Download - 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
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Outline
IntroductionMean field approachColloid-colloid interactions
Quadrupolar interactionsDipolar interactions
Key-LockSoft ColloidsConclusions
NM Silvestre CFTC Seminar - April 14th 2010
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Introduction
Figure: Blood
Figure: Fish oil droplets
Figure: Ink
Figure: Fog at 25th of April bridge
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
What’s a colloid?
Figure: Colloid
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Liquid crystal colloids
Figure: Water droplets dispersed in nematic liquid crystal drops.
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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).
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Important?
1. Self-assembly
2. Colloidal optical materials
3. Super-capacitors
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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 chain
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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: Size monodispered colloidalparticles
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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: Multi-shell colloidal particlesand particles encapsulation
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Topological defects
Broken continuous symmetry
Strong variations of order Cosmology
Crystalline solids
Liquid crystals
...
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Isolated colloidal particles
Figure: P.Poulin et al, PRE 57, 626 (1998).
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Isolated colloidal particles
Figure: Y. Gu et al, PRL 85, 4719 (2000).
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Isolated colloidal particles
Figure: P.Poulin et al, PRE 57, 626 (1998).
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Elastic deformations
Figure: Splay: k1
2(∇ · ~n)2 Figure: Bend: k3
2(~n ×∇× ~n)2
Figure: twist: k2
2(~n · ∇ × ~n)2
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Oseen-Zocher-Frank free energy
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Oseen-Zocher-Frank free energy
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
And yet again ... topological defects!
Close to defects:
‖∇θ‖ =q
r(4)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
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
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Finite Elements Method (FEM)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Finite Elements Method (FEM)
1 2 30.18655
0.18660
0.18665
0 1 2 3iteration
0.185
0.190
0.195
F/k
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Interacting colloidal particles
Figure: Quadrupolar colloidal particles self-assembling. I. Musevic et al,Science 313, 954 (2006).
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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)]
Figure: Nematic configurations forseveral separations and parallelalignment α = 0.
Figure: Nematic configurations forseveral separations and perpendicularalignment α = π/2.
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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)]
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.
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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)]
Figure: Nematic configurations forseveral separations and parallelalignment α = 0.
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.
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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)]
Figure: Nematic configurations forseveral separations and parallelalignment α = 0.
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().
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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)]
Figure: Nematic configurations forseveral separations and parallelalignment α = 0.
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
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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).
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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: 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
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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. 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.
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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
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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
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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 bendelastic constant k3.
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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 bendelastic constant k3.
Important to consider: κ = k3/k1
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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)/(k1d) obtainedfrom computer simulations yielding dipole chains with homochiral colloid pairs,for various κ = k3/k1.
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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)
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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 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.
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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)/(k1d obtainedfrom computer simulations yielding quadrupoles with heterochiral pairs, forvarious κ = k3/k1.
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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 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
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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?
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Capturing colloidal particles
Figure: FR Hung et al, J. Chem. Phys. 127,124702 (2007)
Figure: NMS et al, PRE69, 061402 (2004)
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Introduction Mean field approach Colloid-colloid interactions Key-Lock Soft Colloids Conclusions
Capturing colloidal particles [NMS et al, PRE 69, 061402 (2004)]
¯
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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.
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Capturing colloidal particles [NMS et al, PRE 69, 061402 (2004)]
Figure: Interaction energy F = F/k profile parallel to the wall, for severaldistances s/R.
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Deforming colloids
Figure: P.V. Dolganov et al, EPL 78,66001 (2007).
Figure: NMS et al, PRE 74, 021706(2006).
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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).
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Deforming colloids
Figure: Shape diagram: lines of constant eccentricity. σ = γR/k versus ωR/k.NMS et al, PRE 74, 021706 (2006).
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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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