake & Shikhmurzaev 02

1 0.3se apart from

230mPas

Coalescence: Free surface profiles

Interface formation theory

Conventional theory

Water-glycerolmixture of 230cP

Time: 0 < t < 0.1

s is the distance from the contact line.

Disappearance of the Internal Interface

4

2

0 : 0

1: 10

2 : 10

3: 1

t

t

t

t

Free Surface Evolutions is the distance from the contact line.

4

2

0 : 0

1: 10

2 : 10

3: 1

t

t

t

t

Coalescence: Models vs Experiments

Interface Formation

Parameters from Blake & Shikhmurzaev 02

apart from

Conventional

48mPas

1 0.45se

Widergap

Coalescence: Models vs Experiments

3.3mPas

Interface Formation

Conventional Wideninggap

Parameters from Blake & Shikhmurzaev 02

3 3

2

1.2kg m , (10 )

18 Pa s, (10 )

airair

airair

O

O

For the lowest viscosity ( ) liquid: 3.3mPa s

Influence of a Viscous Gas

Eggers et al, 99: gas forms a pocket of radius

3/2br r

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

ConventionalBlack: inviscid passive gasBlue: viscous gas

3.3mPas

Outstanding QuestionsHow 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)?

FundingFunding

This presentation is based on work supported by:

Early-Time Free Surface ShapesHow large is the initial contact?

Eddi, Winkels & Snoeijer (preprint)

Initial PositionsConventional model takes Hopper’s solution:

for and chosen so that .

IFM is simply a truncated sphere:

Notably, as we tend to the shape

2 2 1/2 2 1

2 2 1/2 2 1

( ) 2 (1 )(1 ) (1 2 cos(2 ) ) (1 )cos( ),

( ) 2 (1 )(1 ) (1 2 cos(2 ) ) (1 )sin( )

r m m m m m

z m m m m m

0 u m min(0)r r

min 0r 2 2( 1) 1r z

2 2 2 2min min( ) ( ) , 1 (1 ) / 2c c cr r z z z z r

Influence of GravityOn the predictions of the conventional model.

Benchmark SimulationsConsider a steady meniscus propagating through a capillary.

To validate the asymptotics for take (with ):2 4 3 2 3 4( , ) 1:(5 10 ,5 10 ), 2 :(10 ,2.5 10 ), 3 :(10 ,5 10 )Ca

2 0.1V , 0Ca

Profiles of Interface FormationProfiles along the free surface for:

2 4 3 2 3 4( , ) 1:(5 10 ,5 10 ), 2 :(10 ,2.5 10 ), 3 :(10 ,5 10 )Ca

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

2 4 3 2 3 4( , ) 1:(5 10 ,5 10 ), 2 :(10 ,2.5 10 ), 3 :(10 ,5 10 )Ca

Value of the Dynamic Contact AngleFor

we obtain

compared to an asymptotic value of (Shikhmurzaev 07):

Outside region of applicability of asymptotics ( ):

2 4 3 2 3 4( , ) 1:(5 10 ,5 10 ), 2 :(10 ,2.5 10 ), 3 :(10 ,5 10 )Ca

1: 104.1 , 2 : 107.7 , 3 : 102d

102.1d

2 (1)V O

Capillary Rise: Models vs ExperimentsInterface formation & Lucas-Washburn ( ) vs

experiments of Joos et al 90

Silicon oil of viscosity 12000cP for two capillary sizes (0.3mm and 0.7mm)

( )h t

d e

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 SimulationFull Simulation

Washburn Washburn

JES & YDS 2013, J. Comp. Phy.

Meniscus height h, in cm, as a function of time t, in seconds.

h

tt

h

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

Microdrop Impact

60e

Velocity Scale

Pressure Scale

-15ms

25m water drop impacting at 5m/s.