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 ...
Bo 
 grp

2
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
Finite weight particles
 ga 2
Bo 
small

 h  Bo h
2
h  aK o (
ALV   1 
A
r
lcap
)  b  A ln(r ) r  lcap
h  h
dA ~ Aplane  Aexcess
2
Small slope; superposn
h  h  (hh)  h 2 h
Ep 
Ep 

2
  hA  hB  dA
Particles move in potential energy gradient
created by their neighbor (or by a boundary)
A

 h h
2
A
 n  ds
B
S
E p  hAo

 h

2
B
 n  ds
Like beads on a strong- slide to low potential
energy site
Nicolson 1949; Chan et al 1981
S
hAo
Ep  
Fz
2
Bubbles, cheerios, froth flotation, ..
Bo  0?
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)
Cheerios effect
Capillary interactions-thin films
Kralchevsky, Nagayama and collaborators 1990sWasan: argues not formed by capillary attraction
nuclei
Whitten, Deegan, Dupont: coffee rings...
Negligible Bond number- capillary interactions
Lucassen, Colloids and Surfaces 1992
Stamou et al: long range qp deflections Phys. Rev. E 2000
Dietrich, 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 microparticlesform qp
Lewandowski et al Langmuir 2008, Soft Matter 2009
Interfacial deflections created by particle
2 H    gh
recast in non  d form; L c  rp
Stamou, PRE 62, 2000
Quadrupolar deflection:
long range perturbation
2 H  Bo h
Bo 
 grp 2

small slope, small Bond number
2h  0

A 
hlong range   22   cos(2   2 
r 
h(r , )  A0 ln r   Ak r  k cos(k   k )
k 1
  tds  0
Stamou, Duschl, Johannsmann, PRE 2000
Kralchevsky et al Langmuir 2001
Far field interactions
h h
Stamou, PRE 62, 2000
A   1
dS ~ A
2
LV
Aexcess  
AAB
plane
S
h h
dA
2
 2  hAhB  n A d
r12
Superposition approx.
CA
 2hB :
curvature
tensor of
B at A
Interaction Energy
Force of Attraction
 Aexcess
A
weighted
integral
particle A's
deformation
Stamou
E12   Aexcess
F12  
 rp 
2
 12 H p cos 2(1  2 )  
 r12 
4
dAexcess
dr12
r
2
1  p 
 48 H p cos 2(1  2 )rp  
 r12 
1  2
5
Excess area drives interactions
but no preferred orientation
Undulated contact lines: pronounced for non-spherical particles
interferograms Micro-Ellipsoids: Loudet
and Yodh . 2005, 2006
Floating poppy seed ~1mm
(Hinsch, 82)
Rennie: curved particles
Micro-Cylinders: Stebe lab
Lithographic Fabrication of Particles
SU-8 photoresist
Silicon Wafer
UV light
Mask
SU-8 photoresist
Expose resist
through mask
Silicon Wafer
Develop
photoresist
SU-8 Particles
Silicon Wafer
Sonicate in EtOH to free particles
Cylinders at fluid interfaces:
Two mechanically stable states
End On
Bo 
 grp 2

negligible
Preferred orientation: GCP
Compare SiAi for each state
Side On
Orientation of partially wet cylinders
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
2rL sin    r 2
2L
1
x

 r sin 
Neglect Gravity
Bo  1;
(P  L )
L
Bo  1
Phase diagram
Lewandowski, et al JPC B 2006;
Langmuir in press
x=1.2, =80o,r=3.5mm
x=0.2, =110o,r=150nm
x=1.3, =80o, r=3.5mm
x=2.8, =110o,r=150nm
End-to-end chaining of cylinders
Undulated contact line owing to particle shape
r12init ~ 180mm
L/D ~ 2.5
50μm
Lewandowski et al, Soft Matter, 5, 2009
Shape of interface around isolated cylinder
=80o
Interface topology satisfying contact angle not unique
Surface evolver simulation, const P, Neumann conditions far field
Minimum surface energy configuration
Environmental SEM
Interferometric Measurement of Interface:
V. Garbin, J. Crocker,
interferometry
Far field: Quadrupolar Attraction
r12  C  t  tc 

F12   Fdrag
dr12
r12 ~
dt
dt ~ r125 dr12
5
dr12
 Cd 6 Rcyl m
dt
r12  C  t  tc 


1

 
1
2



6
E (r12 )  r  

ELLIPSOIDS: C. Loudet et al, PRL, 018301, 2005
Extract magnitude of far field interaction energy
Viscous dissipation
i
E Drag
 6m R  v (r ')dr '  2.16  0.65 x105 kT
CD
rf
r
0.6E Drag  2.24  0.67 x 105 kT
CD=1.73 for L=3
Heiss and Coull
Youngren and Acrivos
Cylinder~ 60% immersed
Capillary interaction energy
 ( L / D  1)2  4  1
1 
5


E  12 H 1 
R



0.985x10
kT

2
4
4


 ( L / D  1)   r12, f r12,i 
Asymptotic exp
2
p
predicted
Isoheight
contours
around
cylinder
a  2L
Divide deformation field into 2
domains:
exterior: elliptical quadrupolar
deformation: 2-3 radii outside of
ellipse circumscribing cylinder
– (very) far field: cylindrical polar qp
b  2D
excess
area
map
Elliptical
quadrupolar
deformation
L:2R=5
near field: large area
concentration at ends
Quadrupoles in Elliptical Coordinates: End-to-end until nr contact
Trajectory computed as:
t  E 
x 


6m RfT  x 
n
φA
n

n 1
t
 E 
 


8m R3 f R   
n
φA+φB
n
(used experimentally measured drag coeffs ft & fr)
Angle
x
n 1
Black line: simulation
Colored symbols: experiment
φB
Dynamic simulation and experiment
Time (secs)
Simulation
Experiment
Rotation: very local; decays
steeply
Not in real time (slowed down X4)
Lewandowski et al Langmuir 2010
Quadrupoles in Elliptical Coordinates: Side-to-Side on close
approach
Charged?
Ellipsoids: Loudet
Vermant
uncharged
Our analysis (ellip qps):
Tip-to-tip preferred for separations >major axis
Side-to-side preferred for separations < major axis
EQP not full
story
Interface near contact
gradient magnitude
cylinder 1
cylinder 2
in-plane bending
capillary bridge
Lorenzo Botto, KJS, in prep
Critical torque and yielding
PREDICTIONS:
yield torque Tc
Constant torque experiment
T>Tc
strain softening
(stress)
critical bending moment
should break chain
f (strain)
cylinder should snap to side-to-side
Surfactant Mediated Arrest
and Recovery of Capillary Interactions
PDA on pH 2: Insoluble Surfactant
Brewster Angle
Microscopy
Nguyen et al. PRL 1992, 4, 419
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
Microstructure: rod-like particles
cylinders
ellipsoids
“Polygonal” networks
“Bamboo ”
“Wormy ” chains- Jan Vermant
Private commun
vs. sphero-cylinders
Rectangular arrays
no deformation
no interactions
Other shapes: Fourier modes
• Lucassen
Colloids and Surfaces 65, 1992
– Interaction between
sinusoidal contact lines
– liquid-vapor surface area
minimized
f
Frequency
Amplitude
In phase
– Particles end face registry
Particle Recognition
f 0
Complex Shapes: Registry
Far field interactions
Quadrupolar in nature
b = -3.75
Complex Shapes: Registry
Interfacial deflections around cylinder
n
t
nparticle  t  sin 
h   h
dS
S
2
Aexcess  
1/2
 L flat 


L
 curve 
hcurved Lcurved ~ h flat L flat
Steepest slope always on shortest face
AEXCESScurve
~
AEXCESSflat
Aspect ratio dictates preferred location:
shortest face preferred
L flat
Preferred alignment
Lcurve
ALVcurve
~
ALVflat
Curved side to curved side
L flat
Lcurve
1
 L flat

 4

L
curve


AEXCESS curve
1
AEXCESS flat
L flat
Lcurve
1
Flat side to flat side
AEXCESS curve
1
AEXCESS flat
Preferred location is shortest face
 L flat

 0.66 

L
 curve

Surface Evolver Results: Confirm Slope Argument
Steepest slope always on shortest face
h
1
0.66
2.0
4.0
d
Langmuir 2008
Glass Walls
Glass Walls
Cylinder alignment on curved interfaces
Cylinder alignment on curved interfaces
ALV depends on cylinder alignment

ALV for a quadrupole on curved interface in
small slope limit
ALV
C
 ALV o  cos 2
4
Tcyl   LV
dALV
  LV C cos  sin 
d
Torque
Two mechanical equilibria:
0 perpendicular to wall
/2 parallel to wall
Stable state: depends on sign of C
d 2 ALV
 C cos 2
2
d
C   4 rp
2
Hp
R
in agreement with experiment
Alignment of ‘biscotti’ shaped particles
A saddle on a saddle
Alignment as a function of particle size
E / kT
-Background curvature 103 times particle radius
Lewandowski et al. Langmuir 2008
Rparticles=3.5mm
Cylinder assembly on curved interfaces
1
2
3
5
6
weak
curvature
1
4
Strong
curvature
1
2
3
In summary,the particle contribution to the total energy is
1
1
E p  hcm Fz  S  Txy  h0 ( x p ) : Π
2
2
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
Ep  h0 ( x p ) : Π
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
shapes with
corners
Current work:
• complex particles
• repulsion
• crowded surfaces-gels
• docking sites
• mechanics of assemblies
•
scale
Open issues: gels, networks, rheology, dense packings
Charged Ellipsoids
-percolating networks
-open flower like structures
-elastic, brittle interfaces
Jan Vermant,
Gerry Fuller
at water-decane
interface,
-becomes denser
with time
on a water drop in oil
Cylinders
-rectangular lattices
-ropes of chains
-open networks
at air water interface,
spread, compressed
to collapse
compression
isotherms, rheology,
role of charge
Other shapes
at air water interface,
with DPPC spread,
compressed
on a water drop in oil
Far field: cylindrical polar quadrupolar mode
Extract Fourier modes from numerical solution:
r > 9Rcyl
Hp 
r2


2
0
h(r , ) cos  2  d
r ~ 9Rcyl
At r ~ 9Rcyl , higher modes 5% contribution
4
L
6
R
L
L
2R
Rate of approach: far field
Fixed Aspect Ratio
Varied Aspect Ratio
L
(t(t-t
c)
c-t)
Faster approach as L increases:
consistent with Hp increase
Interactions of elliptical quadrupoles vs. r12
r12  2L
Solid line ends at tip-to-tip contact
end-to-end alignment favored
r12/R
Torque enforces end to end alignment
Steric effects
Steric effects imposed by
anisotropic hard core
repulsion
Potential= Ellip
Quadrupoles+
Repulsion preventing contact

 x  2/  y  2/  
F         1  0
 b  
 a 
  0 superellipses  cylinder
surface of revolution
=0.2
Asymptotics of interaction energy
L/D
Expansion in powers of 1/r12 :
Eside to side
4
 L / D  12  L2
 R 
 L / D  12  R 
2
1 
 2  1 

40

H




 12H p 1 
2
 r 
p 

2 

 12 
 L / D  1  D
 L / D  1  r12 
Eend toend
 L / D  12  L2
 L / D  12  R 
 R 
2
1 
 2  1 



40

H


 12H 1 
2
 r 
p 
2 


 12 
 L / D  1  D
 L / D  1  r12 
E
6
~ 1 / r12

Torque ~
6
2
4
6
2
p
Torque decays faster (as 1/r6) than force (1/r5)
Torque has strong aspect ratio (L=L/D) dependence
T ~ 80
2
He
R2
L(L  1)  R 


2

r
 L  1  12 
2
3
6
Anisotropic pair potential
after contact
before
contact
Tip contact
104 kT
 H p2 ~  R 2 ~ 107 kT
End-to-end favored until tip-to-tip contact
(Langmuir 2010)