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. Patr´ (ISEL/CFTC)
ıcio
M. M. Telo da Gama (UL/CFTC)

NM Silvestre CFTC Seminar - April 14th 2010
Liquid crystal colloids: a 2d picture 1/27


Outline

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



Introduction

Figure: Blood Figure: Ink

Figure: Fish oil droplets
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) Binding
ˇ
potential measured in kB T . 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 weak
external ﬁelds
2. Topological defects
3. Microﬂuidics
3.1 size monodispersity
3.2 multi-shell particles
3.3 particles encapsulation

Figure: Dipolar colloidal crystal




external ﬁelds
3. Microﬂuidics

Figure: Dipolar colloidal chain




external ﬁelds
3. Microﬂuidics
Figure: Size monodispered colloidal
particles




external ﬁelds
3. Microﬂuidics

Figure: Multi-shell colloidal particles
and particles encapsulation



LC host + colloidal particles

Figure: P.Cluzeau et al, PRE 63,
elastic constants 031702 (2001)
surface tension
size and shape
boundary conditions

Figure: V.G. Nazarenko et al, PRL
87, 075504 (2001)


LC host + colloidal particles

Figure: P. Poulin et al, PRE 59,
elastic constants 4384 (1999)
surface tension
size and shape
boundary conditions

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


Topological defects

Broken continuous symmetry
Cosmology
Strong variations of order
Crystalline solids
Liquid crystals
...



Isolated colloidal particles

Figure: Circular inclusions in smectic
C ﬁlms (2d systems). a) Single
surface defect; b) two boojums. PRE
Figure: Spheres in nematic LCs (3d 73, 041706 (2006)
systems). Top: hedgehog defect;
bottom: saturn-ring defect. PRL 85,
4719 (2000)




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




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



Elastic deformations

Figure: Splay: k1
2
( · n)2 Figure: Bend: k3
(n × × n)2
2

Figure: twist: k2
2
(n · × n)2



Oseen-Zocher-Frank free energy
k1 2 k2 2 k3 2
F = d3 x ( · n) + (n · × n) + (n × × n) (1)
Ω 2 2 2

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




k1 2 k2 2 k3 2
F = d3 x ( · n) + (n · × n) + (n × × n) (1)
Ω 2 2 2

One-constant approximation ki = k:
k 2 2
F = d3 x ( · n) + ( × n) (2)
2 Ω




k1 2 k2 2 k3 2
F = d3 x ( · n) + (n · × n) + (n × × n) (1)
Ω 2 2 2

One-constant approximation ki = k:
k 2 2
F = d3 x ( · n) + ( × n) (2)
2 Ω

Director constrained to 2d, n = (cos θ, sin θ):
kl
F = d2 x θ)2 (3)
2 Ω



And yet again ... topological defects!

Close to defects:
q
θ = (4)
r



And yet again ... topological defects!

Close to defects:
q
θ = (4)
r
Defect core radius:

rc = |q|ξ (5)

Core energy:
π 2
Fcore = q k (6)
2



Landau-de Gennes free energy
For uniaxial nematic LC: Qαβ = Q (nα nβ − δαβ /3)

F = d3 x (fbulk + felastic ) (7)
Ω
Bulk term:
a b c
fbulk = Qαβ Qβα − Qαγ Qγβ Qβα + (Qαβ Qβα )2 (8)
2 3 4
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 (nα nβ − δαβ /3)

F = d3 x (fbulk + felastic ) (7)
Ω
Bulk term:
a b c 2
fbulk = Qαβ Qβα − Qαγ Qγβ Qβα + (Qαβ Qβα ) (8)
2 3 4
Elastic term (one-constant approximation):

L
felastic = ∂γ Qαβ ∂γ Qβα (9)
2



Surface energy (anchoring)

Rapini-Papolar Nobili-Durand

ω 2 W 2
Fω = ds (n · ν) (10) FW = dsQαβ − Qsαβ
∂Ω 2 ∂Ω 2
(11)
ν - preferred molecular orientation Qs = Qs (να νβ − δαβ /3) -
αβ
at the surface preferred tensor order parameter
at the surface
Wglass = 1 × 10−2 J/m2 for 5CB liquid crystal
Weak anchoring: ωR/k < 10
Strong anchoring: ωR/k > 10



Finite Elements Method (FEM)



Finite Elements Method (FEM)

0.195
0.18665

0.18660

F/k
0.190 0.18655
1 2 3

0.185
0 1 2 3
iteration



Interacting colloidal particles

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




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

Figure: Nematic conﬁgurations for
Figure: Nematic conﬁgurations for several separations and perpendicular
several separations and parallel alignment α = π/2.
alignment α = 0.





12
α=0
α = π/4
α = π/2
Long range: θ ≈ θ1 + θ2 11.5

F- Fu
11 11.5

1 − 2 sin2 2α 11.3
Fint ∝ (12) 10.5
R4 11.1
4 5 6 7

3 4 5 6 7
R/a
Repulsion: α = nπ/2
Attraction: α = (2n + 1)π/4
Figure: Interaction free energy for
relative orientations α = 0, π/4, π/2.





12.5 0.3
R = 4.0a
R = 3.0a
12 R = 2.4a
0.2
11.5

α /π
F - Fu

∗
11
0.1
10.5
10 0
0 0.1 0.2 0.3 0.4 0.5 2 4 6 8 10 12
α/π R/a

Figure: Left: Interaction free energy
¯
(F = F/k) for several separations
Figure: Nematic conﬁgurations for
R/a = 4.0( ), 3.0(♦), 2.4( );
several separations and parallel
Right: Preferred orientation α∗ as a
alignment α = 0.
function of the separation.





10.5 11.9
α=0 α = π/2
10.3 11.7

10.1 11.5
11.3

F-Fu
9.9
9.7 11.1

9.5 10.9
10.7
9.3
2 2.1 2.2 2.3 2.4 2.5 2 2.1 2.2 2.3 2.4 2.5
a R /a b R /a

Figure: Interaction free energy
Figure: Nematic conﬁgurations for ¯
(F = F/k) for several anchoring
several separations and parallel strengths ωR/k = 250( ), 10(♦),
alignment α = 0. 7.5( ).





10.5 11.9
α=0 α = π/2
10.3 11.7

10.1 11.5
11.3

F-Fu
9.9
9.7 11.1

9.5 10.9
10.7
9.3
2 2.1 2.2 2.3 2.4 2.5 2 2.1 2.2 2.3 2.4 2.5
a R /a b R /a

Figure: Interaction free energy
Figure: Nematic conﬁgurations for ¯
(F = F/k) for several anchoring
several separations and parallel strengths ωR/k = 250( ), 10(♦),
alignment α = 0. 7.5( ).
Self-assembling: long-range attraction
Equilibrium 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: a) Equilibrium separation
Figure: Energy proﬁles for several smin and b) equilibrium orientation
anchoring strangths ωR/k = 0.1, 1, αmin as functions of anchoring
10, 100. strength ωR/k



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

Boulder, Colorado]

Figure: Chiral colloidal particles in a freely standing smectic film. Depolarized
reflected light microscope images of a smectic C ∗ film of racemic MX8068
showing (a) two colloidal particles with same handedness and (b) two colloidal
particles with opposite handedness. Equilibrium director field around two
islands 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 bend
elastic constant k3 .





Spontaneous polarization P (x) Additional contribution to bend
elastic constant k3 .
Important to consider: κ = k3 /k1




Homochiral inclusions

Figure: Colloid-defect geometry and interaction energies U (D)/(k1 d) obtained
from 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 colloidal
particles interacting on a ﬁlm of
25% chirally doped MX8068. (a)
The quadrupolar structures is in
equilibrium when the particles
almost touch. (b) The equilibrium
separation between the defects
increases as the particles are
separated using optical tweezers.
(c) When the separation is
suﬃciently large, the quadrupolar
symmetry is broken. (d) When the
islands are forced even further
apart, the quadrople evolves into
two separate dipoles.





Figure: Colloid-defect geometry and interaction energies U (D)/(k1 d obtained
from computer simulations yielding quadrupoles with heterochiral pairs, for
various κ = k3 /k1 .





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





How important are the thermal ﬂuctuations?



Capturing colloidal particles

Figure: NMS et al, PRE
Figure: FR Hung et al, J. Chem. Phys. 127, 69, 061402 (2004)
124702 (2007)



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




¯
Figure: Left: Equilibrium interaction free energy F = F/k for depth
d/R = 0.01 as a function of the width of the cavity. Right: Equilibrium
position 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 several
distances 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/h
versus major axis H. h -
minor axis. P.V. Dolganov
et al, EPL 78, 66001 Figure: Optimal eccentricity,
p
(2007). e = 1 − (h/H)2 versus σ = γR/k. γ is the
surface 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.


Liquid Crystal Colloids - CFTC Seminar 2010

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