Liquid crystals: Lecture 2

Topological defects, Droplets and

Lamellar phases

Support: NSF

Oleg D. Lavrentovich

Liquid Crystal Institute

Kent State University, Kent, OH

Boulder School for Condensed Matter and Materials Physics,

Soft Matter In and Out of Equilibrium,

6-31 July, 2015

2

Nematic LC: nema=thread; aka “disclination”

1922, G. Friedel:

Named “nematics”, the simplest

LC, after observing linear defects,

nema=thread, under a polarized

light microscope

Frank’s model of disclinations in N

k=1/2 k=1

2 21

ˆ ˆdiv curl2

f K =

n n

cos sin 0r zn , n , n , , =

The simplest form (one constant) approximation:

2 22

1 2

0 0

1 1 11 1

2 2

r R

length

S r

drF K rd dr K d

r r

= =

= =

= =

2

core B NI core AF k T T r N / M =

Energy per degree of

freedom

# degrees of freedom

Kleman, ODL Soft Matter Physics, p. 393

2~ ; ~core core

A B NI

MKr k few molecular lengthes F k K

N k T T

=

r

n

1 cosˆdiv 1r

dn n d

r d dr r d

= =

n

1 sinˆcurl 1r

z

ndn d

r d r r d

= =

n

2

20

=

A c =

2 0 1 2 1d k A , / , ,... = =

Euler-Lagrange equation

, cos 1 , sin 1rn n k c k c = , cos , sinx yn n k c k c =

2 2

1

0

ln

r R

length core

corer

dr RF k K k K F

r r

=

=

= =

Escape into the 3rd dimension

4

k=1

k=1/2 k=1

P. Cladis, M. Kleman (1972), R.B. Meyer (1972)

nr = cos r , n = 0, nz = sin r

r = R = 0

r = 0 = / 2

P. E. Cladis, M. Kleman J. Physique 33, 591 (1972), R.B. Meyer, Phil. Mag. 27, 405 (1972)

2 21

ˆ ˆdiv curl2

f K =

n n

1 cosˆdiv sin

rd rn d

r dr dr r

= = n ˆcurl coszdn d

dr dr

= = n

22 2

1 2

0 0

1 cos 1sin 2

2

r R

length

r

dF K d rdr

r r r dr

= =

= =

=

expr

2

2

1

0

cos sin 2

r R

length

r

dF K d

d

=

=

=

Euler-Lagrange equation 2

2

0

cos ; 0 0r

const const

= =

2

2cos sin

=

0

coscos

R

r

dy dp

y p

= =

2arctanR r

R r

=

1 3escaped

lengthF K= 1 lnsin gular

length core

core

RF K F

r=

10 coreR rEscape is preferred when

Escape into the 3rd dimension

r = R = 0

r = 0 = / 2

Point defects in the nematic bulk

-hedgehogs

Escape into the 3rd dimension

Homotopy classification: Uniaxial nematic Nu

7

aaa 21= nnsQ

0

1

n=-n

Degeneracy space

Topological stability is established by mappings from real space onto

the degeneracy (or order parameter) space

k=1

Homotopy classification: Lines in Nu

k=1/2

n -n

G. Toulouse, M. Kleman J. Phys. Lett. 37, L149 (1976), G.Volovik, V. Mineev, JETP 46, 1186 (1977)

Topologically unstable Topologically stable

Topologically stable defect: A non-uniform configuration of the order parameter

that cannot be reduced to a uniform state by a continuous transformation. In

practical terms, to destroy a topological effect, one needs an energy exceeding

the self energy of the defects by many orders of magnitude; e.g. melt the entire

sample.

12

k = 12

k =

G. Toulouse, M. Kleman J. Phys. Lett. 37, L149 (1976),

G.Volovik, V. Mineev, JETP 46, 1186 (1977)

Homotopy classification: Lines in Nu

Disclinations in N are described by

the 1st homotopy group,

comprised of two elements

2 121 2 2/ 0,S Z Z = =

12

k = 12

k =

All semi-integer disclinations are

topologically equivalent to each other and can

be smoothly transformed one into another

G. Toulouse, M. Kleman J. Phys. Lett. 37, L149 (1976),

G.Volovik, V. Mineev, JETP 46, 1186 (1977)

Homotopy classification: Lines in Nu

11

2

2 2/ 0, 1, 2,...S Z N = = Hedgehogs (point defects) in uniaxial N

Homotopy classification: Points in 3D Nu

Radial hedgehog

Hyperbolic hedgehog

1?N =

1?N =

1?N =

1?N =

1

1

1

1 11 1

1

1 1

1

t

t

t

t t

S

t

t t

n ... ... n

n n... ...

u uN du ...du... ... ... ...

n n... ...

u u

=

Topological charges of points, vector fields, t-space

Definition of t-dimensional topological charge:

E. Dubrovin et al, Modern Geometry, Springer, 1984

1 2 tn ,n ,...n=n

t-dimensional vector field:

1 2 1tu ,u ,...u

(t-1)-coordinates

specified on the sphere

around the defect

Point defects in 2D:

2 1

2 1 211 2

2loop

dn dnN k n n dl , , ...

dl dl

= =

1k =

1k =

Circle of all

possible

orientations of

magnetization

Point defects in 3D: Ferromagnetic vs Nematic Sphere of all possible orientations of

magnetization

Topologically stable point defect; to remove it, one

needs to destroy ferromagnetic order on the entire line

Topologically charge: how many times the

magnetization vector goes through all

possible orientations

M. Kleman, Phil. Mag. 27 1057 (1973).

N. Mermin et al PRL 36, 594 (1976)

ˆ u,v sin u,v cos u,v ;sin u,v sin u,v ;cos u,v =m

3 1

4N sin dudv

u v v u

=

1N =

ˆ ˆ=m r

1N =

ˆ ˆ=m r

3 1

4

ˆ ˆˆN dudv

u v

=

m m

m

3 ˆ ˆ1ˆ 1

4N dudv

u v

= =

n n

nUniaxial nematic

N=0

N=2

2

0

NN N

=

Homotopy: Points in 3D Nu

Result of merger of 2 hedgehogs in a uniaxial N in presence of a disclination

depends on the pathway of merger

Similar example for dislocations

in presence of disclinations:

G. Toulouse, M. Kleman J. Phys. Lett. 37, L149 (1976),

G.Volovik, V. Mineev, JETP 46, 1186 (1977)

15

Typical texture of a (thick) Nu

n

singular

disclination nonsingular

disclination

n

200 μm

Eventually, all the defects will disappear, except

maybe one line and one point defect,

mostly through annihilation, as ½+½=0 and 1+1=0!

16

Defects in equilibrium: LC droplets

Anisotropic surface energy Rapini-Papoular surface anchoring potential

S. Faetti et al, PRA 30, 3241 (1984): N-I thermotropic interface is weakly anisotropic:

Elasticity:

ˆ ˆsurface oF f= n υ

~elasticF KR

2 2

2 2ˆ ˆ ˆ ˆ cosf W W = =n υ n υ

5 2 6 2

2 2~ 10 J/m ; ~ 10 J/m ; / ~ 0.1 0.01o oW W

~ 5 pNiK

2 2

0 210; 1R W R

KR KR

10 μmR

The droplets of thermotropic N in isotropic melt are

spherical and contain defects to satisfy surface anchoring

conditions

The structure is determined by the balance of

anisotropic surface tension and internal elasticity n̂

υ̂

Balance of elasticity and surface anchoring

leads to the following expectation for scaling behavior:

KRFelastic ~2~ WRFanchoring

μm101.0~/ WK

WK //K R

ˆ ˆ||n n00 =

ˆ ˆn n

2/0 =

n̂

n̂

Liquid Crystals 24, 117 (1998)

Defects correspond to the equilibrium state of the (large) system

Defects in equilibrium: LC droplets

(a)

(b)

(c)

(d)

(e)

υ

n̂

υ

n̂

ˆ||υ n

N droplets in glycerine with temperature

varied anchoring axis; topological dynamics

of boojums, disclination loops and

hedgehogs

NB: Defects correspond to the equilibrium

state of the system; they help to minimize

the sum of the anisotropic surface tension

and bulk energy.

Defects in equilibrium: LC droplets

Do we really want to minimize the

energy for each and every surface

angle?!

Topological dynamics of defects in LC drops

G.E. Volovik et al, Sov. Phys. JETP 58 1159 (1983)

,

1

4 2k N

kC d d N

= =

n nn n n

1k =

1/ 2C =

Bulk point defect hedgehog: 1N =

2 0cos2

C

= 1C =

Replace director with a vector

Surface point defect boojum: two topological charges, 2D and 3D;

υ

n̂

Hedgehog spreadable into

a disclination loop:

1N =

(a)

(b)

(c)

(d)

(e)

υ

n̂

υ

n̂

ˆ||υ n

b

ii

k E =

Poincare theorem: conservation law for vector

fields tangential to the surface

1/ 2

h b

jj

N E

= =

2E =Euler characteristic for sphere

G.E. Volovik et al, Sov. Phys. JETP 58 1159 (1983)

Gauss theorem: conservation law for vector

fields perpendicular to the surface

,1 1 1 1

11 1 0

2i i

b h b b h b

k N s j i ji j b i j b

C C N k N

= = = =

= =

n n

Topological dynamics of defects in LC drops

Applications of LC drops

Privacy windows:

Polymer Dispersed Liquid Crystals

(JW Doane et al, LCI, Kent)

0=E

effective polymern n

0E

o polymern n=

Droplets as Biosensors:

Microscale LC Droplets

(Oil-in-Water Emulsion)

H2O H2O

‘Caged’ LC Droplets

(Droplets in Polymer Capsules)

H2O

LC

Immobilized LC Droplets Internalized LC Droplets

Sensing in Biological Environments:

Abbott and Lynn (UW-Madison)

Transitions triggered by:

Cationic surfactants

Anionic surfactants

Bacterial endotoxin

HTAB (10 μm)

H2O

LC

Angew. Chem. 2013; Langmuir 2014

Not by:

Proteins, serum, etc.

Abbott and Lynn (UW-Madison)

24

Disclinations are not “material” lines and can cross each other.

Two disclinations connecting opposite plates of a nematic cell;

plates are twisted; disclination ends reconnect

2/11 =k

2/12 =k

T.Ishikawa et al, Europhys. Lett. (1998)

2/11 =k 2/12 =k2/11 =k 2/12 =k

Reconnection of disclinations in Nu

Topological test for biaxiality of a nematic:

Two disclinations ½ belonging to different classes cannot

cross each other, the trace is a third line of strength 1; the

two separated disclinations interact through a potential that

resembles interaction of …. Quarks!

G. Toulouse, J. Phys. Lett. 38, L67 (1977)

Reconnection of disclinations in biaxial Nbx

U KLL

3

1 2/ 0;1;1/ 2 ;1/ 2 ;1/ 2x y zS D Quaternion units = =

Degeneracy space: Solid sphere; each point describes a state of three

directors, n,m,l. In biaxial nematics, the strength 1 disclinations cannot

escape (strength 2 can). There are three different classes of ½

disclinations with k semi-integer and one with k=1

1k = does not escape!

P. Dirac, Proc. Roy. Soc. (London) A133, 60 (1931)

Ch droplet, called Robinson spherulite or Frank-Price

structure,

C. Robinson et al, Disc. Faraday Soc. 25, 29 (1958);

Kurik et al, JETP Lett 35, 444 (1982)

Cholesteric drops and Dirac monopole

@ 2director each layerk =

1helical axisN =

T. Orlova et al, Nat. Comm. 6, 7603 (2015)

Two orthogonal

vector fields: helical

axis and director

(magnetic field and

vector-potential)

Point hedgehog in helical axis (magnetic) field and an

attached disclination (Dirac string) in the director field

27

Rare exception of round cohesive droplets:

Ch-SmA phase transition

Ch droplets in glycerine+lecithin; cooling down leads

to extended shapes, then nucleation of spherical SmA

sites; the process often results in division of droplets

(Nastishin et al Sov Phys JETP Lett 1984;

EurophysLett 1990) 84.0 C

83.4 C

82.8 C

82.7 C

82.6 C

82.5 C

82.4 C

82.0 C

28

(LC)2: Lyotropic Chromonic Liquid Crystals

n

7 6 2~ ~ 10 10 J/mBo

k T

LD a

5 2~ 10 J/mW

Lyotropic I-N interface might

be strongly anisotropic and

be influenced by elasticity

Model of surface tension: P. van der Schoot J. Phys.

Chem B 103, 8804 (1999)

J. Bernal and I. Fankuchen, J. Gen. Physiol. (1941): tactoids as N nuclei in tobacco mosaic virus dispersions

29

Droplets of chromonic N in isotropic melt

(tactoids)

30

Surprise #1: Surface-located, not bulk

Vertical cross-section image;

fluorescent confocal

microscopy

Tortora et al., PNAS 108, 5163 (2011)

31

Surprise #2 (mild): Contact angle changes along

the perimeter

Vertical cross-section image;

fluorescent confocal

microscopy

Tortora et al., PNAS 108, 5163 (2011)

32

Surprise #3: Twist

Right-twisted tactoid

Left-twisted tactoid

Tortora et al., PNAS 108, 5163 (2011)

33

Mechanism: “Geometrical” anchoring+large K1/K2

Nematic between two

isotropic parallel

boundaries:

Degenerate in-plane

orientation

One plate tilts, the other

sets “physical anchoring”

say, along the long axis of

the tactoid’s footprint:

balance of twist and splay

establishes a twist angle

One plate tilts: alignment

perpendicular to the

thickness gradient is the

only one without

distortions; other directions

cause splay

x

Tortora et al., PNAS 108, 5163 (2011)

34

Mechanism: “Geometrical” anchoring+large K1/K2

2 2

1 2f K Kz z

2

2 1/K K Condition for the twist:

Elastic energy of a tilted element:

L. Tortora et al., PNAS 108, 5163 (2011)

Twisted tactoids: An example of chiral symmetry braking in a molecularly non-chiral

system; only spatial confinement and elastic anisotropy are needed to produce

macroscopic chiral purity.

2 2 21 212 2

K KF

d d

Bulk equilibrium:

2 2

2 20; 0

z z

= =

2arcsin sin cos 1 / 2 ;z d z d= = =

Easy to fulfill as in

chromonics, 2 1/ ~ 0.1 0.03K K

S. Zhou et al., Soft Matter 10, 6571 (2014)

Twist

deformations

reduce the cost

of splay

deformations

2D tactoids and Kibble mechanism Isotropic-Nematic transition: Anchoring-induced topological defects in each and every

nuclei of the N phase, as long as it is large enough, /R K W

60 mm

(b) υn̂

n̂

a

Kibble (1976) Model of formation

of cosmic domains and strings

cosmic

string

36

2D tactoids and Kibble mechanism in (LC)2

Anchoring-

produced surface

defects-boojums

T = 33.9 oC T = 33.7 oC, t = 0 s T = 33.7 oC, t = 21 s T = 33.4 oC

Kibble-like production of disclinations as a result of tactoids merger

2 1n

k

k

c c m =

Conservation law for

positive and negative cusps:

negative cusp

50 μm

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

37

2D tactoids and k=1 disclinations

0 nm

136 nm

(a) T = 32.7 oC (c) T = 29.7 oC, t = 0 s (d) T = 29.7 oC, t = 603 s (b) T = 29.8 oC

50 μm

1

2

1

2 1

2

1 2 1 2 1 2

1 1 0

1

ln 2 2 22 2

pair

K Lf f r w

r

=

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

38

Drops of isotropic phase in N environment with

distroted director: A balance of surface tension,

anchoring, and elasticity

39

Equilibrium shape+director?

Difficult problem, requires to minimize both the anisotropic surface energy and elastic

interior/exterior

First step: Assume “infinite” elasticity (frozen director); then calculate the equilibrium shape

of the I tactoid at the core, using the angular dependence of the surface tension for each

disclination

k = -1/2

20 1 cos 1w k =

2w =

Polar plot of surface tension of an isotropic

island inside a nematic region representing a

disclination of -1/2 strength;

Problem: Find the shape that minimizes the

anisotropic surface tension

?

?

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

2

ˆ ˆ minFO

V s

f dV W dS n r n υ

Radius of curvature:

2 2 2' ' "R r r = =

" 0 Missing orientations and cusps:

" 0 Round shape, all orientations of I-N interface:

2 221 1 4 1 cos 1 2 1 0w k k k w

40

Equilibrium shape by Wulff construction for

distorted director

υ

x

A

y

Equilibrium shape

Polar plot of surface tension

O

rB

Wulff construction for crystals:

L. Landau, E. Lifshits, Statistical

Physics (1964)

20 1 cos 1w k =

cosr =

cos

sin 'r

r

= =

2 '2 ; arctan'

r

= =

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

Shape of

Domain

Wulff construction by Mathematica

The interior envelope describes the shape in equilibrium

Wulff

Construction

Draw the tangent lines at each point in a polar plot of σ (θ)

Cusp

3c N tactoid

Cusp

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

42

Equilibrium shape by Wulff construction for

distorted director

Missing orientations and cusps:

2 221 1 4 1 cos 1 2 1 0w k k k w

1/ 2; 2k w= =1/ 2; 2k w= = 0; 2k w= = 1/ 2; 20k w= =1; 2k w= =

5 10 15 20

10

5

5

10

1 1 2 3

2

1

1

2

Never the case for k=1/2 and k=1; for k=0, wc=1; for k=-1/2, wc=2/7; for k=-1, wc=1/7

YK Kim et al., J Phys C ond Matt 25 404202 (2013)

43

Summary/What have you learned

Disclinations: Frank model, line energy ~ln of size, ~k2

Integer disclinations: Escape into the third dimension

Semi-integer: Stable

Homotopy classification: A natural language to describe

defects in any medium, LCs and superfluid He-3 including

Surface anchoring: Controls topology and energy of defects;

Defects occur as equilibrium features in LC droplets,

…Including the nuclei during the phase transition from the

isotropic phase

Cholesteric: Dirac monopoles

Chromonic droplets: Spontaneous chiral symmetry broken;

shape is strongly dependent on director deformations, the

problem of full energy (elastic+surface tension+anchoring)

minimization not solved yet

Liquid crystals: Lecture 2.2

Lamellar phases

Support: NSF

Oleg D. Lavrentovich Liquid Crystal Institute and

Chemical Physics Interdisciplinary Program,

Kent State University, Kent, OH 44242

Boulder School for Condensed Matter and Materials Physics,

Soft Matter In and Out of Equilibrium,

6-31 July, 2015

45

Lamellar Phases

1. Free Energy Density

2. Weak distortions: Dislocations, Undulations

3. Strong distortions: Focal conic domains, grain boundaries

Content

46

Smectics A

(thermotropic)

Smectic A

(lyotropic)

Cholesteric

(chiral N)

1-10 nm >0.1 mm

Twist-bend

nematic

Lamellar phases

47

Stripe magnetic domain

Layered structure in magnetic fluid

(M. Seul et. al., P.R.L., 68, 2460 (1992))

(C. Flament et. al., Europhys. Lett., 34, 225 (1992))

Weak perturbations: Dislocations, undulations

48

Elasticity of lamellar phase; 1D translational order

Curvature:

2

2

x

u

Compression/dilation:

z

u

x

z

2

2

22

2

1

2

1

=

x

uK

z

uBf

cos/d'd =

d’ d

x

z

Correction to make the model invariant w.r.t. uniform rotations: 2

2

1

x

u

z

u

,u x z

49

2

2

222

2

1

2

1

2

1

=

x

uK

x

u

z

uBfEnergy density

=

3m

JB

== N

m

JK

~ ?K

periodB

=Material length

Elasticity of lamellar phase; 1D translational order

50

Elasticity of Smectics: Dislocation

By fitting u(x,z), one can measure

Brener and Marchenko PRE‘99

=

z2

xerf1

2

14/bexp12y,xu

ln

15mm

1 2 3

1 2 3

4 4 -100 -50 0 50 100

20

40

60

80

100

120

140

Horizontal Position (mm)

T.Ishikawa, ODL Phys Rev E (1999)

K

B=

14.9period m= m

mm 7.2=

0.2 period

Long range effect of layers deformations

Look for solution of the type:

qxuzu q cos0 == 0=zu

=

L

zqxuu q expcos

4 2

4 20

u uK B

x z

=

E-L equation

qxuzu q cos0 ==

0=zu

01

2

4

1 =L

BqK

Saint-Venaint principle is not applicable to SmA materials;

Smectics: A good model of “la Princesse sur la graine de pois”

If a pea is 1 mm, layer thickness 10 nm, then the pea is felt over 100 m!

~L

2macroscopic scale

microscaleL

== =

G. Durand (1968)

,u x z

2

2

1 BL

q K

=

Undulations in Cholesteric caused by E field

Photo by Luba Kreminska

Senyuk et al PRE 74, 011712 (2006)

Undulations in Cholesteric (2D)

3D version: Senyuk et al PRE 74, 011712 (2006)

22 2 222

02

1 1 1 1

2 2 2 2a

u u u uf K B E

x z x x

=

e e

2

0,

1

2anchoring

z d

uf W

x=

=

xqzuzxu xsin, 0= 2 2 2

0'' 0x x aB q Kq E = e e

2/

2 0'az

xWqB=

= zqconst zcos=

22xxz qqq =

2

0 /aE B = e e

2 2

2 2cot2

x

xx x

B qW

dqq q

=

0

2c

a

KE

a

e e

2

0 2

81

3 c

Eu W

E

=

2

0 2

81

3 c

Eu

E=

Normalized field (H/Hc)

0.98 1.00 1.02 1.04 1.06 1.08 1.10

Dis

pla

ce

me

nt

(mm

)

0

2

4

6

8

10

12

14

= (W=2.4X10-6J/m2)

= (W=2.2X10-6J/m2)

Undulations in Cholesteric (2D)

Ishikawa et al PRE 63, 030501(R) (2001)

2

0 2

81

3 c

Eu W

E

=

2

0 2

81

3 c

Eu

E=

Senyuk et al PRE 74, 011712 (2006)

Undulations in Cholesteric above threshold

Immediately above

the threshold

Well above the threshold,

3D,

Anchoring takes over,

forcing parabolic domain

walls

Well above the

threshold, 2D cell;

broken surface

anchoring

56

mm40

Strong perturbations: Focal conic domains

57

principal radii

Compression/Dilation: Curvature:

2

0

21

21

2

21

21 1

111

=

d

dB

RRK

RRKf

2R

1R

2R

1R

Elasticity of Smectics: Strong distortions

58

2

3~ ~ 1curv

dilat

F KR

F BR R

if R

=K

B~ d0

22

01 12 2

1 2 1 2 0

1 1 1curv dilat

d df f f K K B

R R R R d

= =

At large scales of deformations,

the curved layers are equidistant

d

R d

Elasticity of Smectics

2D focal surfaces

shrink into lines

layers

2D focal surface,

high energy (infinite

curvatures)

Curved smectic layers: Dupin Cyclides

C.P. Dupin, Applications de Géométrie (Paris, 1822)

J. Maxwell, On the cyclide, Q. J. Pure Appl. Math 9, 111 (1868)

To reduce the energy, focal surfaces in SmA

degenerate into 1D focal lines, that can be only of three types:

(a) circle and straight line, or

(b) confocal ellipse and hyperbola,

(c) two confocal parabolae

producing a Focal Conic Domain (FCD); the smectic layers are Dupin cyclides

Si

1F

2F

1C

2C

2R

1R

0S

n̂ r

1910, G. Friedel, F. Grandjean:

Deciphered SmA structure

from observation of

focal conic domains;

X-ray was not available

Smectics and focal conic domains

mm40

60

61

Toroidal Focal Conic Domains

Smoothly fits into the system of

parallel layers

62

Classification of FCDs by Gaussian curvature

00

0

0

FCD-I FCD-III FCD-II

0

How to describe the FCD analytically?

z = 0,x2

a2

y2

b2 = 1 y = 0,

x2

a2

b2

z2

b2 = 1

M x = a cosu,

y = b sin u,

0 u 2 ,

" coshM ,

" sinh

x c vv

z b u

=

=

M"M' cosh cosa v c u=

c2

= a2

b2

M’’

M’

z

x

1

3

3

2

2

M

P

Q

Parametrization of conics with coordinates u and v

defined within the cyclides

Ellipse Hyperbola

Curvilinear coordinates r, u, v

1

3

3

2

2

M

P

Q

M’

M”

R1 = ccosu r 0

R2 = acosh v r 0 2

12

1 2 1 2

1 1 1curv

FCD

F K K dVR R R R

=

2

2

bdV dr d dr du dv

= =

Curvature energy functional

2

1 1" 0

coshR a v r = =

1

1 1' 0

cosR c u r = =

Kleman et al PRE 61, 1574 (2000)

2 1M"M' cosh cos cosh cosa v c u R R a v r r c u= = =

Let us introduce the third coordinate r that “counts” the layers

2 cosh 0R a v r= 1 cos 0R c u r=

d(r) =b2

2dudv

0 0

d(r ) = ABdudv

A du= |uM| = u ( r

M M) =

b

du ,

= r' ' r'

B dv = b

dv

|a coshv ccosu| ex' e1

x' 'M’M”= =

Surface elements of FCD I and FCD II

1 2

1 2(1 )

cos cosh

du dv dF K e a

e u v

=

2

2 22 (1 )

cosh cos

du dv dF K K e a

v e u

=

= r / a

cosh2 v e2

1 e2 =

1

cos2

x

rcutoff = a coshv rc

rcutoff = ccosu rc

f =1

2K

2 K

2

2curv

FCD

bF f dudvdr

=

f =1

2K

2 2K K 1 2curvF F F=

Mathematica©

cannot do it….

substitutions such as lead to the desired result...

Curvature energy of FCD-I

Kleman et al PRE 61, 1574 (2000)

0 e 1

2

2 2 2 14 1 ln 2curv core

c

a eF a e e K K F

r

=

K 28coreF a K e E

, E x K x complete elliptic integrals of the first and second kind

valid for any

Curvature energy of FCD-I

curvF

Kleman et al PRE 61, 1574 (2000)

Circular FCDs-I

a = be = 0 = 0

22 ln ln 2 2c

aF a K K K

r

=

Home assignment: Derive F for a circular FCD

L, sample size

Black: tangential anchoring

White: Normal anchoring

F b ~ Kb b2

F

b= 0 b

*~ K /

Sov Phys JETP’83

Appolonius

filling

Anchoring-Controlled FCD Assembly

The smallest FCDs

are macroscopic

40 μm

R1

R2

FCDs form families with common apex; Bragg, Nature (1934)

Associations of FCDs

R1

R2

Friedel, 1922: When two coplanar ellipses are in contact at M,

the two corresponding hyperbolae have two points of intersection P and Q

and the domains have two generatrices MP and QM of contact.

Law of Corresponding Cones

fcurv = 12 K

1

R1

1

R2

2

K 1

R1R2

At the line of contact of two FCDs,

the curvature energy densities are equal

since the radii of curvature are the same; there

is no line energy at the line of contact

0 e 1

2

2 2 2 14 1 ln 2curv core

c

a eF a e e K K F

r

=

K 28coreF a K e E

valid for any

Curvature energy of FCD-I

Energy decreases as eccentricity increases.

However eccentricity is not a minimization

parameter as it is often fixed by the geometry of

layers outside the FCD curvF

Grain Boundaries in SmA

0 dislocation wall / 2

de Gennes, 1970

curvature wall

FCDs:

intermediate

FCD-filled Grain Boundary-Ist hint

FCD-filled Grain Boundary-Ist hint

To calculate the energy of the FCD wall, we need to know:

•FCD energy (curvatures and defect cores)

•Spatial filling pattern, size distribution

•The energy of residual areas

M. Kleman et al, Eur. Phys. J E 2, 47 (2000)

Residual Areas with Dislocations

Wresidual

disloc~ Bb

2e

L

b

n

1 e2

1 / 2

per unit length ~ ~uf Bb Bae

1/2

2 2~ 1

n

disloc

residual

LF Be b e

b

~ ~disloc

residual

bF f b Number of gaps Bae b Be b

ab

=

bu = 2ae

Dislocations with

total Burgers vector

(Boltenhagen et al, J. Physique (1991)

M. Kleman et al, Eur. Phys. J E 2, 47 (2000)

Residual Areas with Curvatures

curv = 2B tan

2

2

cos

22

2

Arccos~ 2 1

1

n

curv

resid

e e LF Bb

be

C. Blanc, M. Kleman, 1999

M. Kleman et al, Eur. Phys. J E 2, 47 (2000)

g b ~L

b

n

P b ~E e2 1 e2

L

b

n

b

b ~1

1 e2

L

b

n

b2

g b ~L

b

n

P b ~L

b

n

b b ~L

b

n

b2

n 1.32

From Circles to Ellipses

n 1.32

Projective properties of conic sections lead to:

M. Kleman et al, Eur. Phys. J E 2, 47 (2000)

disloc

bulk core residualF F F F=

1 2

2 2

1~ 1 ln 2 2

nL L

bulk bulk b

x b b

x Ldg x f x K e e n dx

xF

=

=

Ka

1 2

2 2

1~ 1

nL L

core core c

x b b

Ldg x f x K e e n dx

xF

=

=

Ea

1/2

2 21

n

disloc

residual

LF Be b e

b

=

FCD wall with Dislocations

M. Kleman et al, Eur. Phys. J E 2, 47 (2000)

/ 0disl

bulk core residF F F b =

bdisl

*

~

1

e

n

2 nab 1- e

2 K e2

ln

2bdisl

*

2

acE e2

FCD

disle ~

1

n 1 L2

L

n

B3

bdisl

*

n1

1 e2

1/ 2

ebdisl

*

ab

n

n 11 e

2 K e2

01

23

4

Log y

0.2

0.4

0.6

0.8

e

0

5

10

15

W

01

23

4

Log ylogb

macroscopic bdisl

*

FCD wall with Dislocations

Photo: Claire Meyer

Grain boundaries with FCDs

Domain walls in SmA

= 0 = / 2 =

dislocations FCDs, gaps with

dislocations

FCDs, gaps with

curvatures curvatures

85

Conclusions/What have you learned

Lamellar phases

Both compressibility and curvatures are generally important in weak elastic

deformation, such as dislocations, surface perturbations, undulations

Strong deformations such as focal conic domains are described sufficiently well

by the curvature term; layer thickness is preserved everywhere except at singular

lines (confocal pairs) that are remnants of singular focal surfaces

Observation of focal conic domains led to correct identification of smectics as

1D periodic stacks of fluid 2D layers

Focal conic domains participate in relaxation of surface anchoring and grain

boundaries