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A de Gennes Legacy: Liquid Crystals as Inspiration for
Fundamental Physics
40 Years of Liquid Crystal Physics
P.G. de Gennes, C.R. Acad. Sci. Paris. 266, 15 (1968): Nematic Fluctuations and Light Scattering
de GennesDays
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Grand Synthesis: Liquid Crystals Are Ideal
1. Equilibrium states of matter characterized by symmetry and conservation laws
2. Broken continuous symmetrya. Elasticity with q2
b. Topological defectsc. Goldstone hydrodynamics with 0 as q 0
3. Fluctuations a. Modify MF critical behavior below dc
b. Destroy long-range order for d<dL
c. Modify elasticity and hydrodynamics in low dd. Unbinding of topological defects destroys elastic
rigidity especially at d=dL
4. Order produces Higgs bosons in gauge theories
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De Gennes Deux Chevaux, Rue Froidvaux -1969
Les Houches – 1967 – Guitar
Orsay – 1969 – from magnetism to Liquid crystals
Superconductors and smectic liquid crystals – 1973
Polymers, branched polymers, gels, and percolation – 70’s
Twist-grain boundary phase – 1989
Liquid Crystal Elastomers – 1980-89 and now
PGG’s car – lent to me for summer
My Wife
Outline
• Nematics
• Nematic Elatomers
• Smectic
From PGG’s early contributions to modern questions
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LC Mesogens
de GennesDays
7S5S
O
C5H11C7H15O CS
O
C5H11C7H15O C 30A
fd Virusfd Virus
Rod-like:calamatic
Disc-like:discotic
Carbon NanotubeBent Core or
Banana NOBOW
Chemistry is king
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Liquid Crystal Phases
Decreasing Symmetry = soap= soap
ThreadThread
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ChiralityChiral molecule: no combination of translations and rotations can superpose molecule on its mirror image
Achiral mirror images Chiral mirror images
Chirality favors twist
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Chiral LC Phases
Cholesteric
Blue Phase
New length: P~500nm>>d
Fig: B. Pansu
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Lyotropic – Mixed Rods and Layers
DNA-Lipid Complexes: Gene Therapy
O’Hern, C.S., and Lubensky, T.C., Phys. Rev. Lett. 80, 4345-4348 (1998).
L. Golubovic and M. Golubovic, Phys. Rev. Lett. 80, 4341 (1998).
Sliding Columnar Phases
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Nematic I• Homogenous but anisotropic• All origins are equivalent but not all
directions• Order parameter (deGennes-Maier-
Saupe):
• Rotations about n leave phase unchanged• Rotations perp. to n take nematic from
one state to another with equal energy on the ground state-manifold
• Slowly Varying, spatially non-uniform rotations cost elastic energy with = Kq2; q=2/length.
• Two new hydrodynamic modes with =-i(K/)q2 = frequency – but decays
( )13
13
ij i j ij
i j ij
Q S nn d
nn d
= -
=á - ñ
n = director
P.G. de Gennes, Physics Lett. 30A, 454 (1969)
P.G. de Gennes, C.R. Acad. Sci. Paris. 266, 15 (1968)Groupe d’Etudes des Cristaux Liquides (Orsay), J. Chem. Phys. 51, 816 (1969)
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de Gennes-Landau Energy
First-order phase transition
Short Range Order Effects in the Isotropic Phase of Nematics and Cholesterics, Mol. Cryst. Liq. Cryst.12, 193 (1971)
First appearance of my name in a Physics Journal:
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Nematic Distortions
Splay Twist
Bend
State invariant under n to –n. Energy invariant under uniform rotations of n – depends only on gradients of n ~ (n)2 ~ n2/R2 )( nn n
)( nn
{ ( ) ( )[ ]
( )[ ] }
22312 1 2
2
3
F d x K K
K
= Ñ × + ×Ñ ´
+ ´ Ñ ´
ò n n n
n n
Frank Energy
2
2( ) ~ Bk Tn
KqqFluctuations:
P.G. de Gennes, C.R. Acad. Sci. Paris. 266, 15 (1968)
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Nematic Defects• Disclinations: Mapping from a closed loop in nematic from any point to its antipode in the ground-state manifold
• Hedgehogs: Mappings from surface enclosing defect to 2D surface of the ground-state manifold
Loops
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PSLW, PSLW, ScienceScience 275,275,
1770 (1997).1770 (1997).
Nematic Emulsions: mixtures of water, mesogen, and surfactant
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Disclinations and colloids
Musevic, I; Skarabot, M; Tkalec, U, et al., SCIENCE 313 954 (2006)
Ravnik, M; Skarabot, M; Zumer, S, et al., Phys Rev Lett 99, 247801 ( 2007)
Each sphere produces one hedgehog: disclination lines carry hedgehog density
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Nematic ElastomersNematogens in a crosslinked network
Spontaneous rotational symmetry breaking in an isotropic elastic medium: nematic order drives stretch
M. Warner and E.M. Terentjev, Liquid Crystal Elastomers (Oxford University Press, New York, 2003)
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Nematic ElastomersConsidered effects crosslinking in LC solvents: Smectics and Cholesterics freeze in anisotropy
Other Elastomer publications:“Réflexions sur un type de polymères nématiques,” C.R. Acad. Sc. Paris B t. 281, 101 (1975)
In Liquid Crystals of One- and two-dimensional order (Springer, Berlin, 1980)
In Polymer Liquid Crystals (Academic Press, New York, 1982)
P.G. de Gennes, M. Hebert, et al. “Artificial muscle based on nematic gels,” Macromolecular Symposia 113, 39 (1997).
M. Hebert, R. Kant, et al. “Dynamics and thermodynamics of artificial muscles based on nematic gels,” J. de Physique 7, 909 (1997).
P.G. de Gennes and K. Okumura, “Phase transitions in nematic rubbers,” Europhysics Letters 63, 76 (2003).
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Thermoelastic Effect
• Large thermally induced strains - artificial muscles
Courtesy of Eugene Terentjev
300% strain
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Soft and Semi-soft Response
Finkelmann, et al., J. Phys. II 7, 1059 (1997);Warner, J. Mech. Phys. Solids 47, 1355 (1999)
Vanishing xz shear modulus
Soft or semi-soft stress-strain for stressperpendicular to order
Soft: spontaneseous symmetry breaking
Semi-soft: frozen-in nematic order with second crosslinking
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Elasticity of Nematic Elastomers( )22 2 3 21
2 Tr Tr Trii
f Bu u C u D um= + - +% % % 213ij ij ij ii
u u ud= -%
21 12 21 2 3 4el
2 2 2 25 1 3
( ) ( )
zz zz aa aa aa ab ab
R Rza za z a z
f C u C u u C u u C u u
C u u K u K u^
¢ ¢ ¢ ¢ ¢ ¢ ¢= + + +
¢ ¢+ + ¶ + ¶
0:
0ij
u
m<
¹%
Transition to stretched
nematic phase with
Broken continuous symmetry: soft or Goldstone mode, vanishing C5
a,b=x,y
Golubovic, TCL, PRL 63, 1082 (1989);Olmsted, J. Phys. II 4, 2215 (1994)
( )/ 2ij i j j i i jk k
u u u u u= ¶ +¶ +¶ ¶
nonlinear strain
P.G. de Gennes, C.R. Acad. Sc. Paris B t. 281, 101 (1975)
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Hydrodynamics, NonlinearitiesThree hydrodynamic sound modes with some velocities vanishing in symmetry directions
Stenull, TCL, PRE 69 ,051801( 2004)
24 4
2
| | ( / | | )
~| | ( / | | )K
d
K d
C s q gq q
K q g q q
h w
h w
+^ ^
- +^ ^
= 4(3 )
5938
(3 )59
42(3 )
59
K
d
d
d
h
h
w
= -
= -
= - -
38/ 591
4/ 5914
~ 1 ln | |
~ 1 ln | |
K a q
C a q
-^
--^
é ù+ë û
é ù+ë û
d<3
d=3
Nonlinearities lead to renormalization of elastic constants
Stenull, TCL, Europhys. Lett. 61, 776 (2003); PRE 69, 02180 (2004); Xing Radzihovsky, Europhys. Lett. 61
G. Grinstein and R. Pelcovits, Phys. Rev. Lett. 47, 856 (1981); Phys. Rev. A 26, 915 (1982)
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Smectics I• Layered structure: periodic in one
dimension• Order parameter:
• Uniform increments of u translate layers and leave energy unchanged
• Directions of molecules and layer normals locked
• Ground-state manifold: line with u=u+nd
00
(x) | | iq uey y -=
Smectic-A
00
( ) . .iqe ccr r y ×= + +nxx
du
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Smectics II
( ) ( )2
3 212sm n||
F d x B u D u Fd^
é ù= Ñ + Ñ - +ê ú
ë ûò n
2 2
||22
Bq q
qw
r^=z
u Fv
t u¶ d
z¶ d
= -
( )212zz zu u u¶= - Ñ
( )23 2 212sm 1zzF d x Bu K u
^é ù= + Ñê úë ûò
Invariance w.r.t. simultaneous rotation of layers and director: Elastic Energy locks u to n; The Higg’s mechanism (D not 0) leaves only one elastic variable
Hydrodynamics
Liao, Clark, Pershan, PRL 30, 639 (1973)
PGG, J. Physique (Paris) Colloq 30, C4 65 (1969)
2
2( ) B
k Tn q
D Kqd =
+F. Brochard, J. Physique 34, 411 (1973)
Fluctuations
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Smectic Topological Defects
Edge Dislocation
Screw Dislocation
Screw dislocation: helicoid orRenaissance staircase
Smectics III
2 ~ln 0u L yá ñ ® ¥ á ñ®
( )
( )
4/ 512 2 4
||
2/ 512 4
1 2||
(q) ~ ln q
(q) ~ ln q
B q
K q
l
l
--
^
-
^
é ù+ê ú
ë ûé ù
+ê úë û
Fluctuation destruction of long-range order:
Non-linearities modify elasticity
2
2
20||
4 2
||
( ) 0( ) ~
0
n
n
q nq qI q
q q
h
h
- +^
- +^
ìï - =ïïíï =ïïî
Safinya, et al. PRL 57, 2718 (1986)
Grinstein & Pelcovits, PRL 47, 856 (1981)
Als-Nielsen et al., PR B 22. 312 (1980)
A. Caillé, C. R. Acad. Sci. Ser. B 274, 891 (1972).
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Smectics and Superconductors
3 2 2
|| ||
2 4120
[ | | | |
|( ) | | | ]
F d x r C
C iq g
yy y
d y y^ ^
= + Ñ
Ñ - +
òn
= Ñ × + ×Ñ ´ + ´ Ñ ´ò 3 2 2 21 2 3
{ ( ) ( ) [ ( )] }nF d x K K Kn n n n n
= ×Ñ ´ò 3*F h d xn n
y= + + *nF F F F
de Gennes, Solid State Commun. 10, 753 (1972).
Frank
Chiral
3 2 2 412[ | | |( 2( / ) ) | | | ]F d x r C i e c g
yy y y= + Ñ - +ò Ah
pm= Ñ ´ò 3 2
0
1( )
8AF d x A
p= - ×Ñ ´ò 31
4HF d xH A
HAF F F F
y= + + Landau-Ginzburg
free energy
LC
SC
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SC in a Field
T
H H
T
Hc
MeissnerHc1
Hc2
Normal Metal
Meissner
Abrikosov
Normal Metal
Hc H
B
HHc2Hc1
B
Normal Metal
Meissner:Type I
Abrikosov:Type II
Type I: < Type II: >
H
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SC’s and LC’sSuperconductor||ei=Cooper-Pair
A= Vector potential
H= magnetic intensity
B=A= microscopic field
B= Maxwell field
normal metal
normal metal in a field
Meissner Phase
Meissner effect
London Penetration depth
coherence length
Vortex
Abrikosov Flux lattice
Liquid Crystals||ei = Mass-density-wave
n = nematic director
h = molecular chirality
k0= n(n) = twist
k0= n(n) = average twist
nematic
cholesteric phase
smectic-A phase
twist expulsion
twist penetration depth
smectic coherence length
screw dislocation
TGB phase
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TGB Phase DiagramDirect analogy with SC
Type II
Renn-TCL, PR A 38, 2132 (1988)Goodby et al., Nature 337, 449 (1989)
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More TGB
H.T. Nguyen et al. J. Phys. II (France) 2 1889 (1992).[10] L. Navailles, P. Barois, and H.T. Nguyen, Phys. Rev. Lett71, 545 (1993); L. Navailles, P. Barois, and H.T. Nguyen,Phys. Rev. Lett. 72, 1300 (1994).
TGBC phase
Quasicrystalline TGBC phaseL. Naivailles, P. Barois, H.T. Nguyen, PRL 71, 545 (1993)
Arindam Kundagrami and T.C. Lubensky, Phys. Rev. E 68, 060703 (2004).
I. Luk'yanchuk, Phys. Rev. E 57, 574(1998).
T. C. Lubensky and S. R. Renn, Mol. Cryst. Liq.Crys. 209, 349-355 (1991).
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More TGB II
C.D. Santangelo and R.D. Kamien, Phys. Rev. E 75 (2007) 011702.
R.D. Kamien and T.C. Lubensky. Phys. Rev. Lett. 82 (1999) 2892.
E.L. Thomas, D.M. Anderson, C.S. Henkee, and D. Hoffman, Nature 334, 598 (1988); S.P. Gido, J. Gunther,E.L. Thomas, and D. Hoffman, Macromolecules 26, 4506 (1993).
( )
3 2 212
2
1 2
1 1;
zz
zz z
F d x Bu KH
u u u HR R
é ù= +ê úë û
= ¶ - Ñ = +
òNonlinearities important
Triply Periodic SmecticScherk’s surface
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Soft SmC ElastomerDirector-strain coupling induces stain in SmC phase
Tilt in xz-plane; stress along y; Soft rotation of tilt direction to yz plane; then “hard” response
Sound velocities of SmC elastomerOlaf Stenull and T.C. Lubensky, Phys. Rev. E 75, 031711 (2007).
J. M. Adams and M. Warner, PR E 71, 021708PR E 72, 011703 (2005).
Olaf Stenull and T.C. Lubensky, PR E 74, 051709 (2006).
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Cosmic Microwave Background2D Nematic liquid crystal
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Polarization of the CMB