liquid droplet dynamics: variations on a theme daniel m. anderson department of mathematical...

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


• the location of the free surface is part of the solution - surface waves in oceans, lakes

wind-driven waves canine-driven waves


• free surface with moving contact lines – LARGE SCALE: floods, lava flows (gravity)

Fluids Spreading on Solids


“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 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:


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


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)]


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


)(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


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


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


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


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


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


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)

liquid droplet dynamics: variations on a theme daniel m. anderson department of mathematical...

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