Instabilities of films, drops and rivulets

Lou Kondic

Department of Mathematical Sciences, NJIT

Collaborators:Javier Diez, UNCPBA, ArgentinaNebo Murisic, NJIT/UCLAYehiel Gotkis, KLA-Tencor, San JosePhilip Rack, Oak Ridge National Lab

Overview • Examples of some applications involving thin

films and drops

• General physical and mathematical issues related

to thin film flows

• Review of flows involving contact lines

• Examples of various flow geometries involving

instabilities

• Problems with evaporation and thermal effects

• Breakup of finite size films and rivulets

• Interaction of different instability mechanisms

• Finite size effects

Numerous applications

• Microfluidics

• Electrowetting, flow control

• Surface cleaning, photolithographic applications

• Flows with particles: internal convection,

deposits

• Some examples of recent experiments showing

dynamics of thin films and drops

Experiments involving electrowetting

• Apply electric field and modify fluid wetting properties

(contact angle change)

F. Mugele's group at Twente University

Challenges

• Understand the details of the physics at the

contact line

• Extend to complex fluids

• Properly account for additional forces

(electrowetting, thermal effects, evaporation, ...)

• Bridge the scales: from micro to meso to macro

• Compute accurately the flow: multiscale problem

From Navier-Stokes to a single PDE

• Free surface flows are very difficult to address

due to continuously changing domain of interest

• Need to simplify the formulation as much as

possible

• Use the fact that the films are thin, fluids are

incompressible and Newtonian

• Ignore thermal effects and evaporation

• Assume simple models for fluid-solid

interaction

Assumptions

• Fluid is thin and all gradients are smalI

• Inertial effects can be ignored

• Capillary number is small

• No-slip boundary condition at liquid-solid

interface (to be discussed)

• Consider first completely wetting fluids; extend

later to partial wetting

!h

!t= ! 1

3µ" · ["h3""2h! #gh3"h cos $ + #gh3 sin$ i]

h: film heightµ: viscosity!: surface tension": densityg: gravity#: inclination angle

Thin film equation

capillarity out-of-plane

gravity

in-plane

gravity

Fourth order nonlinear PDE for the fluid height:

Scales

• scale fluid thickness by thickness far from the contact line,

• in-plane length scale:

• velocity scale:

• time scale:

• Capillary number:

Non-dimensional equation:

Ca = µU/!

l

Ut = l/U

H

!h

!t= !" ·

!h3""2h

"+ D(")" ·

!h3"h

"! !h3

!x

D(!) = (3Ca)1/3 cot !

Contact line singularity• Add precursor film

• spreading on a prewetted surface• introduces new length-scale• Dussan & Davis, JFM '74• de Gennes, RMP '85• Goodwin & Homsy, PoF A '91

• Relax no-slip boundary condition

• For the purpose of understanding macro-behavior,

the models are consistent• For the purpose of computing,

precursor model is more efficient

Computational methods

• Very demanding problem due to high degree, nonlinearity and presence of short scales (precursor)

• Efficiency, accuracy and stability are crucial even with state-of-the-art computers

• Finite difference methods

• ADI methods

• Eres etal PoF '00, Witelski Appl. Num. Math.'03

• Fully implicit methods

• Diez & Kondic PRL '01, PoF '02

• Spectral and pseudospectral methods

• Thiele etal PRE '01 etc

Implemented computational methods

• Implicit time discretization

• Time step: accuracy requirement

• Spatial linearization: Newton method

• Iterative biconjugate gradient solver for linear problem

• Second order in space and time• Details: Diez & Kondic, JCP '02

Examples(wetting fluid; zero contact angle)

• Flow down an incline

(Diez, Kondic, PRL 2001)

Combine with some in-house experiments....

Examples • Flow on patterned substrates

(Kondic, Diez, PoF 2006)

Some more in-house experiments

Or one can put it upside down...

Solitary waves? Work in progress...

Examples • Merge of drops

(Diez, Kondic, JCP 2002)

Flows that include thermal and phase change effects

• Motivation: surface cleaning in semiconductor

applications

• Evaporative effects lead to Marangoni forces

and influence particle deposition

• Related effect: instabilities of evaporating

water/alcohol mixtures

• Joint project with Y. Gotkis from KLA-Tencor

Modeling evaporation

• Include thermal and mass transport across

liquid-gas interface

• Include liquid-solid interaction

• Develop new models for evaporative flux

!"#

$"%

&

&

&

&

'

t

$()*!"+,

-$(-

$(!.

/ 0123.

///435#3.

/ 678

Simulating evaporating drops

Breakup of a rivulet:

Final state in horizontal plane

Experiment: Breakup of a rivulet

• Fluid rivulet beads up into drops

Fluid: PDMS; viscosity: 20 St; jet diameter: 0.046 cm;

Solid: glass coated by FC-725 or EGC-1700 - contact angle

57 – 63 degrees

Main features of instability• Instability develops on a long time scale (minutes,

or longer)

• Breakup propagates from the ends of the fluid

rivulet

• Capillary instability is not observed

• Different instability mechanism from infinite

rivulet case (Davis, JFM ’80, Langbein, JFM

’90, Roy and Schwartz JFM ’99, Yang and

Homsy, PoF ‘07)

• Number of drops and the time scale depend on

fluid and solid properties (viscosity, contact

angle, fluid volume)

Motivation• Numerous applications involving (often

unstable) nano-, micro- and macro- films and

rivulets

• Coatings, printers’ instability, lab-on-a-chip, electro-

chemically driven flows,…

• Typically, infinite fluid domains are considered;

however, nature usually deals with finite

objects…

• Obvious questions:

• Are finite effects important?

• If they are, how to understand them?

• In `real’ problems, which instability mechanism is

relevant?

Modeling assumptions

• Inertial effects can be ignored

• Capillary number is small

• Wetting effects can be modeled using disjoining

pressure model

• Lubrication approximation is appropriate

• (even for large contact angles)

Disjoining pressure

Fourth order nonlinear PDE for the fluid height:

Modify Laplace Young condition at the liquid/air

interface

To include solid-liquid interaction (Frumkin-Deryaguin

model)

where

p = !!" at z = h(x, y)

!!"" !!"!!(h)

!(h) =S

Nh!

!"h!h

#n

!"

h!h

#m$

S = !(1! cos "); N =(n!m)

(m! 1)(n! 1)

Thin film equation with disjoining pressure model

capillarity disjoining pressure

K =S

Nh!; g(!) =

n

!n! m

!m; ! =

h

h!

!h

!t+! ·

!h3!!2h

"+ K! ·

#h2g(")!h

$= 0

Approach:

• Consider first an infinite film: can linear stability analysis

predict instability properties?

(ignore finite size effects and transverse curvature)

• Next consider finite film and relate stability properties of

infinite and finite films

(ignore transverse curvature)

• Finally, bring back the transverse curvature into the

problem.

Linear stability analysis of 1D equation

• Consider infinite film: ignore end effects and transverse

curvature

Linear stability analysis (LSA) predicts existence of stable and unstable

regions in considered parameter space

Thiele etal PRE ’01, Colloids and Surfaces ’02;

Diez & Kondic, PoF ‘07 There is more…

h(x, t) = h0 exp(ıkx) exp(!t)

Absolute Stability and Metastability

Energy-based analysis shows different response to finite size

perturbations; even films that are linearly stable may be unstable with

respect to finite size perturbations: metastability

Important: emerging distance between the drops is different for the two

instability mechanisms

Thiele etal PRE ’01,

Colloids and Surfaces ’02;

Diez & Kondic, PoF ’07

Nucleation dominated

Surface dominated

Connection of finite and infinite cases

• Recall that experiment considers fluid

configuration of finite length

• What is the connection of the stability properties

of a finite and infinite film?

• Compute both and compare

2D (x,z) simulations of a finite film

Connection of finite and infinite film instabilities

(good news)

• Main conclusion: Finite size effects act as a finite-

amplitude perturbation of an infinite film

• The outcome is that resulting distance between the

drops may be significantly different for the finite and

infinite films

• Addition of surface perturbations of a finite film may

lead to change of regime from nucleation-dominated to

surface-dominated, if the perturbations have time to

grow: emerging lengthscales depend on the film size

(Diez & Kondic, PoF ’07)

Connection between instabilities of films and rivulets

(not so good news)

• Stability analysis of the films predicts stability for the

film thicknesses comparable to the experimental

rivulets

• Bring back the transverse curvature and consider a `real’

fluid rivulet

• Computational results show promising agreement with

the experiments, but all details still to be understood

3D Simulations: basic features

– Simulate only ! of the

domain

– Formation of drops

– Instability propagates

from the end(s)

3D Simulations and Experiment:

Large fluid volume: Small fluid volume:

Comparison of 2D and 3D simulations of finite films and rivulets with experiment

• Both 2D and 3D simulations show instability

propagating from the fluid ends (`end-pinching')

• 3D simulations are in good agreement with experiment

regarding number of drops produced

• 3D simulations show different range of instability

compared to 2D

• Dimensionality of the problem important! - transverse curvature present in the 3D simulations modifies the stability properties

Application: breakup of nano-scale liquid metal films

• Experiments: thin metal lines melted by laser-pulses

• Relevance: nano-scale patterning

with Rack, Guan, Fowlkes, Diez, submitted

Simulating nano-scale dewetting

• Choose appropriate

scales and physical

parameters

• Consider only fluid-

phase

• Ignore the details of

melting and

solidification

55 nm

13 nm

Predicting distance between nano-particles

• Computations allow to

predict the scaling of the

particle distance with line

thickness

• Scaling different from free

standing jets, or from fluid

films: experimental

verification pending

0 50 100 150 200 250

h (nm)

0

1000

2000

!m

(n

m)

LSA rivuletRayleigh jet (!

m

LSA flat film

Conclude• Asymptotic methods (lubrication theory) are

finding good application in thin film flows

• Good understanding of the experimental

results, and often quantitive agreement can be

reached

• Adding additional forces: gravity, thermal,

chemical, etc is relatively straightforward

• Future work will include more efficient

computations, improved lubrication

approximation, and more careful modeling of

liquid-solid interaction