liquid crystal colloids - cftc seminar 2010

Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions

Liquid crystal colloids: a 2d picture

Nuno M. Silvestre

CFTC - University of Lisbon

April 14th, 2010

Collaboration: P. Patrıcio (ISEL/CFTC)M. M. Telo da Gama (UL/CFTC)

NM Silvestre CFTC Seminar - April 14th 2010

Liquid crystal colloids: a 2d picture 1/27


Outline

IntroductionMean field approachColloid-colloid interactions

Quadrupolar interactionsDipolar interactions

Key-LockSoft ColloidsConclusions




Introduction

Figure: Blood

Figure: Fish oil droplets

Figure: Ink

Figure: Fog at 25th of April bridge




What’s a colloid?

Figure: Colloid




Liquid crystal colloids

Figure: Water droplets dispersed in nematic liquid crystal drops.




Liquid crystal colloids

Figure: (a) and (b) Self-assembling colloidal particles in 5CB LC. (c) Bindingpotential measured in kBT . M. Skarabot et al, PRE 77, 031705 (2008).




Important?

1. Self-assembly

2. Colloidal optical materials

3. Super-capacitors




How much more ideal can you get?

1. Easily manipulated by weakexternal fields

2. Topological defects

3. Microfluidics

3.1 size monodispersity3.2 multi-shell particles3.3 particles encapsulation

Figure: Dipolar colloidal crystal







3. Microfluidics


Figure: Dipolar colloidal chain







3. Microfluidics


Figure: Size monodispered colloidalparticles







3. Microfluidics


Figure: Multi-shell colloidal particlesand particles encapsulation




LC host + colloidal particles

elastic constants

surface tension

size and shape

boundary conditions

Figure: P.Cluzeau et al, PRE 63,031702 (2001)

Figure: V.G. Nazarenko et al, PRL87, 075504 (2001)




LC host + colloidal particles

elastic constants

surface tension

size and shape

boundary conditions

Figure: P. Poulin et al, PRE 59,4384 (1999)

Figure: S.P. Meeker et al, PRE 61,R6083 (2000)




Topological defects

Broken continuous symmetry

Strong variations of order Cosmology

Crystalline solids

Liquid crystals

...




Isolated colloidal particles

Figure: Spheres in nematic LCs (3dsystems). Top: hedgehog defect;bottom: saturn-ring defect. PRL 85,4719 (2000)

Figure: Circular inclusions in smecticC films (2d systems). a) Singlesurface defect; b) two boojums. PRE73, 041706 (2006)





Figure: P.Poulin et al, PRE 57, 626 (1998).





Figure: Y. Gu et al, PRL 85, 4719 (2000).





Figure: P.Poulin et al, PRE 57, 626 (1998).




Elastic deformations

Figure: Splay: k1

2(∇ · ~n)2 Figure: Bend: k3

2(~n ×∇× ~n)2

Figure: twist: k2

2(~n · ∇ × ~n)2




Oseen-Zocher-Frank free energy

F =

∫

Ω

d3x

(

k1

2(∇ · ~n)

2+

k2

2(~n · ∇ × ~n)

2+

k3

2(~n ×∇× ~n)

2

)

(1)

Figure: Elastic constants of PAA liquid crystal in units of 10 pN. in The

Physics of Liquid Crystals, P.G. de Gennes and J. Prost





F =

∫

Ω

d3x

(

k1

2(∇ · ~n)

2+

k2

2(~n · ∇ × ~n)

2+

k3

2(~n ×∇× ~n)

2

)

(1)

One-constant approximation ki = k:

F =k

2

∫

Ω

d3x(

(∇ · ~n)2

+ (∇× ~n)2)

(2)





F =

∫

Ω

d3x

(

k1

2(∇ · ~n)

2+

k2

2(~n · ∇ × ~n)

2+

k3

2(~n ×∇× ~n)

2

)

(1)

One-constant approximation ki = k:

F =k

2

∫

Ω

d3x(

(∇ · ~n)2

+ (∇× ~n)2)

(2)

Director constrained to 2d, ~n = (cos θ, sin θ):

F =kl

2

∫

Ω

d2x(

∇θ)2)

(3)




And yet again ... topological defects!

Close to defects:

‖∇θ‖ =q

r(4)




And yet again ... topological defects!

Close to defects:

‖∇θ‖ =q

r(4)

Defect core radius:

rc = |q|ξ (5)

Core energy:

Fcore =π

2q2k (6)




Landau-de Gennes free energy

For uniaxial nematic LC: Qαβ = Q (nαnβ − δαβ/3)

F =

∫

Ω

d3x (fbulk + felastic) (7)

Bulk term:

fbulk =a

2QαβQβα −

b

3QαγQγβQβα +

c

4(QαβQβα)2 (8)

a = −0.172× 106 J/m3

b = 2.12 × 106 J/m3

c = 1.73 × 106 J/m3

Table: Typical values for 5CB liquid crystal




Landau-de Gennes free energy

For uniaxial nematic LC: Qαβ = Q (nαnβ − δαβ/3)

F =

∫

Ω

d3x (fbulk + felastic) (7)

Bulk term:

fbulk =a

2QαβQβα −

b

3QαγQγβQβα +

c

4(QαβQβα)

2(8)

Elastic term (one-constant approximation):

felastic =L

2∂γQαβ∂γQβα (9)




Surface energy (anchoring)

Rapini-Papolar

Fω =

∫

∂Ω

dsω

2(~n · ~ν)

2(10)

~ν - preferred molecular orientationat the surface

Nobili-Durand

FW =

∫

∂Ω

dsW

2

(

Qαβ − Qsαβ

)2

(11)Qs

αβ = Qs (νανβ − δαβ/3) -preferred tensor order parameterat the surface

Wglass = 1 × 10−2 J/m2 for 5CB liquid crystal

Weak anchoring: ωR/k < 10Strong anchoring: ωR/k > 10




Finite Elements Method (FEM)




Finite Elements Method (FEM)

1 2 30.18655

0.18660

0.18665

0 1 2 3iteration

0.185

0.190

0.195

F/k




Interacting colloidal particles

Figure: Quadrupolar colloidal particles self-assembling. I. Musevic et al,Science 313, 954 (2006).




Quadrupolar interactions

Colloids in 2d nematics [M. Tasinkevych et al, EPJ E 9, 341 (2002)]

Figure: Nematic configurations forseveral separations and parallelalignment α = 0.

Figure: Nematic configurations forseveral separations and perpendicularalignment α = π/2.






Long range: θ ≈ θ1 + θ2

Fint ∝1 − 2 sin2 2α

R4(12)

Repulsion: α = nπ/2Attraction: α = (2n + 1)π/4

3 4 5 6 7R / a

10.5

11

11.5

12

F- F

u

α = 0 α = π/4 α = π/2

4 5 6 711.1

11.3

11.5

4 611.1

11.3

11.5

Figure: Interaction free energy forrelative orientations α = 0, π/4, π/2.







0 0.1 0.2 0.3 0.4 0.5 α / π

10

10.5

11

11.5

12

12.5

F -

Fu

R = 4.0a R = 3.0a R = 2.4a

2 4 6 8 10 12R / a

0

0.1

0.2

0.3

α∗ / π

Figure: Left: Interaction free energy(F = F/k) for several separationsR/a = 4.0(©), 3.0(♦), 2.4();Right: Preferred orientation α∗ as afunction of the separation.







2 2.1 2.2 2.3 2.4 2.5R /a

9.3

9.5

9.7

9.9

10.1

10.3

10.5

F-F u

2 2.1 2.2 2.3 2.4 2.5R /a

10.7

10.9

11.1

11.3

11.5

11.7

11.9

a b

α = 0 α = π/2

Figure: Interaction free energy(F = F/k) for several anchoringstrengths ωR/k = 250(©), 10(♦),7.5().







2 2.1 2.2 2.3 2.4 2.5R /a

9.3

9.5

9.7

9.9

10.1

10.3

10.5

F-F u

2 2.1 2.2 2.3 2.4 2.5R /a

10.7

10.9

11.1

11.3

11.5

11.7

11.9

a b

α = 0 α = π/2

Figure: Interaction free energy(F = F/k) for several anchoringstrengths ωR/k = 250(©), 10(♦),7.5().

Self-assembling: long-range attractionEquilibrium colloidal structure stability: short-range repulsion





Quadrupolar inclusions in Smectic-C films

Figure: Inclusions in Smectic C film with parallel anchoring and surface defects.P. Cluzeau et al, JEPT Letters 76, 351 (2002).





Quadrupolar inclusions in Smectic-C films [NMS et al, Mol. Cryst.

Liq. Cryst. 495, 618 (2008)]

Figure: Energy profiles for severalanchoring strangths ωR/k = 0.1, 1,10, 100.

Figure: a) Equilibrium separationsmin and b) equilibrium orientationαmin as functions of anchoringstrength ωR/k




Dipolar interactions

Dipolar colloidal particles [in collab. with J. Maclennan and N. Clark,

Boulder, Colorado]

Figure: Chiral colloidal particles in a freely standing smectic film. Depolarizedreflected light microscope images of a smectic C∗ film of racemic MX8068showing (a) two colloidal particles with same handedness and (b) two colloidalparticles with opposite handedness. Equilibrium director field around twoislands with (c) the same handedness and (d) opposite handedness.





Behond the one-constant approximation

Chiral Smectic C∗:

one-elastic-constant approximation







one-elastic-constant approximation NOT VALID








Spontaneous polarization ~P (~x) Additional contribution to bendelastic constant k3.








Spontaneous polarization ~P (~x) Additional contribution to bendelastic constant k3.

Important to consider: κ = k3/k1





Homochiral inclusions

Figure: Colloid-defect geometry and interaction energies U(D)/(k1d) obtainedfrom computer simulations yielding dipole chains with homochiral colloid pairs,for various κ = k3/k1.





Homochiral inclusions

Figure: Dipolar chain. Bar: 20 µm. P.Cluzeau et al, PRE 63, 031702 (2001)





Heterochiral inclusions [ NMS et al PRE 80, 041708 (2009)]

Textures of heterochiral colloidalparticles interacting on a film of25% chirally doped MX8068. (a)The quadrupolar structures is inequilibrium when the particlesalmost touch. (b) The equilibriumseparation between the defectsincreases as the particles areseparated using optical tweezers.(c) When the separation issufficiently large, the quadrupolarsymmetry is broken. (d) When theislands are forced even furtherapart, the quadrople evolves intotwo separate dipoles.






Figure: Colloid-defect geometry and interaction energies U(D)/(k1d obtainedfrom computer simulations yielding quadrupoles with heterochiral pairs, forvarious κ = k3/k1.






Figure: Equilibrium vertical separation S between defects as a function of thecolloid center-to-center separation D in the quadrupolar configuration regime,for racemic and 25% chirally doped films of MX8068, compared with the resultsof numerical calculations for systems with elastic anisotropies κ = 0.2, 1.0, 2.4






How important are the thermal fluctuations?




Capturing colloidal particles

Figure: FR Hung et al, J. Chem. Phys. 127,124702 (2007)

Figure: NMS et al, PRE69, 061402 (2004)




Capturing colloidal particles [NMS et al, PRE 69, 061402 (2004)]

¯





Figure: Left: Equilibrium interaction free energy F = F/k for depthd/R = 0.01 as a function of the width of the cavity. Right: Equilibriumposition of the colloidal particle as a function of the width of the cavity.





Figure: Interaction energy F = F/k profile parallel to the wall, for severaldistances s/R.




Deforming colloids

Figure: P.V. Dolganov et al, EPL 78,66001 (2007).

Figure: NMS et al, PRE 74, 021706(2006).




Deforming colloids

Figure: Aspect ratio H/hversus major axis H . h -minor axis. P.V. Dolganovet al, EPL 78, 66001(2007).

Figure: Optimal eccentricity,e =

p

1 − (h/H)2 versus σ = γR/k. γ is thesurface tension. NMS et al, PRE 74, 021706(2006).




Deforming colloids

Figure: Shape diagram: lines of constant eccentricity. σ = γR/k versus ωR/k.NMS et al, PRE 74, 021706 (2006).




Conclusions

Self-assembling of liquid crystal colloids is driven by long-range

anisotropic attractions

Equilibrium colloidal structures are stabilised by short-range

repulsions that appear in the presence of topological defects

Elastic anisotropy influences the behavior of the topological

defects surrounding the colloidal particles.

Colloidal particles can be captured by self-similar surfaces

The shape of colloidal particles strongly depends on the elasticity

of the LC, the surface tension, and its size.



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