[PPT] - Behavior of a Droplet on a Thin Liquid Film Ellen R. Peterson PowerPoint Presentation

SLIDE 1

Behavior of a Droplet on a Thin Liquid Film

Ellen R. Peterson

Carnegie Mellon University Department of Mathematical Sciences Center for Nonlinear Analysis

February 23, 2012 Funded by National Science Foundation

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 1 / 35

SLIDE 2

Introduction

Project Motivation

Goal: Improve treatment for Cystic Fibrosis

use surfactant to spread medicine over mucus in lungs

Research Group: CMU Physics: Steve Garoff, Fan Gao CMU Chemical Engineering: Todd Przybycien, Robert Tilton, Roomi Kalita, Ramankur Sharma, Amsul Khanal

U. Pitt Medical Center: Tim Corcoran

Questions:

How does the fluid spread?
Where is the surfactant?
How do multiple droplets interact?

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 2 / 35

SLIDE 3

Introduction

Background on Spreading

substrate r fluid D z H H D <<1 Drop spreading on a solid substrate: Multivalued velocity profile-precursor layer:

Dussan, Davis (1974) Bertozzi (1998) Glasner (2003)

Coarsening of droplets:

Otto, Rump, Slepcev (2005) Witelski, Glasner (2003)

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 3 / 35

SLIDE 4

Surfactant

Background on Spreading Surfactant

Surfactant spreading on liquid subphase: Model: Gaver and Grotberg (1990) ht + 1

r

1

2rh2σ(Γ)r

r = β 1

r

1

3rh3hr

r − κ 1

r

1

3rh3 1 r (rhr)r

r
r

Γt + 1

r (rhΓσ(Γ)r)r = β 1 r

1

2rh2Γhr

r − κ 1

r

1

2rh2Γ

1

r (rhr)r

r
r + δ 1

r (rΓr)r

Asymptotic spreading behavior/similarity solutions: Jensen, Grotberg (1992), Jensen (1994) ht − 1 r 1

2rh2Γr

r = 0

Γt − 1 r (rhΓΓr)r = 0 Similarity scaling: h(r, t) = H(ξ), Γ(r, t) = 1 t

1 2

G(ξ), ξ = r t

1 4

Similarity solutions: H(ξ) = 2ξ2, G(ξ) = −1 8 log ξ, ξ = r t

1 4 Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 4 / 35

SLIDE 5

Surfactant

Numerical similarity solutions: ERP, Shearer (2012)

Equations: ht − ξ ˙ r0(t) r0(t) hξ = 1 r0(t)2

ξ 1

2h2Γξ

ξ

Γt − ξ˙ r0(t) r0(t) Γξ = 1 r0(t)2 (ξhΓΓξ)ξ ˙ r0 = − 1 r0 h(1, t)Γξ(1, t) ξ = r r0(t) Solutions: h(r, t) = H(ξ), Γ(r, t) = 1 t

1 2

G(ξ), ξ = r r0(t) H(ξ) = 2ξ2, G(ξ) = −1 8 log ξ, ξ = r t

1 4

0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2

r/r0(t) h

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4

r/r0(t) Γ

H G ξ ξ 1000<t<10000

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 5 / 35

SLIDE 6

Surfactant

Inner Solution: ERP, Shearer (2011)

h(r, t) = t− 2

9

H0 +

1 18G0 r t

1 6

2 + A r t

1 9

2 + O(r 4) Γ(r, t) = t− 4

9

G0 + µ

H0 r t

1 6

2 + O(r 4)

0.1 0.2 0.3 0.4 0.02 0.04 0.06 0.08 0.1 0.12

η h(η,t)

0.2 0.4 0.6 0.55 0.6 0.65

η t4/9Γ(η,t)

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 6 / 35

SLIDE 7

Surfactant

Experimental Goals

Experimental spreading: Bull et al (1999), Fallest et al (2010) Visualize the height of the film (using laser line) Visualize the location of the surfactant (fluorescence) Match the experimental data to the PDE model

20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

r [mm] (a) t = 1 sec 20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

r [mm] (b) t = 2 sec 20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

r [mm] (c) t = 3 sec 20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

r [mm] (d) t = 6 sec ring

Karen Daniels’s Lab NC State University

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 7 / 35

SLIDE 8

Surfactant

Spreading Behavior

10 10

1

10

1

10

2

r0 [mm]

λ=0.25

20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

r [mm] (b) t = 2 sec

tracking

10

−1

10 10

1

10

1

102

t [sec]

rM [mm] λ=0.295

20 40 1 2

H [mm]

20 40 2 4

Γ [μg/cm2]

(b) t = 2 sec

tracking

r [mm] t [sec]

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 8 / 35

SLIDE 9

Surfactant

Surfactant Initial Condition

Using time scale t = t∗ :

2 4 6 0.5 1 1.5

r h

2 4 6 0.5 1 1.5 2

r Γ

2 4 6 0.5 1 1.5 2

r Γ

2 4 6 0.5 1 1.5 2

r Γ

2 4 6 0.5 1 1.5

r h

2 4 6 0.5 1 r Γ 2 4 6 0.5 1 1.5

h r

2 4 6 0.2 0.4 0.6

Γ

2 4 6 0.2 0.4 0.6 0.8 1

r Γ

2 4 6 0.5 1 1.5

r h r

Surfactant at t=9 Height at t=9 Initial Conditions

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 9 / 35

SLIDE 10

Spreading Droplet

How do we predict the spreading behavior?

Spreading Parameter: (Harkins 1952) S = ΣF − ΣDF − ΣD

Complete spreading (S > 0)
Spreading on a Deep Layer:

Dipietro, Huh, & Cox, 1978; DiPietro & Cox, 1980; Foda & Cox, 1980

Spreading Power Law:

Fraaije & Cazabat, 1989

Lens-shape (S < 0)
Numerical axisymmetric solution,

static lens : Pujado & Scriven, 1971

Σ Σ Σ

D DF F

S>0 S<0

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SLIDE 11

Spreading Droplet

Assumptions

Newtonian drop on Newtonian fluid
Fluids immiscible
Lubrication Approximation H

D << 1

S < 0
axisymmetric spreading

Issue: Contact Line Force Balance

substrate r fluid drop D z H r

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 11 / 35

SLIDE 12

Spreading Droplet

Deriving Droplet Spreading Equations

pD

r = µDuD zz,

pDF

r

= µDFuDF

zz ,

pF

r = µF uF zz

pD

z = −ρDg,

pDF

z

= −ρDFg, pF

z = −ρFg

pD(a, t) = patm − ΣDarr, pDF(b, t) = pD(b, t) − ΣDF brr, pF(h, t) = patm − ΣF hrr

u =0

F

u =0

DF

u =0

F z DF

u =uD u =0

D z

μ u =μ u

D D DF DF z z

h(r,t) a(r,t) b(r,t)

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SLIDE 13

Spreading Droplet

Droplet Spreading Equations - axisymmetric

ht = 1 r

r
Bohr −

1 r (rhr)r

r
h3
r

bt = 1 r

r
b3
Bo(1 − β)br − σDF

1 r (rbr)r

r
+
Boβar − σD

1 r (rar)r

r

b2 2 (3a − b)

r

at = 1 r

r
Bo(1 − β)br − σDF

1 r (rbr)r

r

b2 2 (3a − b) + 1 − λ λ (a − b)3 + a3 Boβar − σD 1 r (rar)r

r
r

Bo: Bond number β: density ratio λ: viscosity ratio σD: surface tension of drop and air σDF: surface tension of drop and fluid related work: Kriegsmann & Miksis 2003

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 13 / 35

SLIDE 14

Equilibrium Solutions

Bohr − 1 r (rhr)r

r

= 0 Boβar − σD 1 r (rar)r

r

= 0 Bo(1 − β)br − σDF 1 r (rbr)r

r

= 0. If β = 1 these equations are Bessel equations. We’ll assume β < 1 h(r) = Ch + ChhI0 √ Bor

+ ChhhK0

√ Bor

a(r)

= Ca + CaI0

Boβ

σD r

+ CaaaK0
Boβ

σD r

b(r)

= Cb + CbbI0

Bo(1 − β)

σDF r

+ CbbbK0
Bo(1 − β)

σDF r

Related studies: Kriegsmann & Miksis (2003), Pujado & Scriven (1971)

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SLIDE 15

Equilibrium Solutions

Boundary Conditions

Edge Conditions: ar(0, t) = 0, br(0, t) = 0, hr(L, t) = 0 Continuity of Interface: a(r0, t) = h(r0, t), b(r0, t) = h(r0, t) Continuity of Pressure:

pDF = pF|r=r0
∆h − σD∆a − σDF∆b = 0
r=r0

Height of Film: 2π r0 b(r, t)rdr + 2π L

r0

h(r, t)rdr = MassF Location of r0(t): MassD = 2π r0 (a(r) − b(r))r dr h(r,t) a(r,t) b(r,t) r (t)

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 15 / 35

SLIDE 16

Equilibrium Solutions

Boundary Conditions

Balance of Surface Tension Forces: ΣF cos θF + ΣD cos θD + ΣDF cos θDF = 0 ΣF sin θF + ΣD sin θD + ΣDF sin θDF = 0 Apply approximations: θF ≈ εhr, θD ≈ −εar, θDF ≈ −εbr

ΣF

1 + ε2h2

r

2 + . . .

− ΣD
1 + ε2a2

r

2 + . . .

− ΣDF
1 + ε2b2

r

2 + . . .

= 0

ΣF

εhr + ε3h3

r

3 + . . .

− ΣD
εar + ε3a3

r

3 + . . .

− ΣDF
εbr + ε3b3

r

3 + . . .

= 0

O(1) : 1 − σD − σDF = 0 = ⇒ S ∼ O(ε2) O(ε) : hr − σDar − σDFbr = 0 O(ε2) : h2

r −σDa2 r −σDFb2 r = −2 θF θ

D

2π-θDF Σ Σ Σ

D F DF

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 16 / 35

SLIDE 17

Equilibrium Solutions

Applying Boundary Conditions

Apply boundary conditions at edges of domain: ar(0, t) = 0 br(0, t) = 0 hr(L, t) = 0 Resulting Equations: h(r) = Ch + ChhhK0 √ Bor

a(r)

= Ca + CaaI0

Boβ

σD r

b(r)

= Cb + CbbI0

Bo(1 − β)

σDF r

Find coefficients using remaining boundary conditions and conservation of

mass of droplet: V = 2π r0 (a(r) − b(r))r dr

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 17 / 35

SLIDE 18

Equilibrium Solutions

Profiles - fix S, vary β

Bo = 0.1, σD = 0.7, S = −0.1

5 10 0.5 1 1.5

r h

5 10 0.5 1 1.5

h

5 10 0.5 1 1.5

r h

β=0.1 β=0.7 β=3.0

r =2.014 r =2.059 r =2.066

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SLIDE 19

Equilibrium Solutions

Profiles - fix β, vary S

Bo = 0.1, σD = 0.7, β = 0.5

5 10 0.5 1 1.5 2 2.5

r h

5 10 0.5 1 1.5 2 2.5

r h

5 10 0.5 1 1.5 2 2.5

r h

S= -0.001 S= -0.01 S= -0.1 S= -1

r =3.013 r =2.062 r =1.394

5 10 0.5 1 1.5 2 2.5

r h

r =4.494

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SLIDE 20

Experiment

Evidence of a Lens - Fan Gao

Water on Polyacrylimide
Lens remains for over 10 minutes

camera microsope lens pipet drop substrate ronchi ruling light source pipette

Fan Gao: CMU Physics

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SLIDE 21

Experiment

Experimental Systems

Substrate Drop S See Lens?? Glycerol (75%)/water (25%) Hexadecane 7.53 no Glycerol(10%)/water (90%) Hexadecane −3.71 yes Glycerol(75%)/water (25%) Oleic Acid 20.18 yes Water Oleic Acid 32.25 yes Σ Σ Σ

D DF F

S=Σ

F Σ D Σ DF

Interface Interface Type Surface Tension (dynes/cm) Glycerol (75%)/Air ΣF 63.63 Glycerol (10%)/Air ΣF 71.5 Water/Air ΣF 72.75 Hexadecane/Air ΣD 26.18 Oleic acid/Air ΣD 27.34 Hexadecane/Glycerol (75%) ΣDF 29.92 Hexadecane/Glycerol (10%) ΣDF 45.61 Oleic acid/Glycerol (75%) ΣDF 16.11 Oleic acid/Water ΣDF 13.16

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 21 / 35

SLIDE 22

Experiment

Autophobing

Is fluid escaping the lens? Will the lens spread on its own monolayer?

lens Wilhelmy pin monolayer ??

On SOLID surface: Langmuir (1917) - first suggested the mechanism of autophobing Zisman (1964) - adsorption of solution outside the bulk Frank & Garoff (1995) - advancement and retraction of spreading drop

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SLIDE 23

Experiment

Hexadecane on glycerol (75%)

ΣF: Glycerol/Air: 63.63 dyne/cm ΣD:Hexadecane/Air: 26.18 dyne/cm ΣDF: Hexadecane/Glycerol: 29.92 dyne/cm S = 7.53> 0

ΔΣ

No Lens!!

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 23 / 35

SLIDE 24

Experiment

Hexadecane on glycerol (10%)

ΣF: Glycerol/Air: 65.76 dyne/cm ΣD:Hexadecane/Air: 26.18 dyne/cm ΣDF: Hexadecane/Glycerol: 43.29 dyne/cm S = −3.71< 0

145

5

65 220 110

ΔΣ s

Lens!!

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 24 / 35

SLIDE 25

Experiment

Oleic acid on glycerol (75%)

ΣF: Glycerol/Air: 63.63 dyne/cm ΣD: Oleic acid/Air: 27.34 dyne/cm ΣDF: Oleic acid/Glycerol: 16.11 dyne/cm S = 20.18> 0 (1)

ΔΣ

Lens!!

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SLIDE 26

Experiment

Oleic acid on water

ΣF: Water/Air: 72.75 dyne/cm ΣD: Oleic acid/Air: 27.34 dyne/cm ΣDF: Oleic acid/Water: 13.16 dyne/cm S = 32.25> 0 (2)

ΔΣ

Lens!!

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 26 / 35

SLIDE 27

Experiment

Comparison of spread area

System: Oleic acid spreading on Glycerol (75%)

Original spreading parameter:

S = 63.63 − 27.34 − 16.11 = 20.18

Allow surface tension of substrate (ΣF ) to be altered by escaping

monolayer: ΣF = 38.11 dyne/cm S = 38.11 − 27.34 − 16.11 = −5.34 For the equilibrium solution corresponding to this altered system, find the spread area: 2r0 = 1.91 − 2.22 mm (Theory) 2r0 ≈ 1.6 − 1.72 mm (Experiment)

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 27 / 35

SLIDE 28

Experiment

Discussion of Thin Film Approximation

Recall the solution form: h(r) = Ch + ChhI0 √ Bor

+ ChhhK0

√ Bor

a(r)

= Ca + CaI0

Boβ

σD r

+ CaaaK0
Boβ

σD r

b(r)

= Cb + CbbI0

Bo(1 − β)

σDF r

+ CbbbK0
Bo(1 − β)

σDF r

where

Bond = ρF gL2

ΣF

β = ρD

ρF

σD = ΣD

ΣF

σDF = ΣDF

ΣF

Additional Restriction: S ∼ O(ε2)

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SLIDE 29

Experiment

Varying S comparison

0.5 1 1.5 2 0.8 1 1.2 1.4 1.6 1.8 2

r (mm) h (mm)

0.5 1 1.5 2 0.8 1 1.2 1.4 1.6 1.8 2

r (mm) h (mm)

0.5 1 1.5 2 0.8 1 1.2 1.4 1.6 1.8 2

r (mm) h(mm)

Blue: Thin Film Approximation Red: Full Curvature, Pujado & Scriven r =1.588 r =1.586 r =0.765 r =0.778 r =1.066 r =1.042 S=-0.01 S=-0.1 S=-1

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 29 / 35

SLIDE 30

Type II Equilibrium Solutions

What happens if drop touches bottom surface?

Type II

Equations: h(r) = Ch + ChhI0 √ Bor

+ ChhhK0

√ Bor

a(r)

= Ca + CaI0

Boβ

σD r

+ CaaaK0
Boβ

σD r

b(r)

= Cb + CbbI0

Bo(1 − β)

σDF r

+ CbbbK0
Bo(1 − β)

σDF r

New Force Balance Conditions where drop touches bottom surface

(r = ¯ X): Cb + CbbI0

Bo(1 − β)

σDF ¯ X

+ CbbbK0
Bo(1 − β)

σDF ¯ X

= 0

br( ¯ X) =

−2T

σDF

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SLIDE 31

Type II Equilibrium Solutions

Spreading Coefficients

2 4 6 8 10 0.2 0.4 0.6 0.8

r h

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8

r h

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8

r h

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8

r h

T=-0.000001 T=-0.0001 T=-0.01 T=-0.09 S=-0.096

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SLIDE 32

Type II Equilibrium Solutions

Mass comparison

500 1000 1500 2000 2500 3000 3500 50 100 150 200 250 300

MassF MassD

β=.4 β=.6 β=.7 β=1 β=1.2 β=1.4 0.005 0.01 0.015 0.02 0.04 0.06 0.08

MassD

Y X

0.2 0.4 0.6 0.8 1 0.5 1 1.5

X MassD

0.02 0.04 0.06 0.08 0.1 0.005 0.01 0.015

Type II Type I

Type I Type II Type I Type II

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SLIDE 33

Type II Equilibrium Solutions

Planar Drop

Only consider x ≥ 0, solution symmetric

Type I solution: h(x, t) = Ch + Chhe−

√ Bondx

a(x, t) = Ca + Caacosh

βBond

σD x

b(x, t) = Cb + Cbbcosh
(1 − β)Bond

σDF x

Type II solution:

h(x, t) = Ch + Chhe−

√ Bondx

a(x, t) = Ca + Caacosh

βBond

σD x

b(x, t) = Cb + Cbe−
(1−β)Bond

σDF

x + Cbbe

(1−β)Bond

σDF

x

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SLIDE 34

Type II Equilibrium Solutions

Mass Comparison

5 10 15 20 25 1 2 3 4 5 6 7

MassF MassD

β=0.2 β=0.4 β=0.6 β=0.8

Type I

5 10 15 20 25 1 2 3 4 5 6 7

MassF MassD

β=0.2 β=0.4 β=0.6 β=0.8

Type II

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 34 / 35

SLIDE 35

Future Work

Summary

Conclusions: Found equilibrium solutions for droplet on thin film Explored drop touching bottom surface Compared with experiment to confirm mechanism of autophobing Future (short term) Work: Stability analysis of equilibrium solution Comparison between thin film approximation and thicker film solution Spreading behavior of drop Compare experimental shape of the drop with mathematical predictions Future (long term) Work: Non-Newtonian underlying fluid Surfactant-laden droplet

Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 35 / 35