surface tension, droplets, and contact lines...
TRANSCRIPT
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Surface Tension, Droplets, and Contact Lines
Lecture III
Boulder Condensed Matter Summer School 2015
Eric DufresneYale
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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
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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
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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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