Swansea 2007

Some Aspects of Drops Impacting on Solid Surfaces

Some Aspects of Drops Impacting on Solid Surfaces

J.E Sprittles

Y.D. Shikhmurzaev

EFMC7

Manchester 2008

Motivation

• Drop impact and spreading occurs in many industrial processes.

• 100 million inkjet printers sold yearly.

• Recent experiments show standard models of drop impact and spreading will be inadequate for microdrops.

Photos courtesy of Romain Rioboo

The Moving Contact Line Problem

dθU

Fails to provide a solution for a drop spreading steadily over a solid substrate.

The classical recipe of:

Bulk

Incompressible Navier Stokes

Solid

No-Slip

Free Surface

Stress balance with capillary pressure.

Flow Of Liquids Over Solids

Two Issues:– Allow For A Solution

– Describe The Angle Between The Free Surface and the Solid (The Contact Angle).

dθ U

Standard Solution:– Allow Slip Between

Solid and Liquid

– Let dθ = ( U )f

Standard Modelling of Axisymmetric Drop Impact and Spreading

• Bulk:

Incompressible Navier-Stokes

• Boundary:

Classical conditions apart from no-slip being replaced by Navier Slip:

• Contact Angle

Related to contact line speed using empirical relation (Jiang et al 79)

u.t n.P.t

Slip Tangential Stress

Finite Element Modelling:The Spine Method (Scriven and co-workers)

The Spine

Nodes fixed on solid.

Nodes define free surface.

Q) Does The Standard Model Work?Pyramidal Drops (Millimetre Size)

Experiment Renardy et al. Simulation Sprittles.

A) Qualitatively OKQuantitatively Not

ExperimentallyPrediction of

Standard Model

Standard Recipe’s Problems:

• Incorrect Kinematics

• Logarithmic Pressure Singularity at the Contact Line.

• But perhaps worst of all….

dθU

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

dθ ( U )f

U, m/s

dθ

Experiments answer, NO!

“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)

“There is no universal expression to relate contact angle with contact line speed”. (Bayer and Megaridis 06)

As in Curtain Coating (Used To Industrially Coat Materials)

Standard models:

Fixed Substrate Speed => Unique Contact Angle

U, cm/s

dθ

dθ

U

Dynamic contact angle as a function of coating speed for different flow rates (Blake & Shikhmurzaev 02).

Liquid Drops Spreading on Solids:Process of Interface Formation

lg

slSolid

Gas

Liquid

In Frame Moving With Drop

Interfaces are shown with finite thickness for representation

only.

The Interface Formation 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

Simplest Model of Interface Formation

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

s s1 1 1 2 2 2

1 3 2

v e v e 0

cos

s s

d

2

* 12 || ||2

s 2 22

s 2 222 2

12|| || || 2 22

21,2 1,2 1,2

u 1u 0, u u u

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

(u v ) n

( v )

v (u U ) , v U

( )

s se

s sss e

s s

s s

pt

t

a b

In the bulk:

On liquid-solid interfaces:On free surfaces:

At contact lines:

θd

e2

e1

n

n

f (r, t )=0 • Generalisation of standard/classical model

Qualitative Results:IFM In Frame Moving With Contact Line

lg

sl

Solid Speed U

Gas

Liquid

Qualitative Results:IFM In Frame Moving With Contact Line

lg

sl

Solid

Gas

Liquid

Solid Speed U

• Increasing impact speed• Changes flow field

Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces

• Pattern a surface to ‘correct’ deposition.

Courtesy of Professor Roisman

Chemically Patterned Surfaces

• Flow over a transition between solids of differing wettabilities.

• Standard model predicts no effect.

1 2

2

Solid 1 Solid 2

What happens in this region?

Shear flow in the far fieldShear flow in the far field

Molecular Dynamics Simulations

Courtesy of Professor N.V. Priezjev

More wettable CompressedMore wettable CompressedLess wettable RarefiedLess wettable Rarefied

Results - Streamlines

Solid 2 less wettableSolid 2 less wettable

Qualitative agreementQualitative agreement

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

Sprittles & Shikhmurzaev, EPJ (2008), In Print

Thanks

Drop Impact on a Hydrophobic (non-wettable) Substrate

Re=100, We=10, β = 100, .120s

Q) Does The Standard Model Work? Rebound on a Hydrophobic Substrate

Q) Does The Standard Model Work? Impact of a Microdrop

Radius = 25 m, Impact Speed = 12.2 m/sRe=345, We=51, β = 100, .67s

Experiment Dong 06. Simulation Sprittles.

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

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 flow.