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