1. 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
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20. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
NM Silvestre CFTC Seminar - April 14th 2010
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21. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Oseen-Zocher-Frank free energy
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
NM Silvestre CFTC Seminar - April 14th 2010
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22. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Oseen-Zocher-Frank free energy
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
NM Silvestre CFTC Seminar - April 14th 2010
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23. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
And yet again ... topological defects!
Close to defects:
q
θ = (4)
r
NM Silvestre CFTC Seminar - April 14th 2010
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24. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
NM Silvestre CFTC Seminar - April 14th 2010
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25. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
NM Silvestre CFTC Seminar - April 14th 2010
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26. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
NM Silvestre CFTC Seminar - April 14th 2010
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27. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
NM Silvestre CFTC Seminar - April 14th 2010
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28. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Finite Elements Method (FEM)
NM Silvestre CFTC Seminar - April 14th 2010
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33. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Quadrupolar interactions
Colloids in 2d nematics [M. Tasinkevych et al, EPJ E 9, 341 (2002)]
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.
NM Silvestre CFTC Seminar - April 14th 2010
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34. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Quadrupolar interactions
Colloids in 2d nematics [M. Tasinkevych et al, EPJ E 9, 341 (2002)]
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( ).
NM Silvestre CFTC Seminar - April 14th 2010
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35. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Quadrupolar interactions
Colloids in 2d nematics [M. Tasinkevych et al, EPJ E 9, 341 (2002)]
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
NM Silvestre CFTC Seminar - April 14th 2010
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36. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Quadrupolar interactions
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).
NM Silvestre CFTC Seminar - April 14th 2010
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37. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Quadrupolar interactions
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
NM Silvestre CFTC Seminar - April 14th 2010
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38. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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. 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.
NM Silvestre CFTC Seminar - April 14th 2010
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39. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Behond the one-constant approximation
Chiral Smectic C ∗ :
one-elastic-constant approximation
NM Silvestre CFTC Seminar - April 14th 2010
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40. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Behond the one-constant approximation
Chiral Smectic C ∗ :
one-elastic-constant approximation NOT VALID
NM Silvestre CFTC Seminar - April 14th 2010
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41. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Behond the one-constant approximation
Chiral Smectic C ∗ :
one-elastic-constant approximation NOT VALID
Spontaneous polarization P (x) Additional contribution to bend
elastic constant k3 .
NM Silvestre CFTC Seminar - April 14th 2010
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42. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Behond the one-constant approximation
Chiral Smectic C ∗ :
one-elastic-constant approximation NOT VALID
Spontaneous polarization P (x) Additional contribution to bend
elastic constant k3 .
Important to consider: κ = k3 /k1
NM Silvestre CFTC Seminar - April 14th 2010
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43. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
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 .
NM Silvestre CFTC Seminar - April 14th 2010
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44. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Homochiral inclusions
Figure: Dipolar chain. Bar: 20 µm. P.Cluzeau et al, PRE 63, 031702 (2001)
NM Silvestre CFTC Seminar - April 14th 2010
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45. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
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.
NM Silvestre CFTC Seminar - April 14th 2010
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46. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Heterochiral inclusions [ NMS et al PRE 80, 041708 (2009)]
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 .
NM Silvestre CFTC Seminar - April 14th 2010
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47. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Heterochiral inclusions [ NMS et al PRE 80, 041708 (2009)]
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
NM Silvestre CFTC Seminar - April 14th 2010
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48. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Dipolar interactions
Heterochiral inclusions [ NMS et al PRE 80, 041708 (2009)]
How important are the thermal fluctuations?
NM Silvestre CFTC Seminar - April 14th 2010
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49. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Capturing colloidal particles
Figure: NMS et al, PRE
Figure: FR Hung et al, J. Chem. Phys. 127, 69, 061402 (2004)
124702 (2007)
NM Silvestre CFTC Seminar - April 14th 2010
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50. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Capturing colloidal particles [NMS et al, PRE 69, 061402 (2004)]
NM Silvestre CFTC Seminar - April 14th 2010
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51. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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.
NM Silvestre CFTC Seminar - April 14th 2010
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52. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Capturing colloidal particles [NMS et al, PRE 69, 061402 (2004)]
¯
Figure: Interaction energy F = F/k profile parallel to the wall, for several
distances s/R.
NM Silvestre CFTC Seminar - April 14th 2010
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53. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Deforming colloids
Figure: P.V. Dolganov et al, EPL 78,
66001 (2007).
Figure: NMS et al, PRE 74, 021706
(2006).
NM Silvestre CFTC Seminar - April 14th 2010
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54. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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).
NM Silvestre CFTC Seminar - April 14th 2010
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55. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Deforming colloids
Figure: Shape diagram: lines of constant eccentricity. σ = γR/k versus ωR/k.
NMS et al, PRE 74, 021706 (2006).
NM Silvestre CFTC Seminar - April 14th 2010
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56. Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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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