Elasticity mediated long range attraction of particles

Aditi Chakrabarti* and Manoj K. Chaudhury

Lehigh University

Department of Chemical Engineering

Bethlehem

Pennsylvania-18015

Self Assembly of Particles

• Nanoscale-particle assembly via Molecular Interactions

Hydrogen Bonding, Van der Waals interactions, etc

• Externally directed self assembly

Electric, Magnetic fields– particles are polarized

• Assembly induced by – combination of fields

Capillary interactions, Assembly in Liquid crystals

Nicolson, M. M. 1949,Gifford & Scriven 1971, Bowden, Terfort, Carbeck, Whitesides Science 1997.

Poulin, Stark, Lubensky, Weitz Science 1997.Cavallaro, Gharbi, Beller, Čopar, Shi, Baumgart, Yang,Kamien, Stebe, PNAS 2013

Ken Ishikwa

Small Scale:

Van der

Waals forces

Large Scale:

Gravitational

Forces

Directed Self Assembly- Role of Elasticity ?

Monch and Herminghaus, EPL, 53, 2001.Ghatak and Chaudhury, Langmuir, 19, 2003.

H

l

Solid Support

Particles

H1, (m1)

l

Soft Support (m2)H2

Elastic Films ? Elastic Half Space ?

Role of Adhesion and Friction

Systems used to study Self Assembly of Particles

0 s 20 s 26 s

3mm

II. Spheres on Gel Surface

0s 4s 10s

III. Spheres inside Gel

Chakrabarti, A.; Chaudhury, M. K Langmuir 2014, 30, 4684.Chakrabarti, A.; Chaudhury, M. K Langmuir 2013, 29, 15543.

Elastic Bond No.

m

RgEBo

For Gels

II., III.

H

gREBo

liqm

2

2

3

Elastic Bond No.

Case I.

Attraction of:

I. Cylinders on elastic film

Liquid(Incompressible)

LiquidLiquid

0

2

0

2

2dx

dx

dTdxg

LUT

Gravitational

Free Energy of

Liquid

Elastic Energy

of the Stretched

Film

T

gTE )(1

Decay Length

0/ TU

Functional

Minimization:xe 0

PDMS

(1:1 Sylgard 184 & 186) )152( mm1:1

Glycerol-Water)/29( mmNd mmN /22

Thin Elastic Film supported on a Pool of Liquid

PDMS-

grafted

Steel cylinder

nm5

~mm radius

Tension in the Film

)1()( 00

xe

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

12 16 20 2440 2 6

4

3

168

1

20 24

2

0(0

)

(mm

)

x (mm)

1.8 mm

12.1 mm

5mm

0

0.00002

0.00004

0.00006

0 0.0000050.000010.0000150.000021.00 0.5 1.50

2

4

2.0

6 10-5

10-5

- 2

(m2)

H (m)

2.4

mmN /2756

mmNTE /34070

2mm

-3

-2

-1

0

-10 -5 0 5 10

0

0 -5x’ (mm)

-10-5-10

1

2

3

’

(mm

) 5.7 mm

)2/cosh(

)cosh('' 0

x

gTE )(1

Decay Length

Experimental Investigation of Decay Length

1

Estimation of Energy of Attraction

( )0

* gmTotal

Energy,

UT

Implicit

Explicit( )( )

2cosh2

sinh1

2

2

0

gL

( ) ( ) ho ,0/ 0 TUVertical Stability:

:

:

1

L Length of cylinder

Decay Length

2/

2/

2/

2/

22

0

2

0

2* ''

2'

2

'2

dxdx

dTdx

gLdx

dx

dTdxgLghmU E

T

Gravitational

Free Energy

of Displaced

Liquid

Elastic Energy

of the

Stretched Film

Gravitational

Energy

of Particles on

Film

Elastic

Film Glycerol-

Waterh

G.E. and E.E for

Intermediate Profile

Role of Adhesion Hysteresis

Difference in energy of CLOSING and OPENING a crack

5mm

BA

Forced rolling against a spring to estimate the hysteresis:

LFW / 2/100 mmJW

She, H.; Chaudhury, M. K. Langmuir 2000, 16, 622-625.

Kendall, K. Wear 1975, 33, 351-358.

)sinh()2(cosh

)sinh()(2

1

Ch

hh

)'()( *

hhgmLW

W

4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2

(h∞–

h)

/ h

∞

5.7 mm10.4 mm12.6 mm

2/135 mmJW

Adhesion Hysteresis

Reducing Adhesion Hysteresis with Hydrogel layer on Elastic Film

5mm0 45 46 s

20 mm

Longer range of attraction

0 s5mm 45

120

74

75 115

0 26 30 s

16 mm5mm

Low friction of Gel +

Flexibility of Elastic film

0 s 3

7 135mm

B

A

Guided interaction with Surface folding

Elastic

Film Glycerol-Water

Thin Gel Layer

Petri

dish5-15 mm

1-2 mm

Self Assembly

Particles on the Surface of GEL (10-50 Pa) ?

0 s 20 s 26 s

3mm

0 s 10 s 16 s

4mm

Hydrophobic Glass Particles ~mm range

Scaling Theory for Elastocapillary Attraction of

Particles on Soft Gels

L h0

h

)/( *

co

o

LLKh

h

Nicolson’s

Superposition

Principle

Nicolson, M. M. Proc. Cambridge Philos. Soc. 1949, 45, 288−295

hgm *

Implicit

)/( **

coo LLKghm

ExplicitChange in

Gravitational

Potential

Energy

0 s 20 s 26 s

3mm

*

cLDecay

Length for

Gels

Pa20

0 Pa

Water

35 Pa

Gel

0

(L)

845 Pa

Gel

)/()( *

00 cLLKL

Elastocapillary Decay Length Lc

*for Soft Gels

Hydrophobic glass sphere

*

cL Decay Length of Gel

GEL

1.2mm radius

10 Pa

Modified Bessel function

Can also be determined

2

200 400 600 80010000

0

1

3

m (Pa)

Lc* (

mm

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

L / Lc*

00

21 3 54

0.1

0.20.30.40.5

0.6

Δh

/h

0

0.710 Pa gel-Air

19 Pa gel-Air

10 Pa gel-Heptane

)/(~ *

co

o

LLKh

h

0 m 15 m 17 m

0 s 20 s 26 s

0 s 10 sA

B

21 s

C

3mm

3mm

3mm

Elastocapillary Attraction of Particles on the

Surface of Soft Gels

Heptane

Air

L h0

h

Schematic showing how the analysis was done

Linear Kinematic Friction in Spheres moving on the

surface of Gels

)/(~ *

00

*

cLKghmU

Chakrabarti, A.; Chaudhury, M. K Langmuir 2014,30, 4684.

0 m 15 m 17 m

0 s 20 s 26 s

0 s 10 sA

B

21 s

C

3mm

3mm

3mm

Heptane

0

*

~ghm

Attractive

Force:

dt

d~Frictional

Force:

t~2

Diffusive

collapse 80

180

280

380

0 250 500 750 1000

0

40

80

0 10 200 10 20

40

80

0

2.3

80

180

280

380

0 250 500 750 1000

0.25

t (s)

l2(m

m2)

H

l

Solid Support

Gravitational Forces

Van der Waals forces,

Externally Applied Forces

Electric field, Magnetic field

Self Assembly of particles inside Gel

At Nanoscale?

BACKUP SLIDES

Attraction and Self Assembly of Particles inside a Soft Gel

Chakrabarti, A.; Chaudhury, M. K Langmuir 2013, 29, 15543−15550.

h0

Hoop Stress

Cooperative Effects of Surface Tension, Elasticity, and Gravity

Excessive Stretching in the Gel

Podgorski, T.; Belmonte, A. J. Fluid Mech. 2002, 460, 337−348 Graham, M. D. Phys. Fluids 2003 15(6), 1702-1710 (for similar phenomenon in viscoelastic liquid)

0s 4s 10s

Lot more energy stored when Particle is inside Gel

0.044 s0.034 s0.03 s

10 mm 0 s 30 s

A B C

hydrophobic

hydrophilic

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 20

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2

(h∞-

h)/

h0

L/Lc*

Ceramic, ϕ4.8mm Ceramic, ϕ3.2mm Cu, ϕ2.4mm Cu, ϕ2mm

0.35

Lc*=√(mh0/g)

Self Organization inside Gels

When a film undergoes both bending and stretching,

0 pgTD xxxxxx

Periodic solution with exponential decay

52 10))(1( H

Ratio of Bending to Stretching,

)1()( 00

xe 0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

12 16 20 2440 2 6

4

3

168

1

20 24

2

0(0

)

(mm

)

x (mm)

1.8 mm

12.1 mm

5mm

x'=0

x=0

Elastic

FilmGlycerol-

Water

h

2/ 2/

Estimation of Energy

of Attraction

2/

2/

2/

2/

22

0

2

0

2* ''

2'

2

'2

l

l

l

l

dxdx

dTdx

gLdx

dx

dTdxgLghmU

)2/cosh(

)cosh('' 0

x xe 0

( )( )

2cosh2

sinh12

2

2

0

0

*

LggmUT

( )( )

2cosh2

sinh1

2

*

gLgm o

)sinh()2(cosh2

)sinh()(2

h

hh5mmWrinkling

Folding

,0/ 0 U

0

* gmUT

)(0 h

Assumption:

ho

C)

A)

D)

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0.000005 0.00001

F/L

(g

)2/3

(N/m

2)1

/3(m

5/3

)

H 1/304/3 (m5/3)

1.00 0.50

24

6

10-4

10-5

10

8

12

108

B)

F Cover slip

Deformed

Elastic FilmGlycerol-

Water0

0

0.1

0.2

0.3

0.4

0.5

0 0.001 0.002 0.003

F/L

(N

/m)

0 (m)0 0.001

0

0.1

0.2

0.3

0.4

0.5

3.9 mm6.2 mm

14.4 mm

0.002 0.003

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 0.000005 0.00001

F/L

(g

)3/4

(N/m

2)1

/4(m

7/4

)

H 1/403/2 (m7/4)

1.00 0.50

2

4

5 10-4

10-5

50

3

1

dxdx

dTLdxgLU f

2

2

0 fUF0

2

g

L

F

23

0

4143 )()( EHgL

F

( )n 0

Estimation of

Elastic

Modulus of

the Thin Film

34

0

3132 )()( EHgL

F

Strain in the film,

n=2 n=1

0 s 3

7 135mm

B

A

Metastable States

Folding Instabilities guiding Particle Interaction

Camera

Glass

Plates

Magnet

Steel disk

0 50 100 150 200

Frequency (Hz)

Po

wer

(a.

u.)

0 50 100 150 200

35 Pa

135 Pa

330 Pa

8.6 Hz

17 Hz

26.4 Hz

555 Pa

845 Pa

34.4 Hz

42 Hz

A) B)

2 4 6

200

400

600

800

1000

0

m (

Pa)

% PAM in Gel3 5

Measurement of the Shear Moduli m of Gel

Bulk Hydrogel– What happens to the Particles on the surface of

GEL (10-50 Pa) ?

Surface Chemistry Matters

10 mm 0 s 30 s

A B C

hydrophobic

hydrophilic

10 Pa

0 s 20 s 26 s

3mm

0 s 10 s 16 s

4mm

contactAhgm *

10 mm 0 s 30 s

A B C

0 s

64 380

19 27

385

5mm

Particle Self Assembly at the

Gel Surface

0 s

5 s 8 s

4 s5mm

Tubulation and Engulfment at

Gel-Heptane Interface

Stress Patterning in Gel to

Guide Attractions on Surfacehydrophobic

hydrophilic

Elastocapillary Force Mediated Attractions in Different Scenarios