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

European Coating Symposium, MonsSeptember 2013

Coating Phenomena

Impact of a solid on a liquidDuez et al 07

Dip coating experimentsCourtesy of Terry Blake

Impact of a liquid on a solidXu et al 05

Questions

?

1) Why is there still so much debate about wetting?

2) Are computational techniques essential?

3) Are the gas’ dynamics important?

4) How can we identify the ‘true’ physics?

Coating ExperimentsAdvantages:Flow is steady making

experimental analysis more tractable.

Parameter space is easier to map:Speeds over 6 ordersViscosities over 3 orders

appclU Liquid

GasSolid

The ‘apparent angle’

Coating ResultsApparent angle measured at resolution of 20microns for

water-glycerol solutions with μ=1, 10, 100 mPas.Increasing μ

clcl

UCa

You only observe the ‘apparent angle’. The actual one is fixed.Free surface bends below the experiment’s resolution (20μm)

Interpretation A: Static Contact Angle

erU

( )app r

The ‘actual angle’

Dynamics of angle cause change in apparent angleDynamic contact angle is a function of speed

Interpretation B: Dynamic Contact Angle

rU

d( )app r

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

Slip ModelsA: Equilibrium contact angleB: Slip - typically Navier-slip

eU ( )app r

l s

B: No-slip => No solution

Often, we have

Asymptotics for the Apparent Angle

2 2

22 2 20

2 2

ln

sin cos sin,

2sin 2 sin sin

( ) sin

app ds

g

l

Lg g Ca l

k K dg k

k K k

K

k

, 1slCaL

3 3 9 lnapp ds

LCa l

lnapp ds

Lg g Ca l

1d

In Cox 86, it was shown that in this case:

And for Voinov (76) has shown:

JES & YDS 2013, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, Journal of Computational Physics, 233, 34-65

Computational Domain

U Gas

Liquid

x1x10x108x102

x104

Resolution:

Arbitrary Lagrangian Eulerian MeshBased on the ‘spine method’ of Scriven and co-workers

Microdrop simulationwith impact, spreading and rebound

Free Surface Profiles With: 67 , 1nme sl

Computations vs Asymptotics

Ca=0.5

Ca=0.05

Ca=0.005

Solid line: ComputationsDashed line: Asymptotic (Cox’s) result

Limitations of Cox’s FormulaChen, Rame & Garoff 95:

“Aspects of the unique hydrodynamics acting in the inner region, not included in the model, project out and become visible in the imaged region.”

0.1Ca 0.5Ca

( )r m

app

( )r m

app

2) Are computational techniques essential?

Yes! To accurately capture high-speed coating flows.

Slip Model vs ExperimentsGas’ viscosity leads to air entrainment at a finite speed.

Decreasing viscosity ratio

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

Hydrodynamic Assist

U, cm/s

dθ

Blake et al 99

-1(ms )U

app

app

Vary Flow Rate

30d

UEffect is not due to free surface bending(Wilson et al 06)

Physics of 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

2u 1u 0, u u upt

s s1 1 1 2 2 2

1 3 2

v e v e 0cos

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

f ftp

t

In the bulk (Navier Stokes):

At contact lines:

On free surfaces:Interface Formation Model

θd

e2

e1

nnf (r, t )=0

Interface Formation Modelling

*2 || ||

s 2 22

s 2 222 2

2|| || || 2

21,2 1,2 1,2

1n [ u ( u) ] (I nn) u U2

(u v ) n

( v )

1v (u U )2( )

s se

s sss e

s

s s

t

a b

Liquid-solid interface

Interface Formation vs ExperimentsApparent angle = Dynamic actual angle

1) Why is there still a debate about wetting?

Fundamentally different models describe experiments(with reasonable parameter values).

+ Viscous bending

Influence of Gas PressureSplashing in DropImpact:Xu, Zhang & Nagel 05

Air Entrainment Speedin Dip CoatingBenkreira & Ikin 10

(Lack of) Influence of Inertia Bulk flow can’t be responsible for the effect.

Re = 0

Re = 100

Rarefied Gas DynamicsSlip at solid-gas interface is due to finite mean free path.Mean free path (hence Kn) depends on gas density.

1KnL

U

uKn u Uy

Gas Dynamics Near Contact Line

U

Atmospheric pressure: mean free path ~ 0.1 microns

/u U

s

s

0.1 m

Gas Dynamics Near Contact LineAt Reduced Pressure: mean free path~ 10microns

/u U

s

U

s

0.1 m 10 m

Delayed Air EntrainmentMean free paths (mfp) are:Atmospheric pressure: mfp ~ 0.1 micronsReduced pressure (10mbar): mfp ~ 10 microns

cCa

( )mfp m

3) Are the gas’ dynamics important?

Yes, its behaviour is key to air entrainment

Microdrop Impact

JES & YDS 2012, The Dynamics of Liquid Drops and their Interaction with Solids of Varying Wettability, Physics of Fluids, 24, 082001.

Coalescence of Liquid DropsDeveloped framework can be adapted for coalescence.

Thoroddsen’s Group: Ultra high-speed imaging

Nagel’s Group:Sub-optical electrical measurements r

Thoroddsen et al 2005

dSimulation

Experiment

Coalescence: Models vs ExperimentsBridge radius versus time: 2mm drops of 220cP water-glycerol.

Interface formation

Conventional

Nagel’sElectrical Measurements

Thoroddsen’sOpticalExperiments

/r R

/t R

4) How can we identify the ‘true’ physics?

By accessing smaller spatio-temporal scales

JES & YDS 2012, Coalescence of Liquid Drops: Different Models vs Experiments, Physics of Fluids, 24, 122105

Microscale Dynamic WettingUltra high speed imaging of microfluidic wetting phenomenon, with Dr E. Li & Professor S.T. Thoroddsen

FundingFundingThis presentation is based on work supported by:

Computations vs Experiments1.5, 10, 104 mPa s Water-glycerol solutions of

& Asymptotics

Asymptotic Formula for Actual Angle in IFM

2 1 0

22

2 2 2 2 20

2 2 2 2

2 22

2 ( , )cos cos

.

sin cos ( ) sin cos ( )( , )

sin cos ( ) sin cos ( )

, ( ) sin

s se e d

e d se

d d dd

d d d d

d

V u k

V V

V ScCa

K k Ku k

K k K

K

When there is no ‘hydrodynamic assist’, for small capillary numbers the actual angle is dynamic:

Moffat 64

IFM vs Experiments

Shikhmurzaev 93

Shikhmurzaev 93 + Cox 86

& AsymptoticsActual angle varies and free surface bends.

‘Hydrodynamic Resist’

Smaller Capillaries

U

d

R

New effect: contact angle depends on capillary size

( m)R

Sobolev et al 01

d

1/3U

Fibre Coating: Effect of Geometry

app

d

Simpkins & Kuck 03

app

4mmd U

2mmd

U

Drop Spreading: Effect of Impact Speed

10.18ms

10.25ms

)

U

app

-1(ms )U

appBayer & Megaridis 06

30d

CoalescenceConventional model: singular as initial cusp is rounded in zero time -> infinite velocities

Interface formation: singularity-free as cusp is rounded in finite time that it takes internal interface to disappear

Forming interface

d

Instant rounding

Infinite bridge speed

90d180d

drdt

r

Gradual rounding

Finite bridge speed

Coalescence: Free surface profiles

Interface formation theory

Conventional theory

Water-Glycerolmixture of 230cP

Time: 0 < t < 0.1

0.01 - 0.360.03 - 0.3650.1 - 0.370.6 - 0.391 - 0.43 - 0.426 - 0.4410 - 0.45

Wetting6 (1489) running 1166,2277 and 0.1microns – saved previous t_info – running for current info

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