Capillary Interactions between Anisotropic Particles
Kathleen J Stebe Chemical and Biomolecular Engineering
University of Pennsylvania
Acknowledgements
• Eric Lewandowski-experiment, analysis• Marcello Cavallaro-curvature gradients, confinement• Lorenzo Botto-simulation, analysis • Valeria Garbin-(Crocker)-interferometry• Lu Yao-crowded surfaces, registry, repulsion• Jorge Bernate-surface evolver• Alice Tseng- environmental SEM
MRSEC facilities at JHU/PENN; NSF
On the attraction of floating particles W. A. Gifford and L. E. Scriven Chem. Eng. Sci. 1971, 26, 287.
Received 8 August 1970; accepted 17 August 1970. Capillary attraction between floating particles, a phenomenon of everyday experience as well as technological importance, is caused by interfacial tension and buoyancy forces ...
2pgr
Bo
Finite Bo: sphere or infinite cylinder• Nicolson Proc Camb Phil Soc 45 (1949)• Gifford and Scriven Chem. Eng. Sci., 26, 287 (1971)• Chan et al J. Colloid Interface Sci. 79 (1981)• Singh and Joseph Journal Fluid Mech. 530 (2005) sphere or disk
Slope Area
2( )
2
2
2
2
p A B
A
p A B
S
p Ao B
S
Aop z
h h h h h h
E h h dA
E h h ds
E h h ds
hE F
n
n
Particles move in potential energy gradient created by their neighbor (or by a boundary)
Like beads on a strong- slide to low potential energy site
Nicolson 1949; Chan et al 1981
Bubbles, cheerios, froth flotation, ..
Finite weight particles
Small slope; superposn
1 ~2LV plane excessA
h hA dA A A
22 small
( ) ln( ) o capcap
gah Bo h Bo
rh aK b A r r l
l
0?Bo
Particles at free-surfaces• Particle-stabilized emulsions Ramsden (1904); Pickering (1907)• Bubbles Nicolson Proc Camb Phil Soc 45 (1949)• Current interest using microparticles: BINKS Special edition PCCP 2008
• Froth flotation Gifford and L. E. Scriven (1971) Chan et al (1981) Singh and Joseph (2005)
Capillary interactions-thin films
Kralchevsky, Nagayama and collaborators 1990s-Wasan: argues not formed by capillary attraction
nuclei
Negligible Bond number- capillary interactionsLucassen, Colloids and Surfaces 1992Stamou et al: long range qp deflections Phys. Rev. E 2000Dietrich, Oettel, and collaborators-ellipsoids Kralchevsky and collaborators-weakly non-spherical shapes Binks and collaborators
Ellipsoids Hilgenfeldt Europhys.Lett. 72, 671 (2005) Loudet* et al. Phys. Rev. Lett. 2005, 2006, 2009 Lehle et al Eur Phys Lett 2008 Vermant, Fuller, Furst-assembly and rheology
Complex shapes Whitesides: Bowden et al. Functionalized mm-particles Science 1997, Langmuir 2001+~20 more Rennie (2000); Fournier (2002): bi-metal microparticles- form qp Lewandowski et al Langmuir 2008, Soft Matter 2009
Cheerios effect
Whitten, Deegan, Dupont: coffee rings...
Interfacial deflections created by particle
0ds tStamou, Duschl, Johannsmann, PRE 2000Kralchevsky et al Langmuir 2001
2
2
2
; L
2
,
0
c p
p
H gh
recast in non d form r
H Bo h
grBo
small slope small Bond number
h
01
( , ) ln cos( )kk k
k
h r A r A r k
2 range 22
cos(2long
Ah
r
Quadrupolar deflection: long range perturbation
Stamou, PRE 62, 2000
Far field interactions
Interaction Energy
Force of Attraction
Excess area drives interactions but no preferred orientation
r12
1212
5
2 11 2
12
1 2
48 cos 2( )
excess
pp p
dAF
dr
rH r
r
1 ~2LV plane excessS
h hA dS A A
4
212 1 2
12
12 cos 2( ) pexcess p
rE A H
r
Stamou, PRE 62, 2000
Stamou
2
2
2 :A
excess
AB A B A
C
B A
curvature weightedtensor of integralB at A particle A's
deformation
h hA dA
A h h d
h
n
Superposition approx.
Floating poppy seed ~1mm (Hinsch, 82)
interferograms
Micro-Cylinders: Stebe lab
Undulated contact lines: pronounced for non-spherical particles
Rennie: curved particles
Micro-Ellipsoids: Loudet and Yodh . 2005, 2006
Lithographic Fabrication of Particles
SU-8 photoresistSilicon Wafer
SU-8 photoresist
Silicon Wafer
MaskUV light
Expose resist through mask
Develop photoresist
SU-8 Particles
Silicon Wafer
Sonicate in EtOH to free particles
Cylinders at fluid interfaces: Two mechanically stable states
Preferred orientation: GCPCompare SgiAi for each state
Side OnEnd On
2pgr
Bo
negligible
Orientation of partially wet cylinders
( )1; 1P L
L
Bo Bo
2 1
sin
Lx
r
22 sinrL r
Side On End On
Analytical assume
-Flat interface along cylindrical body -Ends fully wet or de-wet - Neglect excess L/V area
Minimum surface energy - Surface Evolver, contact angle
L/V interface approximation- Equate holes in interface
Neglect Gravity
x=1.2, q=80o,r=3.5mm x=0.2, q=110o,r=150nm x=1.3, q=80o, r=3.5mm x=2.8, q=110o,r=150nm
Phase diagram
Lewandowski, et al JPC B 2006; Langmuir in press
End-to-end chaining of cylinders
Lewandowski et al, Soft Matter, 5, 2009
12 ~ 180initr m
L/D ~ 2.5
Undulated contact line owing to particle shape
50μm
Shape of interface around isolated cylinder
Environmental SEM
Minimum surface energy configuration
Interface topology satisfying contact angle not unique Surface evolver simulation, const P, Neumann conditions far field
=80o
Interferometric Measurement of Interface:
V. Garbin, J. Crocker, interferometry
Far field: Quadrupolar Attraction
ELLIPSOIDS: C. Loudet et al, PRL, 018301, 2005
1212
5 1212
512 12
6
~
~
drag d cyl
drF F C R
dtdr
rdt
dt r dr
12
12
12
1
6( )
cr C t t
E r r
12 cr C t t
Extract magnitude of far field interaction energy
5
5
6 ( ') ' 2.16 0.65 10
0.6 2.24 0.67 10
D
ER x kT
C
E x kT
i
f
rDrag
r
Drag
v r dr
Viscous dissipation CD=1.73 for L=3 Heiss and CoullYoungren and Acrivos
Cylinder~ 60% immersed
Capillary interaction energy
predicted12, 12,
22 4 5
2 4 4
( / 1) 1 112 1 0.985x10
( / 1)f i
p
L DE H R kT
L D r r
Asymptotic exp
Isoheight contours around cylinder
Divide deformation field into 2 domains:
exterior: elliptical quadrupolar deformation: 2-3 radii outside of ellipse circumscribing cylinder
– (very) far field: cylindrical polar qp
Elliptical quadrupolar deformation
excess area map
near field: large area concentration at ends
2
2
a L
b D
L:2R=5
Dynamic simulation and experiment
Experiment
Simulation
1
6
nn n
T
t Ex x
Rf x
138
nn n
R
t E
R f
Trajectory computed as:
(used experimentally measured drag coeffs ft & fr)
Time (secs)
Angl
e
Black line: simulationColored symbols: experiment
φA
φB
φA+φB
Not in real time (slowed down X4)
Rotation: very local; decays steeply
Lewandowski et al Langmuir 2010
Quadrupoles in Elliptical Coordinates: End-to-end until nr contact
Ellipsoids: Loudet
Our analysis (ellip qps):
Tip-to-tip preferred for separations >major axis
Side-to-side preferred for separations < major axis
EQP not full story
Vermant
Quadrupoles in Elliptical Coordinates: Side-to-Side on close approach
Charged?
uncharged
Interface near contact
cylinder 2cylinder 1
capillary bridge
gradient magnitude
in-plane bending
Lorenzo Botto, KJS, in prep
Critical torque and yielding
critical bending moment should break chain
T>Tc
Constant torque experiment
cylinder should snap to side-to-side
strain softening
yield torque Tc
f (strain)
(stress)
PREDICTIONS:
Surfactant Mediated Arrest and Recovery of Capillary Interactions
Nguyen et al. PRL 1992, 4, 419
PDA on pH 2: Insoluble Surfactant
Brewster Angle Microscopy
1. PDA creates a tangential immobile surface
2. NaOH deprotonates PDA (increased solubility)
3. SU-8 rods form ordered assemblies
Lewandowski et al Soft Matter 2009
Magnet integrated into chain
With Yao LU; w R LEHENY, unpublished
“Polygonal” networks
Microstructure: rod-like particles
“Bamboo ”
Rectangular arrays
water-in-oil emulsion drop
cylinders ellipsoids
“Wormy ” chains- Jan VermantPrivate commun
vs. sphero-cylinders no deformation no interactions
Other shapes: Fourier modes
• Lucassen Colloids and Surfaces 65, 1992
– Interaction between sinusoidal contact lines
– liquid-vapor surface area minimized
FrequencyAmplitudeIn phase
– Particles end face registryParticle Recognition
f
= 0f
Complex Shapes: Registry
Far field interactionsQuadrupolar in nature b = -3.75
Complex Shapes: Registry
sin particle n t
Interfacial deflections around cylinder
~curved curved flat flath L h L
2excess S
h hA dS
1/2
~flat EXCESScurve
curve EXCESSflat
L A
L A
Aspect ratio dictates preferred location: shortest face preferred
t
n
Steepest slope always on shortest face
Preferred alignment
Preferred location is shortest face
~flat LVcurve
curve LVflat
L A
L A
1
1
flat
curve
EXCESS curve
EXCESS flat
L
L
A
A
Curved side to curved side
Flat side to flat side
1
1
flat
curve
EXCESS curve
EXCESS flat
L
L
A
A
4flat
curve
L
L
0.66flat
curve
L
L
Surface Evolver Results: Confirm Slope Argument
1
0.662.04.0
Steepest slope always on shortest face
h
d
Cylinder alignment on curved interfaces
Gla
ss W
alls
Gla
ss W
alls
Langmuir 2008
Cylinder alignment on curved interfaces
cos sinLVcyl LV LV
dAT C
d
cos 24
LV LV o
CA A
Torque
Two mechanical equilibria:
0 perpendicular to wall
/2 parallel to wallALV for a quadrupole on curved interface in
small slope limit
gALV depends on cylinder alignment
2
2cos 2
LVd AC
d
Stable state: depends on sign of C
24 pp
HC r
R
in agreement with experiment
Alignment of ‘biscotti’ shaped particles
A saddle on a saddle
/E kT
Alignment as a function of particle size
-Background curvature 103 times particle radiusLewandowski et al. Langmuir 2008
Rparticles=3.5 m
Strongcurvature
11 2 3
4 5 6
weakcurvature
1 2 3
Cylinder assembly on curved interfaces
0
1 1( ) :
2 2p cm z xy pE h F h S T x Π
0 ( ) :p pE h x Π
In summary,the particle contribution to the total energy is
This form reveals a structure that is very familiar in the study of electrostatics.
•The force "interacts" with the height,• the torque "interacts" with the slope, •The quadrupole moment "interacts" with the curvature tensor.
The first term is leading order for heavy isotropic particles, and corresponds to that derived by Nicholson.
The second term is important for anisotropic particles acted upon by an external torque. For an anisotropic force- and torque-free particle, the first two terms are identically zero and the particle contribution to the energy becomes
Botto
Conclusions• Ellipsoids vs. cylinders Cylinders: hierachy of interactions- elliptical quadrupolar/near field
• Chaining: cemented by near field interactions
• Preferred orientation: f(aspect ratio of particle)
• Curvature gradients: Motion and alignment
• Complex shapes: Registry of end-face features
Cylinders on water drop in oil
Current work: • complex particles• repulsion• crowded surfaces-gels• docking sites• mechanics of assemblies
• scale
shapes with corners
Open issues: gels, networks, rheology, dense packings
on a water drop in oilCharged Ellipsoids-percolating networks-open flower like structures-elastic, brittle interfaces
Jan Vermant, Gerry Fuller
Cylinders-rectangular lattices-ropes of chains-open networks
at air water interface, spread, compressedto collapse
at air water interface, with DPPC spread, compressed
at water-decane interface, -becomes denser with time
compression isotherms, rheology, role of charge
on a water drop in oil
Other shapes
Far field: cylindrical polar quadrupolar mode
Extract Fourier modes from numerical solution:
2
2
0( , ) cos 2p
rH h r d
r ~ 9Rcyl
r > 9Rcyl
At r ~ 9Rcyl , higher modes 5% contribution
4 6L
R
2
L
R
Rate of approach: far field
L
Varied Aspect Ratio
Faster approach as increases:consistent with Hp increase
Fixed Aspect Ratio
(t-tc)(tc-t)
Interactions of elliptical quadrupoles vs. r12
Solid line ends at tip-to-tip contact
end-to-end alignment favored
r12/R
Torque enforces end to end alignment
12 2r L
Steric effects
Steric effects imposed byanisotropic hard core repulsion
Potential= Ellip Quadrupoles+Repulsion preventing contact
2/ 2/
1 0
0
x yF
a b
superellipses cylinder
surface of revolution
=0.2
Asymptotics of interaction energy
4
122
22
1/
1/112
r
R
DL
DLHE
psidetoside
Expansion in powers of 1/r12 :
4
122
22
1/
1/112
r
R
DL
DLHE pendtoend
Torque decays faster (as 1/r6) than force (1/r5)
Torque has strong aspect ratio (=L/D) dependence
L/D
612/1~~ r
ETorque
6
122
2
2
22 1
1/
1/140
r
R
D
L
DL
DLH
p
6
122
2
2
22 1
1/
1/140
r
R
D
L
DL
DLH
p
2 2 3
22
6
12
~( 1)
180 eH
TR
Rr
Anisotropic pair potential
(Langmuir 2010)
End-to-end favored until tip-to-tip contact
Tip contact
after contact before contact
104 kT
2 2 710pH ~ R ~ kT