Swansea 2007

Drop Impact and Spreading on Surfaces of Variable WettabilityDrop Impact and Spreading on Surfaces of Variable Wettability

J.E Sprittles

Y.D. Shikhmurzaev

Bonn 2007

Swansea 2007

Motivation

• Drop impact and spreading occurs in many industrial processes.

• 100 million inkjet printers sold yearly.

• NEW: Inkjet printing of electronic circuits.

• Why study the ‘old problem’ of drops spreading on surfaces?

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Worthington 1876 – First Experiments

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Worthington’s Sketches

Millimetre sized drops of milk on smoked glass.

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Modern Day Experiments (mm drops of water)

Courtesy of Romain Rioboo

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Xu et al 03 Drops don’t splash at the top of Everest!

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Renardy 03 et al - Pyramidal Drops

• Impact of oscillating water drops on super hydrophobic substrates

1

2 3

4

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The Simplest Problem

• How does a drops behaviour depend on:

fluid properties,

drop speed,

drop size,.. etc?

Spread Factor

Apex Height

Contact Angle

a

0U

d

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The Contact Angle

• In equilibrium the angle defines the wettability of a solid-liquid combination.

• How should we describe it in a dynamic situation?

2

More Wettable (Hydrophilic)More Wettable (Hydrophilic)

Less Wettable (Hydrophobic)Less Wettable (Hydrophobic)

Solid 1 Solid 2

1 2

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Modelling of Drop Impact and Spreading Phenomena

• The Moving Contact Line Problem

• Conventional Approaches and

their Drawbacks

• The Shikhmurzaev Model

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The Moving Contact Line Problem

Liquid

Inviscid Gas

Contact line

d

Contact angle

Solid

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The Moving Contact Line Problem

No-SlipImpermeability

Kinematic condition Dynamic condition

Navier-StokesContinuity

d

Contact angle prescribed

• No solution!!!

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The Conventional Approach

These are treated separately by:

1) Modifying the no-slip condition near the Contact Line (CL) to allow slip, e.g.

2) Prescribing the Contact Angle as a function of various parameters, e.g.

y

u

u

One must:1) Allow a solution to be obtained.2) Describe the macroscopic contact angle.

cd u3

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Experiments Show This Is Wrong…

• Can one describe the contact angle as a function of the parameters?

“There is no universal expression to relate contact angle with contact line speed”.

(Bayer and Megaridis 06)

“There is no general correlation of the dynamic contact angle as a function of surface characteristics, droplet fluid and diameter and impact velocity.”

(Sikalo et al 02)

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As in Curtain Coating

Used to industrially coat materials.

Conventional models:

Fixed substrate speed => Unique contact angle

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‘Hydrodynamic Assist of Dynamic Wetting’

The contact angle depends on the flow field.

See: Blake et al 1994, Blake et al 1999, Clarke et al 2006

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Angle Also Dependent On The Geometry: Flow Through a Channel

• The contact angle is dependent on d and U.

(Ngan & Dussan 82)

U

U

d

d

Conclusion:

Angle is determined by the flow field

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The Shikhmuraev Model’s Predictions

Unlike conventional models:

• The contact angle is determined by the flow field.

• No stagnation region at the contact line.

• No infinite pressure at the contact line

=> Numerics easier

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Shikhmurzaev ModelWhat is it?

• Generalisation of the classical boundary conditions.

• Considers the interface as a thermodynamic system with mass, momentum and energy exchange with the bulk.

• Used to relieve paradoxes in modelling of capillary flows such as …..

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Some Previous Applications

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The Shikhmurzaev Model Qualitatively (Flow near the contact line)

lg

sl

Solid

Gas

Width of interfacial layer

Liquid

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Shikhmurzaev Model

• Solid-liquid and liquid-gas interfaces have an asymmetry of forces acting on them.

• In the continuum approximation the dynamics of the interfacial layer should be applied at a surface.

• Surface properties survive even when the interface's thickness is considered negligible.

Surface tension

Surface density

Surface velocity s

s

v

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Shikhmurzaev Model

231

2s221

s11

cos

0evev

d

ss

)u(v4)41(

)v(

n)vu(

0v

0)nnI(n

nnn

||||11

11s11

1

11s1

s1

1

1

s

se

ss

s

se

s

t

ft

f

P

P

Uv

)Uu(v

)v(

n)vu(

)Uu()nnI(n

2

2||||21

||2

22s22

2

22s2

||||221

s

s

se

ss

s

se

s

t

P

On liquid-solid interfaces:On free surfaces: At contact lines:

θd

e2

e1

n

n

f (r, t )=0

22,12,12,1 )( ss ba

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What if (the far field) ?

ed

22,12,12,1 )( s

ees

eeee ba

Uv

)Uu(v

Uu

)Uu()nnI(n

2

||||21

||2

||||

s

s

P On liquid-solid interfaces:On free surfaces: At contact lines:

see

s2,12,1

uv

uv

0u

0)nnI(n

nnn

1

||||1

1

s

s

e

ft

f

P

P

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Summary

• Classical Fluid Mechanics => No Solution

• Conventional Methods Are Fundamentally Flawed

• The Shikhmurzaev Model Should Be Investigated

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Our Approach

• Bulk: Incompressible Navier-Stokes equations

• Boundary: Conventional Model (for a start!)

• Use Finite Element Method.

• Assume axisymmetric motion (unlike below!).

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Numerical Approach

• Use the finite element method:

Velocity and Free Surface quadratic

Pressure Linear

• The ‘Spine Method’ is used to represent the free surface

• ~2000 elements

• Second order time integration

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The Spine Method(Scriven and co-workers)

The Spine

Nodes fixed on solid.

Nodes define free surface.

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Code Validation

• Consider large deformation oscillations of viscous liquid drops.

• Compare with results from previous investigations, Basaran 91 and Meradji 01.

)cos(1)( nn Pfr Microgravity Experiment

• Compare aspect ratio of drop as a function of time.

• Starting position is

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Second Harmonic – Large Deformation

For Re=100, f2 = 0.9

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Second Harmonic – Large Deformation (cont)

• Aspect ratio of the drop as a function of time.

• A damped wave.

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Fourth Harmonic – Large Deformation

For Re=100, f4 = 0.9

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Drop Impact on a Hydrophilic (Wettable) Substrate

Re=100, We=10, β = 100, .30s

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The Experiment – Water on GlassCourtesy of Dr A. Clarke (Kodak)

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Drop Impact on a Hydrophobic (non-wettable) Substrate

Re=100, We=10, β = 100, .120s

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The Experiment – Water on HydrophobeCourtesy of Dr A. Clarke (Kodak)

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High Speed Impact

Radius = 25 m, Impact Speed = 12.2 m/s

Re=345, We=51, β = 100, .67s

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Non-Spherical Drops on Hydrophobic Substrates

Radius = 1.75mm, Impact Speed = 0.4 m/s,Re=1435, We=8, .

175s

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Impact + Spreading of Non-Spherical Dropson Hydrophobic Substrates

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Impact + Spreading of Non-Spherical Dropson Hydrophobic Substrates

The Pyramid!

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Impact + Spreading of Non-Spherical Dropson Hydrophobic Substrates

Experiment shows pinch off of drops from the apex

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Impact + Spreading of Non-Spherical Dropson Hydrophobic Substrates

As in experiments, drop becomes toroidal

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Current Work

• Quantitatively compare results against experiment.

• Incorporate the Shikhmurzaev model.

• Consider variations in wettability ….

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How to Incorporate Variations in Wettability?

• Technologically, why are flows over patterned surfaces important?

• What are the issues with modelling such flows?

• How will a single change in wettability affect a flow?

• How about intermittent changes?

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Using Patterned Surfaces

• Manipulate free surface flows using unbalanced surface tension forces.

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Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces

• Pattern a surface with areas of differing wettability.

• ‘Corrects’ deposition.

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Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces

• Pattern a surface to ‘correct’ deposition.Courtesy of Professor Roisman

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The Problem• What if there is no free surface?

• Do variations in the wettability affect an adjacent flow?

1 2

2

Solid 1 Solid 2

What happens in this region?

Shear flow in the far fieldShear flow in the far field

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Molecular Dynamics Simulations

Courtesy of Professor N.V. Priezjev

More wettable CompressedMore wettable CompressedLess wettable RarefiedLess wettable Rarefied

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Hydrodynamic Modelling:Defining Wettability

• Defining wettability

ee 1lg1 cos

e1

lg

e1

• The Young equation:

The contact lineThe contact line

Solid 1

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Hydrodynamic Modelling:Which Model?

• No-Slip

No effect

• Slip Models (e.g. Navier Slip)

There is no theta!

• A Problem..

We have no tools!

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Qualitative Picture

e1 e2

BulkBulk

• Fluid particles are driven into areas of differing wettability.

• Surface properties take a finite time to relax to their new equilibrium state.

• What happens when flow drives fluid particles along the interface?

• Mass, momentum and energy exchange between surface and bulk.

• The process of interface formation.

0

2

Solid 1 Solid 2

• Consider region of interest.

Finite thicknes: For Visualisation Only

Finite thicknes: For Visualisation Only

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Interface Formation Equations – Hydrodynamic of Interfaces

• Surface density is related to surface tension:

,)0(ss

.2,1;

coslg0 iiess

ie

• Equilibrium surface density defines wettability:

• Surface possesses integral properties such as a surface tension, ; surface velocity, and surface density, . sv

s

Equation of

State

Equation of

State

Input of Wettability

Input of Wettability

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The Shikhmurzaev Model:Constant Wettability

Const se

s

BulkBulk

Interfacial Layer: For Visualisation Only. In the continuum limit..

Interfacial Layer: For Visualisation Only. In the continuum limit..

• If then we have Navier Slip

0n.u

t.utuun

u.t

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Solid-Liquid Boundary Conditions – Shikhmurzaev Equations

.2

tu2

1tv

,tu2

1uun

s

ses

e

s

ses

e

s

BulkBulk

tu

Tangential velocityTangential velocity

Surface

velocity

Surface

velocity

Solid facing side of interface: No-slip

Solid facing side of interface: No-slip

se

se

s

ntLayer is for VISUALISATION only. In the continuum limit…

Layer is for VISUALISATION only. In the continuum limit…

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Solid-Liquid Boundary Conditions – Shikhmurzaev Equations

.v

,nu

s

se

ss

se

s

BulkBulk

nutv s s

es sv

Continuity of surface mass

Continuity of surface mass

Normal velocityNormal velocity

Solid facing side of interface: Impermeability

Solid facing side of interface: Impermeability

Layer is for VISUALISATION only. In the continuum limit…

Layer is for VISUALISATION only. In the continuum limit…

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Problem Formulation

• 2D, steady flow of an incompressible, viscous, Newtonian fluid over a stationary flat solid surface (y=0), driven by a shear in the far field.

• Bulk– Navier Stokes equations:

)vu, (u

• Boundary Conditions– Shear flow in the far field, which, using

gives:

.uuu,0u 2 p

.as0, 22

yxvSy

u

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Results - Streamlines• Consider solid 1 (x<0) more wettable than solid 2 (x>0).

• Coupled, nonlinear PDEs were solved using the finite element method.

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Results – Different Solid Combinations

• Consider different solid combinations.

110,10:3

110,60:2

60,10:1

21

21

21

ee

ee

ee

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Results – Size of The Effect

eeJ 21 coscos

• Consider the normal flux out of the interface, per unit time, J.

• We find:

• The constant of proportionality is dependent on the fluid and the magnitude of the shear applied.

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Results - The Generators of Slip

• Variations in slip are mainly caused by variations in surface tension.

1) Deviation of shear stress on the interface from equilibrium.

2) Surface tension gradients.

1) Deviation of shear stress on the interface from equilibrium.

2) Surface tension gradients.

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Periodically Patterned Surface

• Consider Solid 1 More Wettable.

• Consider a=1 -> Strips Have Equal Width.

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Results - Streamlines

Solid 2 less wettableSolid 2 less wettable

Qualitative agreementQualitative agreement

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Results – Velocity Profiles

Tangential (slip) velocity varies around its equilibrium value of u=9.8.

Tangential (slip) velocity varies around its equilibrium value of u=9.8.

Fluxes are both in and out of the interfacial layer. Overall mass is conserved.

Fluxes are both in and out of the interfacial layer. Overall mass is conserved.

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Further Work

Compare results with molecular dynamics simulations.

Devise experiments to test predictions.

Fully investigate periodic case.

Single transition investigation is in:

Sprittles & Shikhmurzaev, Phys. Rev. E 76, 021602 (2007).

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Thanks!

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Numerical Analysis of Formula for J

Shapes are numerical results.

Lines represent predicted flux

Shapes are numerical results.

Lines represent predicted flux

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Interface Formation Equations + Input of Wettability

se2

se1

se

x

,.tanh2

1

2

11221 l

xse

se

se

se

se

Transition in wettability centred at x=y=0.

Transition in wettability centred at x=y=0.

Input of wettability

Input of wettability

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Surface Equation of State

)( :2

)( :1

0

2

ss

ss ba

Break-up Dynamicwetting

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Deviation of The Actual Contact Angle => Non Equilibrium Surface Tensions

Left: Curtain Coating Experiments (+) vs Theory (lines)Blake et al 1999 Wilson et al 2006

Right: Molecular Dynamics Koplik et al 1989

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Comparison of Theory With Experiment

0.0001 0.0010 0.0100 0.1000 1.0000

0

30

60

90

120

150

180

d

C a

0.0001 0.0010 0.0100 0.1000 1.0000

0

30

60

90

120

150

180

d

Ca

Perfect wetting (Hoffman 1975; Ström et al. 1990; Fermigier & Jenffer 1991)

Partial wetting (□: Hoffman 1975;

: Burley & Kennedy 1976; , ,: Ström et al. 1990)

It has been shown that the theory is in good agreementwith all experimental data published in the literature.

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Mechanism of Relaxation

s/P. 10-103 , 67

0.0 0.1 0.2 0.3

60

90

120

150

180d

C a

0.0 0.1 0.2 0.3 0.4 0.5

60

90

120

150

180

d

Ca

0.0 0.1 0.2 0.3 0.4

60

90

120

150

180

d

C a

Comparison of the theory with experimentson fluids with different viscosity (1.5-672 cP) confirms that the mechanism of the interface formation is diffusive in nature (J. Coll. Interface Sci. 253,196 (2002)). Estimates for parameters of the modelhave been obtained, in particular, showingthat for water-glycerol mixtures one has:

where