Surface Tension, Droplets, and Contact Lines

Lecture III

Boulder Condensed Matter Summer School 2015

Eric DufresneYale

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2015

Dr. Rob Style->Oxford

Elizabeth Jerison, Kate Jensen, Ye Xu, Ross Boltyanskiy, Larry Wilen, John Wettlaufer(c)

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Three Foundational Theories of Interfacial Mechanics

Eshelby (1957)

composites, fracture,

dislocations

JKR (1971)

adhesionwetting

Young-Dupre (18XX)

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Three Foundational Theories of Interfacial Mechanics

Eshelby (1957)

composites, fracture,

dislocations

JKR (1971)

adhesionwetting

Young-Dupre (18XX)

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0.3mm0.3mm

atomized glycerol

3 kPa silicone gel ( 50 𝜇𝜇𝜇𝜇)

glass (coverslip)

1.8 MPa silicone elastomer ( 50 𝜇𝜇𝜇𝜇)

glass (coverslip)

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Atomized spray of glycerol on a flat surface of a soft substrate with a

thickness/stiffness gradient

3 KPa

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Young-Dupre equation relates contact line geometry and material properties in equilibrium

Thomas Young1773-1829

cos𝜃𝜃 =𝛾𝛾𝑠𝑠𝑠𝑠 − 𝛾𝛾𝑠𝑠𝑙𝑙𝛾𝛾𝑙𝑙𝑠𝑠

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Glycerol drops on silicone (E=3kPa)Zygo surface profilometer

On a soft substrate, apparent contact angle depends on droplet size and thickness of soft layer

3um soft layer

38um soft layer

?

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Droplets Deform Soft Substrates

v

l

s

Yonglu Che

3kPa silicone

glass

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Droplets Deform Soft Substrates

v

l

s3kPa silicone

glass

𝛾𝛾𝑙𝑙𝑠𝑠

?

𝜎𝜎𝑧𝑧𝑧𝑧 ≈ −(2𝛾𝛾𝑙𝑙𝑠𝑠/𝑅𝑅)𝜃𝜃(𝑅𝑅 − 𝑟𝑟) + 𝛾𝛾𝑙𝑙𝑠𝑠𝛿𝛿(𝑟𝑟 − 𝑅𝑅)

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Elastic Theories Cannot Balance Contact Line Forces

Reasonable estimates for contact line width lead to unreasonable strains and displacements

-100 -50 0 50 100

0

0.5

1

1.5

u z[ ¹

m]

x [¹ m]position [microns]

posit

ion

[mic

rons

]

𝜎𝜎 = 𝜀𝜀𝜀𝜀 𝜎𝜎 = 𝛾𝛾/𝛿𝛿

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Profiles change dramatically with droplet size

0 50 100 150 200 250 300 350-10

-5

0

5

10

r [µm]

z [ µ

m]

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1) Align the peaks...

0 50 100 150 200 250 300 350-10

-5

0

5

10

r [µm]

z [ µ

m]

Overall profiles change – but how about the ridge?

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1) Align the peaks... 2) Rotate...

0 50 100 150 200 250 300 350-10

-5

0

5

10

r [µm]

z [ µ

m]

Overall profiles change – but how about the ridge?

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61 glycerol dropsradii: 18um - 1000um

Four different substrates: 13.5 - 50um thick

Ridge shape is universal near contact line

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93°

r [µm]

-6 -4 -2 0 2 4 6-2

-1

0

1

r [µm]

heig

ht [ µ

m]

149°

fluorinert fc-7014 drops

radii: 140um - 270umsubstrate: 23 um thick

glycerol61 drops

radii: 18um - 1000umsubstrates: 13.5 - 50um thick

Contact line geometry depends on the wetting fluid

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10 100 1000-50

-40

-30

-20

-10

0

10Ψ

[deg

rees

]

Droplet radius [µm]

h=14µmh=20µmh=30µmh=50µm

Cusp rotates as droplets get smaller

VL

VL VL(c)

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Macroscopic contact angle follows rotation of cusp

10 100 1000-50

-40

-30

-20

-10

0

10Ψ

[deg

rees

]

Droplet radius [µm]

40

50

60

70

80

90

100

θ [d

egre

es]

h=14µmh=20µmh=30µmh=50µm

100

90

80

70

60

50

40

𝜃𝜃 [degrees]

Angles between all three interfaces are fixed!

𝜃𝜃

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While apparent contact angle depends on boundary conditions… microscopic configuration of interfaces is universal

Glycerol

Silicone

Air128.2o

93.4o

Fluorinated oil

Silicone

Air

151o

149o

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Key Experimental Observations

• Young’s law doesn’t work on a 3KPa silicone substrate when glycerol droplets are smaller than about 100 microns. The apparent contact angle drops as the droplet gets smaller.

• For droplets much bigger than 100 microns, you get a size independent contact angle. This contact angle matches that of much stiffer silicone, of order 1MPa, about 90 degrees.

• Within two microns of the contact line, all the interfaces are straight and meet each other at fixed orientations. These orientations depend on the liquid.

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Glycerol

Silicone

Air128.2o

93.4oFluorinated oil

Silicone

Air

151o

149o

Υ𝑠𝑠𝑠𝑠 Υ𝑠𝑠𝑙𝑙

𝛾𝛾𝑙𝑙𝑠𝑠

Υ𝑠𝑠𝑠𝑠 Υ𝑠𝑠𝑙𝑙

𝛾𝛾𝑙𝑙𝑠𝑠

Hypothesis: geometry at contact line is determined by avector balance of interfacial stresses

𝛾𝛾𝑙𝑙𝑠𝑠: l-v surface tension Υ𝑠𝑠𝑠𝑠: s-v surface tension Υ𝑠𝑠𝑙𝑙: s-l surface tension

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Deformation of a linear elastic solid surface

𝑢𝑢𝑧𝑧 = 𝐴𝐴 sin 2𝜋𝜋𝜋𝜋/𝜆𝜆

Elastic restoring force: 𝜎𝜎𝐸𝐸 = 𝜀𝜀𝜀𝜀 ~ 𝐴𝐴𝜀𝜀/𝜆𝜆

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Flattening of a linear elastic solid by surface tension

𝑢𝑢𝑧𝑧 = 𝐴𝐴 sin 2𝜋𝜋𝜋𝜋/𝜆𝜆

Elastic restoring force: 𝜎𝜎𝐸𝐸 = 𝜀𝜀𝜀𝜀 ~ 𝐴𝐴𝜀𝜀/𝜆𝜆

Capillary force (LaPlace): 𝜎𝜎𝛾𝛾 ~ Υ𝐴𝐴/𝜆𝜆2

Υ: solid surface tension

Long Ajdari 1996, Jerison Dufresne 2011, Jagota 2012

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Balance of Elasticity and Capillarity Defines a Length scale

𝜎𝜎𝐸𝐸𝜎𝜎Υ

~𝜆𝜆Υ/𝜀𝜀

l = Υ/𝜀𝜀Elastocapillary Length

𝜆𝜆 ≫ Υ/𝜀𝜀 𝜆𝜆 ≪ Υ/𝜀𝜀

Elasticity Dominates Surface Tension Dominates

𝑢𝑢𝑧𝑧 = 𝐴𝐴 sin 2𝜋𝜋𝜋𝜋/𝜆𝜆

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Capillarity Dominates at Short Length Scales on Soft Materials

Υ = 0.03 N/m

Υ/𝜀𝜀 = 0.1 Å for 𝜀𝜀 = 3 GPa

Υ/𝜀𝜀 = 10 nm for 𝜀𝜀 = 3 MPa

Υ/𝜀𝜀 = 10 µm for 𝜀𝜀 = 3 KPa

𝑢𝑢𝑧𝑧 = 𝐴𝐴 sin 2𝜋𝜋𝜋𝜋/𝜆𝜆

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Linear Elasticity Plus Solid Surface Tension Captures Profiles

h=13.5 umR=216 um

h=19.5 umR=72 um

h=29.5 umR=25 um

h=50 umR=177 um

Sharp features near contact line are controlled by surface tension.

Far-field determined by elasticity

L V

S

L V

S

L V

S

L V

S

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Glycerol drops on silicone (E=3kPa)

3um soft layer

38um soft layer

Linear Elasticity Plus Solid Surface Tension Predicts Change in Apparent Contact Angle

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Υ/𝜀𝜀𝑅𝑅 ≪ 1 Υ/𝜀𝜀𝑅𝑅 ≫ 1

Υ/𝜀𝜀𝐸𝐸 ≫ 1

𝐸𝐸

Neumann(liquid on liquid)

Young-Dupre(liquid on rigid solid)

Wetting on Deformable Solids

𝛼𝛼𝛽𝛽

𝜃𝜃

𝛼𝛼𝛽𝛽

Theory and preliminary expts:Jerison et al Physical Review Letters 2011Style et al Soft Matter 2012

Breakdown of Young-DupreStyle et al Physical Review Letters 2013

Drop movementStyle et al PNAS 2013(c)

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Three Foundational Theories of Interfacial Mechanics

Eshelby (1957)

composites, fracture,

dislocations

JKR (1971)

adhesionwetting

Young-Dupre (18XX)

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Johnson, Kendall & Roberts (1971) – ‘JKR’

Substrate surface tension, Υ𝑠𝑠𝑠𝑠 ,Υ𝑠𝑠𝑝𝑝?

d

Adhesion Energy, 𝑊𝑊 = 𝛾𝛾𝑠𝑠𝑝𝑝 − 𝛾𝛾𝑠𝑠𝑠𝑠 − 𝛾𝛾𝑝𝑝𝑠𝑠

Substrate Elasticity, 𝜀𝜀

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Zero-force indentation for different stiffness substrates

Glass bead15 micron radius

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Glass bead15 micron radius

Zero-force indentation for different stiffness substrates

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Glass bead15 micron radius

Zero-force indentation for different stiffness substrates

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d, indentation

E=3kPa

E=85kPa

E=250kPa

E=500kPa

Glass beadd

Zero-force indentation for different stiffness substrates

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Comparing results with JKR

E=3kPa

E=85kPa

E=250kPa

E=500kPa

d,indentation

W~70mN/m

W~25mN/m

}Bead radius[um]

Inde

ntat

ion

[um

]

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JKR collapses the data, but scaling changes for small particles and soft surfaces

Bead radius x Young’s modulus [N/m]

Inde

ntat

ion

xYo

ung’

s m

odul

us [N

/m]

E=3kPaE=85kPaE=250kPaE=500kPa

𝐸𝐸~𝑅𝑅1/3

JKR:

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Bead radius x Young’s modulus [N/m]

Inde

ntat

ion

xYo

ung’

s m

odul

us [N

/m]

E=3kPaE=85kPaE=250kPaE=500kPa

Indentation proportional to bead radius for small beads

𝐸𝐸~𝑅𝑅

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LiquidSoft solid

Indentation, 𝐸𝐸~𝑅𝑅 implies constant contact angle

For small particles at zero force, the indentation depth is given by Young-Dupre, just like a colloidal particle on a fluid interface

Style et al Nature Communications (2013)Jensen et al …preprint to appear soon on arXiv(c)

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JKR plus surface tension fits data

Bead radius x Young’s modulus [N/m]

Inde

ntat

ion

xYo

ung’

s m

odul

us [N

/m]

E=3kPaE=85kPaE=250kPaE=500kPa JKR

modified JKRwith surface tension

Minimize total energy (a la Carillo/Raphael/Dobrynin 2010):

Elastic energy + Adhesion Energy + Surface Tension

(from JKR) Υ𝑠𝑠𝑠𝑠Δ𝐴𝐴

𝑊𝑊 = 71𝜇𝜇𝑚𝑚𝜇𝜇 , Υ𝑠𝑠𝑠𝑠 = 45

𝜇𝜇𝑚𝑚𝜇𝜇

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In the limit of small beads, our scaling predicts…

𝐸𝐸 = 𝑊𝑊𝑅𝑅/Υ𝑠𝑠𝑠𝑠

Υ𝑠𝑠𝑠𝑠 cos 𝜃𝜃 = 𝑊𝑊 − Υ𝑠𝑠𝑠𝑠

𝛾𝛾𝑠𝑠𝑠𝑠 cos 𝜃𝜃 = 𝛾𝛾𝑠𝑠𝑝𝑝 − 𝛾𝛾𝑝𝑝𝑠𝑠

𝐸𝐸equivalently…

for Υ𝑠𝑠𝑠𝑠 = 𝛾𝛾𝑠𝑠𝑠𝑠,

Young-Dupre with soft substrate in the place of the liquid

‘Smells’ like Young-Dupre

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Adhesion Summary• For large particles, Υ/𝜀𝜀𝑅𝑅 ≪ 1, the classic

balance of surface energy and elasticity by JKR accurately describes contact mechanics of soft substrate

• For small particles, Υ/𝜀𝜀𝑅𝑅 ≫ 1, the indentation depth is given by Young-Dupre, just like a colloidal particle on a fluid interface.

•Again, surface tension swamps elasticity for Υ/𝜀𝜀𝑅𝑅 ≫ 1

Style et al Nature Communications (2013)

Latest coming to the arXiv next week! (c) Er

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Three Foundational Theories of Interfacial Mechanics

Eshelby (1957)

composites, fracture,

dislocations

JKR (1971)

adhesionwetting

Young-Dupre (18XX)

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solid

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fluid(c) Er

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fluid

micron scale glycerol droplets in PDMS

0% 5% 10% 15% 20% v/v

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Micron-scale glycerol droplets soften 100KPa PDMS

Increasing glycerol

CompositeStiffness

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Micron-scale glycerol droplets stiffen 3 KPa PDMS

Elastic theory works for stiff matrices, but not for soft!

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QuickTime™ and aMotion JPEG OpenDML decompressor

are needed to see this picture.

Visualizing single inclusions

~MPa silicone sheet

Ionic liquid drops

3 or 100 KPaPDMS

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33µm

Visualizing single inclusions

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Elastic theory says drop shape should depend on strain…not size or stiffness

Linear elastic solid(Young’s modulus E)

Inclusion

(Eshelby, 1957)(c) Er

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Bigger droplets deform more than little onesSt

rain

6%

19%

29%

Droplet size2.5µm 6.4µm 16.5µm

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Scale-dependent deformation

l

w

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l

w

Scale-dependent deformation

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Far-field strain collapses droplet strain……and a length scale emerges!

Micro strain/Macro strain

Droplet Size(c) Er

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Microscopic response depends on size and stiffness

E = 100 KPaE = 3 KPa

Elastic theory says this response should be independent of size and stiffness

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Classic elastic theories ignore the interface

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More generally, surface tension creates a normal-stress jump across curved interfaces

total curvature: 𝜅𝜅 = 𝜕𝜕𝑖𝑖𝑛𝑛𝑖𝑖

(𝜎𝜎2−𝜎𝜎1) � �𝑛𝑛 = Υ12𝜅𝜅 �𝑛𝑛

Generalized Young-Laplace:

surface tension, Υ12: i.e. surface stress assumed to be isotropic

�𝑛𝑛

𝜎𝜎1𝜎𝜎2

Υ12

Υ12

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Surface tension can drive elastic deformation

Υ𝜅𝜅~𝜀𝜀𝜀𝜀

𝜀𝜀~𝜅𝜅 Υ/𝜀𝜀

Υ𝜅𝜅

𝜀𝜀

materialproperty

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𝜅𝜅Υ/𝜀𝜀 ≪ 1

Υ/𝜀𝜀𝑅𝑅 ≪ 1

bulk elasticity dominates

surface tension dominates

𝜅𝜅Υ/𝜀𝜀 ≫ 1

Υ/𝜀𝜀𝑅𝑅 ≫ 1

Microscopic response to macroscopic strain

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Eshelby with Surface Tension (analytic)

Increasing surface tension

macroscopic strain 0.3

strain independent surface tension, no shear stress at the interface

Υ/𝜀𝜀𝑅𝑅 = 10Υ/𝜀𝜀𝑅𝑅 = 0.1 Υ/𝜀𝜀𝑅𝑅 = 1

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Linear elastic theory with surface tension captures single droplet trend

Micro strain/Macro strain

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Surface tension ‘pulls back’ when the bulk solid attempts to deform embedded droplets

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Composite Stiffness Dilute Limit (Eshelby Method)

3 KPa100 KPa

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Composite Stiffness Dilute Limit (Eshelby Method)

2/3

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• When liquid inclusions are smaller than Υ/𝜀𝜀, surface tension dominates bulk elastic response

• In this limit, fluid inclusions stiffen soft solids

• Need to revisit applications of Eshelby, e.g. fracture mechanics

• More generally, surface tension dominates elastic response when 𝜅𝜅Υ/𝜀𝜀 ≫ 1

• References:• Experiment: Style et al Nature Physics 2015 • Theory: Style et al Soft Matter 2015

Inclusions in Soft Solids

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The Big PictureClassic theories of solid mechanics fail when 𝜅𝜅Υ/𝜀𝜀 ≳ 1

Soft solids can behave very differently than stiff ones.Implications for cellular biomechanics…

Many solid mechanics problems need to be revisited…(c) Er

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