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Liquid Droplet Dynamics: Variations on a Theme
Daniel M. Anderson
Department of Mathematical Sciences
George Mason University
Collaborators:• S.H. Davis, Northwestern University
• M.G. Worster, University of Cambridge• M.G. Forest, University of North Carolina• R. Superfine, University of North Carolina
• W.W. Schultz, University of Michigan• J. Siddique, George Mason University• E. Barreto, George Mason University
• B. Gluckman, George Mason University/Penn. State University
Supported by NASA (Microgravity Science), 3M Corporation and NSF (Applied Mathematics – DMS-0306996)
Free-Boundary Problems in Fluid Dynamics
• the location of the free surface is part of the solution - surface waves in oceans, lakes
wind-driven waves
Free-Boundary Problems in Fluid Dynamics
• the location of the free surface is part of the solution - surface waves in oceans, lakes
wind-driven waves canine-driven waves
Free-Boundary Problems in Fluid Dynamics
• free surface with moving contact lines – LARGE SCALE: floods, lava flows (gravity)
Fluids Spreading on Solids
Free-Boundary Problems in Fluid Dynamics
“About 2 million gallons of raw molasses burst from a storage tank at the corner of Foster and Commercial streets about noon on January 15, 1919. The black wave of the sticky substance was so powerful that it knocked buildings off their foundations and killed 21 people. Newspapers described the cleanup effort as nightmarish …”
• From The Boston Globe, May 28, 1996
The Great Molasses Flood – Boston, MA 1919
Free-Boundary Problems in Fluid Dynamics
• free surface with moving contact lines – SMALL SCALE: micro-fluidics, nano-fluidics (surface tension)
Fluids Spreading on Solids
1mm
• Isothermal Spreading Droplet (‘Plain vanilla’) [Greenspan, 1978]• Non-Isothermally Spreading Droplet [Ehrhard & Davis, 1991]• Migrating Droplet [Smith, 1995]• Evaporating Droplet [Anderson & Davis, 1995]• Freezing Droplet [Anderson, Worster, Davis, Schultz, 1996, 2000]• Melting Droplet [Anderson, Forest & Superfine, 2001]• Imbibing Droplet, Rigid Porous Substrate [Hocking & Davis, 2000]• Imbibing Droplet, Deformable Porous Substrate [Anderson, 2005]• Vibrating Droplet [Vukasinovic, Smith, Glezer, James, 2003, 2004]
Outline of Talk:
Spreading Droplet
Isothermal
Anatomy of a Spreading Droplet
Solid Boundary
Isothermal System
),( trhz
)(tar
)(t0V
• In the liquid: - Navier-Stokes Equations
• Free-Surface Conditions: - Normal and tangential stress balances - Mass balance (kinematic condition) • Conditions at the solid boundary: - velocity normal to interface is zero - ‘slip’ allowed in tangential velocity
• Contact-line conditions: - ‘contact’ (droplet height is zero) - condition on contact angle
Spreading Droplet: ‘Full’ Problem
z
uu
liquid
GOAL: Identify a physical regime that corresponds to experiments and allows isolation of important physical effects. Reduce mathematical model accordingly.
air
solid substrate
Thin Film Equations: Original Form
Thin Film Equations: Rescaled-Dimensionless Form
Thin Film Equations: Lubrication Theory Limit
• Slow flow (Re << 1) and slender geometry, zero gravity• Full problem reduces to an evolution equation for the interface shape
Isothermal Spreading Droplet: Lubrication Theory
C
0h
[Greenspan, 1978; Ehrhard & Davis, 1991, 1993; Haley & Miksis, 1991]
• symmetry conditions at • contact line conditions:
)(tar )(f
dt
da
0r
[Dussan V. 1979; Ehrhard & Davis, 1991, 1993]
01
3
12
22
1
r
h
rr
h
rrhh
rr
C
t
h
mR
mA
f
)(
0
)(
)(
R
AR
A
Capillary number
whereat
Isothermal Droplet Spreading
0C• Analytical formula for interface shape and contact line position
• large surface tension and
Droplet Evolution
m
dt
da
)13/(1
00
0
131)(
m
m
a
tm
a
ta
22
2),( ra
atrh
Spreading Droplet
Non-Isothermal
Anatomy of a Non-isothermally-Spreading Droplet
Hot (or Cold) Solid Boundary
Non-Isothermal System
),( trhz
)(tar
)(t0V
[Ehrhard & Davis, 1991]
• Slow flow, slender geometry, zero gravity, temperature-dependent surface tension• Full problem reduces to an evolution equation for the interface shape
Non-Isothermal Spreading Droplet: Lubrication Theory
M
[Ehrhard & Davis, 1991, 1993]
• quasi-steady temperature:
B
02
1
)1(
1
3
1122
22
r
hh
Bh
Mhr
r
h
rr
h
rrhh
rrt
hC
Marangoni number
at contact line
)(fdt
da0h
Bh
zhBT
1
)(1
• contact line conditions:
capillarity(surface tension)
unsteady term Marangoni effects(surface tension gradients)
Biot number (interface heat transfer)
• Thermocapillary forces on interface (Marangoni effects) drive a flow from warmer regions to colder regions (surface tension decreases with temperature).
• Spreading is enhanced when substrate is cooled.
• Spreading is retarded when substrate is heated.
• Experiments using paraffin oil and silicone oil spreading on glass confirm these predictions.
Non-Isothermal Spreading Droplet: Results[Ehrhard & Davis, 1991, 1993]
Migrating Droplet
Non-Isothermal
Anatomy of a Migrating Droplet[Smith, 1995]
),( txhz
)(tax L
Hot
Non-Isothermal System
)(tax R
)(tR0V
Cold
)(tL
• Slow flow, slender geometry, zero gravity, temp.-dep surface tension• Imposed temperature variation along solid boundary
• Full problem reduces to an evolution equation for the interface shape
Migrating Droplet: Lubrication Theory[Smith 1995]
0)1(1
ˆ
2
1
3
123
32
x
h
Bh
M
Bh
Nhh
x
hhh
xt
hC
at left and right contact lines
)( LL fdt
da
)( RR fdt
da
)( LL axx
h
)( RR axx
h
NxT 1
• contact line conditions:
capillarity(surface tension)
unsteady term Marangoni effects(surface tension gradients)
NOT SYMMETRIC!
• Droplet placed on a non-uniformly heated substrate migrates towards colder temperature region (for sufficiently large temperature gradients).
• Steady-state solutions include motionless drops and drops moving at constant speed (towards cooler regions).
• Thermocapillary-driven fluid flow in the drop distorts the free surface, modifies the apparent contact angle which in turn modifies contact line speed.
Migrating Droplet: Results[Smith, 1995]
Migrating Droplet: Results[Smith, 1995]
[video compliments of Marc Smith, 2006]
COLDHOT
Evaporating Droplet
Anatomy of an Evaporating Droplet
),( txhz
Heated Boundary
Non-isothermal System
)(t)(tV
)(tax
Evaporating Droplet• (Anderson & Davis, 1994; Hocking 1995). • Lubrication theory leads to an evolution equation
Evaporation number
0
31
221
3
22
2
x
h
hK
hhE
x
h
hK
hhMK
3
321
3
1
x
hhhC
xhK
E
t
h
capillarity(surface tension)
Marangoni effects(surface tension gradients)
vapor recoil
E
C
MK
evaporation (mass loss)
Nonequilibrium param.
Capillary number
Marangoni number
Scaled density ratio
Slip coefficient
Evaporating Droplet• [Anderson & Davis, 1994; Hocking 1995].
liquid volume is not constant in time (droplet vanishes in finite time)
)(
fK
E
dt
da
• Lubrication theory leads to an evolution equation
0x
0h )(tax
boundary conditionssymmetry at
0
31
221
3
22
2
x
h
hK
hhE
x
h
hK
hhMK
3
321
3
1
x
hhhC
xhK
E
t
h
)(
0
0ta
dxhK
E
t
h
at
x
h)(tax at
contact line condition
Evaporating Droplet• Small capillary number (large surface tension) [Anderson & Davis, 1994].
global mass balance
where
contact line condition)(
fK
E
dt
da
*1
*
2
tanh6)(
a
a
a
Ea
dt
ad
• Competition between spreading and evaporation
EVAPORATION EVENTUALLY WINS!
aK
aa22*
22
2),( xa
atxh
plus initial conditions
Evaporating Droplet
• contact line position recedes monotonically • contact angle increases initially and remains relatively constant
• strong evaporation, weak spreading
Evaporating Droplet
• contact line position advances initially • contact angle decreases monotonically and has a nearly constant intermediate region
• weak evaporation, strong spreading
• Evaporative effects are strongest near the contact-line region due to largest thermal gradients there.
• Effects that increase the contact angle retard evaporation - thermocapillarity: flow directed toward the colder droplet center - vapor recoil: nonuniform pressure (strongest at contact line) tends to contract the droplet
• Effects that decrease the contact angle promote evaporation - contact line spreading
Evaporating Droplet: Results[Anderson & Davis, 1995]
Freezing Droplet
Freezing Droplet
• This problem is motivated by the need to understand crystal growth problems and ‘containerless’ processing systems such as Czochralski growth, float-zone processing or surface melting.
• The common feature in these systems is the presence of a ‘tri-junction’ – where a liquid, its solid and a vapor phase meet – at which phase transformation occurs.
• Simple Model Problem: WHAT HAPPENS WHEN WE FREEZE A LIQUID DROPLET FROM BELOW ON A COLD SUBSTRATE?
Experimental Investigation (Water/Ice)
Initial, motionless, water droplet at room temperature
[Anderson, Worster & Davis (1996)]
Experimental Investigation (Water/Ice)
Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate)
Initial, motionless, water droplet at room temperature
[Anderson, Worster & Davis (1996)]
Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate)
?
Experimental Investigation (Water/Ice)
Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate)
Initial, motionless, water droplet at room temperature
[Anderson, Worster & Davis (1996)]
Cool bottom boundary (ethylene glycol – anti-freeze – pumped through channels in bottom plate)
Anatomy of a Freezing Droplet
)(tRx ),( trHz L
)(rHz S
Cold Boundary
Non-isothermal System
)(t)(tVL
0Rr )(tRx
),( trhz
Freezing Droplet
LLSS VVMass
• surface tension dominated liquid shape [Anderson, Worster & Davis, 1996].
Mass balance
Capillarity and gravity relate
Assume the solid – liquid interface is planar (1D heat conduction from cold boundary of temperature ; isothermal liquid at temperature )
Tri-junction condition (3 models)
Constant contact angle
‘Fixed’ contact line
Nonzero growth angle
L
S
,,RVL
MTCT
tL
TTcth CM )(
2)(
dt
dhR
dt
dVL 2
L
c
Latent heat
Thermal diffusivity
Specific heat
Freezing Droplet: Constant Contact Angle Model
0
• no inflexion points• solid shape is independent of growth rate
• Solidified Shape = Cone!
• Contact angle in liquid is constant
Droplet Evolution
020
0 13
)(R
r
R
VrH S
Freezing Droplet: Experimental Evidence
• Solidified silicon in crucible of e-beam evaporation system (Phil Adams, LSU, 2005)
Freezing Droplet: Fixed Contact Line Model
• no inflexion points• water/ice predicted to have zero slope at top• solid shape is independent of growth rate
• The tri-junction moves tangent to the liquid – vapor interface; the liqiud contact angle is free to vary
/
Solidified Shapes
concave down (zero slope)
concave down (nonzero slope)
concave up (nonzero slope)
dt
dh
dt
dR
tan
1
Freezing Droplet: Nonzero Growth Angle Model
• no inflexion point• all materials with nonzero growth angle have pointed top• solid shape is independent of growth rate
• The tri-junction moves at a fixed growth angle to the liquid – vapor interface (angle through vapor phase is )
/
Solidified Shapes
concave down (nonzero slope)
concave up (nonzero slope)
dt
dh
dt
dR
i )tan(
1
[Satunkin et al. (1980), Sanz (1986), Sanz et al. (1987)]
i
Freezing Droplet: Nonzero Growth Angle
92.0 1.0i
simulation
Experiment: ice
Freezing Droplet• A two-dimensional model for the thermal field in the solid was obtained by a boundary integral method [Schultz, Worster &
Anderson, 2000].
Freezing Droplet[Schultz, Worster & Anderson, 2000]
Results:
• both peaks and dimples can form at the top of the drop (depending on the growth angle and density ratio)• inflexion points are also possible
Melting Droplet
Melting Droplet:
cos
)(T
• thermal diffusion time ~ 10 – 25 seconds• data collapse if time is scaled with
• Motivated by experiments on polystyrene spheres (1mm radius) by D. Glick [UNC Physics Ph.D. 1998 – with R. Superfine]
t
Glick Contact Angle Data
)(
)()(
T
TRT eff
)(T
)(TReff
= viscosity (varies by 3 orders of magnitude in experiment)
= surface tension (varies by ~ 10%)
= ad hoc length scale, increases with temperature
99C
138C
Anatomy of a Melting Droplet
),( tzRr S
)(tRr
),( tzRr L
Hot Boundary
Non-isothermal System
liquid
solid
)(thz
)(tar
)(tVL
)(tVS
)(t)(t
)(t
Melting Droplet: Model
hRa ,,
- initially spherical solid- no gravity- surface tension dominates – quasi-steady liquid vapor interface- solid-liquid interface assumed planar
[Anderson, Forest & Superfine, 2001]
Liquid Shape – Spherical 2
222 )2
tan()2
(sec),(
azatzRL
pVV LS ,, ,,Nine Unknown Functions of Time
lengths angles volumes and pressure
Melting Droplet: Model
)sin(20 2 RpR
)/( tzTT
)(fdt
da
[Anderson, Forest & Superfine, 2001]
Thermal Problem:
)()( tVtVM LLSS
Geometry:
Differential-Algebraic System: solved by DASSL code [Brenan, Campbell, Petzold, 1995]
Motion of Solid:
Mass Balance:
Contact-line Dynamics:
tth ~)(
1D thermal diffusion, planar solid-liquid interface
Balance of forces – equation of motion for solid
Provides five relations between lengths, angles and volumes
Melting Droplet (medium )
characteristic contact-line speed
Melting Droplet Dynamics
TK
TK
TK
measures competition between
spreading and melting TK
small : dynamics similar to isothermal spreading
large : dynamics deviate from isothermal spreading
characteristic melting speed
Melting Droplet Dynamics• contact angle relaxes faster in spreading/melting configuration• results do not collapse with rescaling of time
• contact line is less mobile in spreading/melting configuration• spreading promotes melting
TK
cos
time
increases
TK
)(ta
time
increases
Melting and Freezing Droplet
Melting and Freezing Droplet Dynamics
Imbibing Droplet
Rigid Porous Substrate
Anatomy of a Droplet Imbibing into a Rigid Porous Substrate
),( txHz L
)(tRx
Isothermal System
liquid
wet/rigid porous material
)(tax )(tVL )(t
)(thz l
0z
dry porous material
[Hocking & Davis, 1999, 2000]
• slender limit (lubrication theory)• imbibition is one-dimensional – liquid penetrates vertically only – no radial capillarity. The porous base is assumed to be made up of vertical pores.
Imbibing Droplet: Rigid Porous Substrate
0)(1
3
32
x
HHH
xPt
h
t
H LLL
S
lL
[Hocking & Davis, 1999, 2000]
LH
l
l
ht
h 1
lh
SP
Evolution equations for liquid shape and penetration depth
porous-base modified slip coefficient
porosity suction parameter
1D capillary suction flow
Imbibing Droplet: Rigid Porous Substrate
)(4
3),( 22
30
0 xaa
VtxH L
[Hocking & Davis, 1999, 2000]
tV
aaa 2
3
4
0
3022
0
thl 2
)(4
3 2230
0
0
xaa
Vhl
ax
0axa
‘central region’ solution
Contact angle cannot be written as a single-valued function of the contact line speed – in contrast to ‘regular’ spreading.
Imbibing Droplet
Deformable Porous Substrate
Imbibing Droplet: Deformable Porous Substrate
• Motivation and Applications: - swelling of paper/print film in inkjet printing - soil science - infiltration - medical science (flows in soft tissue)
Modeling Assumptions
- adopt the simplest description of fluid drop (Hocking & Davis, 2000) - assume 1D imbibition and 1D substrate deformation (Preziosi et al. 1996, Barry & Aldis, 1992,1993) - porous material is initially dry with uniform solid fraction - no gravity
Anatomy of a Droplet Imbibing into a Deformable Substrate
),( trHz L
)(tRr
)(thz s
Isothermal System
liquid
wet/deformable porous material
)(tar )(tVL )(t
)(thz l
0z
dry porous material
Imbibing Droplet: Deformable Porous Substrate
0
z
w
ts
0)1(
lwzt
z
pKww sl
)1(
)(
Equations in wet/deforming porous material [Preziosi et al. 1996]
combine into single PDE for solid fraction
mass conservation for solid and liquid
modified Darcy Eq.
stress equilibrium
zz
p
)(
0
z
K
zztrc
t
)(')(),(0
sw
lw
p
)(K
)(
),(0 trc
solid fraction
liquid velocity
solid velocity
liquid pressure
permeability
solid stress
liquid viscosityrelated to boundary values of solid fraction
Imbibing Droplet: Deformable Porous Substrate
Boundary Conditions
Interior: similarity solution
th ll ~ th ss ~Exterior: numerical solution
Interior
Exterior
l
r zero stress
),( trhz s
),( trhz lat
at
l
0z
),( trhz l
),( trhz s
ls ww no puddles,
no dry-out
at
at
Imbibing Droplet: Deformable Porous Substrate
initial and eventual
swelling initial swelling
eventually undeformed initial swelling
eventual compression
Hocking & Davis
model for liquid droplet
Deformable Substrate: Sponge Problem
water dropped onto an initially dry and compressed sponge (photos by E. Barreto and B. Gluckman)
Deformable Substrate: 1D Sponge Problem With Gravity
Ask Javed!!
Capillary-rise of a liquid into a deformable porous material.
-- How does this compare to the case of a rigid porous material?
-- Does the liquid rise to an equilibrium height?
-- How much deformation occurs?
Vibrating Droplet
Vibrating Droplet: Droplet Atomization[James, Vukasinovic, Smith, Glezer (J. Fluid Mech. 476, 2003)][Vukasinovic, Smith, Glezer (Phys. Fluids, 16, 2004)]
videos compliments of Marc Smith, 2006
0.1 ml water, frequency 1050 HzAmplitude increases linearly in timefield of view (12.5mm X 6mm), 500 frames/sec
From JVSG: ``During droplet ejection, the effective mass of the drop—diaphragm system decreases and the resonant frequency increases. If the initial forcing frequency is above the resonant frequency of the system, droplet ejection causes the system to move closer to resonance, which in turn causes more vigorous vibration and faster droplet ejection. This ultimately leads to drop bursting.’’
• Reactive Spreading [Braun et al., 1995; Warren, Boettinger & Roosen, 1998]• Motion and Arrest of a Molten Droplet [Schiaffino & Sonin, 1997]• Evaporating and Migrating Droplet [Huntley & Smith, 1996]• Spreading of Hanging Droplets [Ehrhard, 1994]
Other Droplet Work:• Isothermal Spreading [Hocking, 1992; de Gennes, 1985; Dussan V. & Davis, 1974; Shikhmurzaev, 1997; Thompson & Robbins, 1989; Koplik & Banavar, 1995; Bertozzi et al. 1998, Barenblatt et al. 1997, Jacqmin, 2000]• Evaporating Drops [Hocking, 1995; Morris, 1997, 2003, 2004; Ajaev, 2005]• Freezing Drops [Ajaev & Davis, 2003]
AND MANY OTHERS!!!
Summary
• The ‘plain vanilla’ droplet spreading problem and its multiple variations lead to interesting scientific, experimental, mathematical modeling and computational problems in the general class of free-boundary problems in fluid mechanics and materials science.
• There are lots of variations still to explore!
The End
• This work has been supported by - National Aeronautics and Space Administration (NASA) Microgravity Science and Application Program - 3M Corporation - National Science Foundation (Applied Mathematics Program, DMS-0306996)