Post on 13-Jul-2020
transcript
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
$()*!"+,
-$(-
$(!.
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Simulating evaporating drops
Breakup of a rivulet:
Final state in horizontal plane
Experiment: Breakup of a rivulet
• Fluid rivulet beads up into drops
Fluid: PDMS; viscosity: 20 St; jet diameter: 0.046 cm;
Solid: glass coated by FC-725 or EGC-1700 - contact angle
57 – 63 degrees
Main features of instability• Instability develops on a long time scale (minutes,
or longer)
• Breakup propagates from the ends of the fluid
rivulet
• Capillary instability is not observed
• Different instability mechanism from infinite
rivulet case (Davis, JFM ’80, Langbein, JFM
’90, Roy and Schwartz JFM ’99, Yang and
Homsy, PoF ‘07)
• Number of drops and the time scale depend on
fluid and solid properties (viscosity, contact
angle, fluid volume)
Motivation• Numerous applications involving (often
unstable) nano-, micro- and macro- films and
rivulets
• Coatings, printers’ instability, lab-on-a-chip, electro-
chemically driven flows,…
• Typically, infinite fluid domains are considered;
however, nature usually deals with finite
objects…
• Obvious questions:
• Are finite effects important?
• If they are, how to understand them?
• In `real’ problems, which instability mechanism is
relevant?
Modeling assumptions
• Inertial effects can be ignored
• Capillary number is small
• Wetting effects can be modeled using disjoining
pressure model
• Lubrication approximation is appropriate
• (even for large contact angles)
Disjoining pressure
Fourth order nonlinear PDE for the fluid height:
Modify Laplace Young condition at the liquid/air
interface
To include solid-liquid interaction (Frumkin-Deryaguin
model)
where
p = !!" at z = h(x, y)
!!"" !!"!!(h)
!(h) =S
Nh!
!"h!h
#n
!"
h!h
#m$
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