Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* - - PowerPoint PPT Presentation

▶
analysis of a ginzburg landau type energy model for
SMART_READER_LITE
LIVE PREVIEW

Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* - - PowerPoint PPT Presentation

Apr 10, 2023 19 likes •512 views

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* Liquid Crystals with Defects

slide-1
SLIDE 1

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* Liquid Crystals with Defects

Sean Colbert-Kelly, joint work with Daniel Phillips and Geoffrey McFadden

Applied and Computational Mathematics Division Seminar Series National Institute of Standards and Technology

May 28, 2013

Colbert-Kelly Analysis of a G-L Energy

slide-2
SLIDE 2

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-3
SLIDE 3

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-4
SLIDE 4

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

GL functional is defined as Eε(u) = 1 2

  • Ω |∇u|2 + 1

2ε2 (1−|u|2)2 dx Introduced in study of phase transition problems in superconductivity (also used in superfluids and mixture of fluid states) u - complex order parameter (condensate wave function/concentration/vector field orientation) ε - coherence length which can depend on temperature (ξ(T))/diffuse interface/core radius

Colbert-Kelly Analysis of a G-L Energy

slide-5
SLIDE 5

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

When in equilibrium, the order parameter u minimizes Eε. Taking variations of u, the following must be satisfied δEε =

  • Ω[−∆u − 1

ε2 u(1−|u|2)]δu dx = 0 = ⇒ −∆u = 1 ε2 u(1−|u|2) Ex: ut = u +tv, δEε = dEε

dt (u +tv)|t=0, δu = v

Colbert-Kelly Analysis of a G-L Energy

slide-6
SLIDE 6

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Example in 1D

The Euler-Lagrange (E-L) equation in 1D then becomes −uxx − 1 ε2 u(1−u2) = 0 Solution: uε = tanh(

x √ 2ε ) given the boundary conditions

u(0) = lim|x|→∞ux(x) = 0.

Colbert-Kelly Analysis of a G-L Energy

slide-7
SLIDE 7

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

The function y = (1−t2)2 (Two-well potential in 1D)

Colbert-Kelly Analysis of a G-L Energy

slide-8
SLIDE 8

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Plot of solutions for various epsilons

Colbert-Kelly Analysis of a G-L Energy

slide-9
SLIDE 9

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-10
SLIDE 10

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

What are LCs

Figure: The molecular orientation of different states of matter. Left - Solid, Middle - Liquid Crystal, Right - Isotropic Liquid

Colbert-Kelly Analysis of a G-L Energy

slide-11
SLIDE 11

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Types of LCs

Figure: Arrangement of Molecules in particular LCs. Left - Nematic LCs, Middle - Cholesteric (Chiral Nematic) LCs, Right - Smectic LCs

Colbert-Kelly Analysis of a G-L Energy

slide-12
SLIDE 12

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Smectic C* Liquid Crystal Molecular Orientation

Figure: Left Two Figures Source: http://barrett-group. mcgill.ca/teaching/liquid_crystal/LC03.htm

Colbert-Kelly Analysis of a G-L Energy

slide-13
SLIDE 13

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Director Projection onto Plane

Colbert-Kelly Analysis of a G-L Energy

slide-14
SLIDE 14

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-15
SLIDE 15

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Introducing a dust particle

Colbert-Kelly Analysis of a G-L Energy

slide-16
SLIDE 16

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Figure: The effect of impurity ions on a thin film Smectic C* liquid crystal[LPM]

Colbert-Kelly Analysis of a G-L Energy

slide-17
SLIDE 17

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Energy Described over Smectic C* Liquid Crystals

Consists of elastic energy, anchoring energy at domain boundary, and anchoring energy at boundary of defect core Energy from core boundary negligible. Anchoring energy at domain boundary results from polarization field.

Colbert-Kelly Analysis of a G-L Energy

slide-18
SLIDE 18

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Effect of polarization field

pn×v = ⇒ p⊥c The elastic energy contribution from the polarization field is described as

  • Ω ∇·pdx =
  • ∂Ω p·ν dσ

where ν is the outer unit normal vector on ∂Ω

Colbert-Kelly Analysis of a G-L Energy

slide-19
SLIDE 19

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Want

  • ∂Ω p·ν dσ to be as negative as possible.

= ⇒ p = −αν, α ∈ R+ on ∂Ω = ⇒ cτ on ∂Ω Introducing boundary values model effect of spontaneous polarization.

Colbert-Kelly Analysis of a G-L Energy

slide-20
SLIDE 20

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

The resulting framework becomes minimizing

  • Ω k1(div u)2 +k2(curl u)2 dA

u = (u1,u2),|u| = 1 div u = ∂x1u1 +∂x2u2, curl u = ∂x1u2 −∂x2u1 splay and bend constants k1,k2 > 0, k1 = k2 to incorporate electrostatic contribution from p. {u ∈ H1(Ω) : |u(x)| = 1 for x ∈ Ω and u = g on ∂Ω} = / for deg g := d > 0.

Colbert-Kelly Analysis of a G-L Energy

slide-21
SLIDE 21

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-22
SLIDE 22

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

We study instead Jε(u) = 1 2

  • Ω k1(divu)2 +k2(curlu)2 + 1

2ε2 (1−|u|2)2 dx =

  • Ω jε(u,∇u)dx

(1) u ∈ H1

g(Ω) = {u ∈ H1(Ω;R2) : u = g on ∂Ω}

where g is smooth on ∂Ω, |g| = 1, and deg g = d > 0

Colbert-Kelly Analysis of a G-L Energy

slide-23
SLIDE 23

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Set k = min(k1,k2). k1(div u)2 +k2(curl u)2 = k1|∇u|2 +(k2 −k1)(curl u)2 +2k1det ∇u = k2|∇u|2 +(k1 −k2)(div u)2 +2k2det ∇u If k = k1, all constants in second line are positive and if k = k2, all constants in third line are positive.

Colbert-Kelly Analysis of a G-L Energy

slide-24
SLIDE 24

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Splay Configuration

us = ± x |x| = ±(x1,x2) |x| curl us = 0 = ⇒ (div us)2 = |∇us|2 = 1 |x|2 for x = 0

Colbert-Kelly Analysis of a G-L Energy

slide-25
SLIDE 25

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Bend Configuration

ub = ± xt |x| = ±(−x2,x1) |x| = ±i x |x| div ub = 0 = ⇒ (curl ub)2 = |∇ub|2 = 1 |x|2 for x = 0

Colbert-Kelly Analysis of a G-L Energy

slide-26
SLIDE 26

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Choose b1,...,bd, fix R > 0 and define a particular test function ˜ uε.

Colbert-Kelly Analysis of a G-L Energy

slide-27
SLIDE 27

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Let uε ∈ H1

g(Ω;R2) be a minimizer to Jε in the set of admissible

  • functions. Then from our construction

Jε(uε) ≤ Jε(˜ uε) ≤ kπd log(1 ε )+C1

Colbert-Kelly Analysis of a G-L Energy

slide-28
SLIDE 28

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Now, note that k

  • Ω det ∇u dx = kπd for all u in the set of

admissible functions. Hence, if k = k1,

  • Ω k1(div u)2 +k2(curl u)2 dx

=

  • Ω k|∇u|2 +(k2 −k)(curl u)2 +2kdet ∇u dx

=

  • Ω k|∇u|2 +(k2 −k)(curl u)2 dx +2kπd

≥

  • Ω k|∇u|2 dx

Colbert-Kelly Analysis of a G-L Energy

slide-29
SLIDE 29

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Jε(uε) ≥ 1 2

  • Ω k|∇uε|2 + 1

2ε2 (1−|uε|2)2 dx ≥ kπd log(1 ε )−C2 The last inequality is due to the work of Bethuel, Brezis, and Helein [BBH] and Struwe [St]

Colbert-Kelly Analysis of a G-L Energy

slide-30
SLIDE 30

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

With the two inequalities, we obtain the following estimate 1 ε2

  • Ω(1−|uε|2)2 dx ≤ C3

From the above inequality, we can show uεC(Ω),ε∇uεC(Ω) ≤ C4 for 0 < ε < 1.

Colbert-Kelly Analysis of a G-L Energy

slide-31
SLIDE 31

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Define v(y) = uε(εy +x0), x0 ∈ Ω,y ∈ B1(0) := B1. = ⇒

  • B1

(1−|v|2)2 dy ≤ C3

Colbert-Kelly Analysis of a G-L Energy

slide-32
SLIDE 32

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

E-L Equations

−k1∇(∇·u)+k2∇×(∇×u) = 1 ε2 u(1−|u|2) Identifying L u = −k1∇(∇·u)+k2∇×(∇×u), then for v defined

  • n B1 solves

L v = v(1−|v|2). = ⇒ vC1(B1/2) ≤ C4 where C4 does not depends on x0, giving the estimates.

Colbert-Kelly Analysis of a G-L Energy

slide-33
SLIDE 33

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Jε(uε) ≤ kπd log( 1

ε )+C1

Jε(uε) ≥ kπd log( 1

ε )−C2 1 ε2

  • Ω(1−|uε|2)2 dx ≤ C3

uεC(Ω),ε∇uεC(Ω) ≤ C4 With these estimates, using the Structure and Compactness results of Lin [L], we obtain a family {uε} of functions that satisfy the following: uεℓ(x) → u∗(x) :=

d

∏

j=1

x −aj |x −aj|eih(x) where aj ∈ Ω, al = aj for l = j and h ∈ H1(Ω) for some subsequence εℓ → 0; convergence is strong in L2 and weakly H1

loc(Ω\{a1,...,ad}).

Colbert-Kelly Analysis of a G-L Energy

slide-34
SLIDE 34

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Because {uε} are minimizers to Jε, we obtain stronger convergence, i.e. uεℓ → u∗ in Cα

loc(Ω\{a1,...,ad}) and

Cm

loc(Ω\{a1,...,ad}). Furthermore |uℓ| → 1 uniformily away

from {a1,...,ad}. This gives us information away from the cores but not much about defects or what is occurring near them.

Colbert-Kelly Analysis of a G-L Energy

slide-35
SLIDE 35

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

We analyzing the canonical map u∗(x) for x near each defect aj Ω = (Ω\∪d

j=1Bρ(aj))∪(∪d j=1Bρ(aj)\Brε(aj))∪(∪d j=1Brε(aj)))

ε << rε = o(1)

Colbert-Kelly Analysis of a G-L Energy

slide-36
SLIDE 36

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

The function u∗ = ∏d

j=1 x−aj |x−aj|eih(x) satisfies

    

  • Ω k1|∇h|2 +(k2 −k1)(curl u∗)2 dx < ∞

if k = k1

  • Ω k2|∇h|2 +(k1 −k2)(div u∗)2 dx < ∞

if k = k2

Colbert-Kelly Analysis of a G-L Energy

slide-37
SLIDE 37

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

x −aj |x −aj| = eiθj(x),x = aj

Colbert-Kelly Analysis of a G-L Energy

slide-38
SLIDE 38

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Fix an. Then set φn = ∑j=n θj +h. Then we have         

  • Bρ(an)

sin2(φn) |x −an|2 dx ≤ C if k = k1

  • Bρ(an)

cos2(φn) |x −an|2 dx ≤ C if k = k2. The constant C does not depend on ρ.

Colbert-Kelly Analysis of a G-L Energy

slide-39
SLIDE 39

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

       1 |∂Bρ(an)|

  • ∂Bρ(an) φn dx → mnπ

for some mn ∈ Z if k = k1 1 |∂Bρ(an)|

  • ∂Bρ(an) φn dx → π

2 +mnπ for some mn ∈ Z if k = k2. In terms of the limit function, the above limit implies that u∗(ρy +an) →

  • ±y

if k = k1 ±iy if k = k2 in L2(∂B1) as ρ → 0. Hence, one pattern has less energy than the other in either case. (k2 < k1 = ⇒ bend pattern has less energy than splay pattern)

Colbert-Kelly Analysis of a G-L Energy

slide-40
SLIDE 40

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Figure: g = eiθ,k2 < k1

Colbert-Kelly Analysis of a G-L Energy

slide-41
SLIDE 41

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Figure: g = e2iθ,k2 < k1

Colbert-Kelly Analysis of a G-L Energy

slide-42
SLIDE 42

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Now we want to show that these locations minimize the energy

  • ver the domain. Again, construct the proper test function ˜

vℓ

Colbert-Kelly Analysis of a G-L Energy

slide-43
SLIDE 43

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Let a = (a1,...,ad), for simplicity, let k = k1. Then we can write k1(div u)2 +k2(curl u)2 = k|∇u|2 +(k2 −k)(curl u)2 +2kdet ∇u Using the constructed test function, we can show lim

ℓ→∞

  • Jεℓ(uℓ)−kπd ln

1 εℓ

  • = kW(a)+H(a,k1,k2)+dγ

where a minimizes kW(b)+H(b,k1,k2), b ∈ Ωd.

Colbert-Kelly Analysis of a G-L Energy

slide-44
SLIDE 44

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Gb = ∑d

n=1 ln(|x −bn|),

W(b) =1 2

  • ∂Ω 2Gb(∂τg ×g)−(∂νGb)Gb dσ +πd

− ∑

m=n

π ln(|bn −bm|) and H (b,φ,k1,k2) = 1 2

  • Ω k1|∇φ|2 +(k2 −k1)(curl v)2 dx if k = k1

H(b,k1,k2) := min

φ H (b,φ,k1,k2)

v =

d

∏

j=1

x −bj |x −bj|eiφ(x)

Colbert-Kelly Analysis of a G-L Energy

slide-45
SLIDE 45

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Outline

1

Ginzburg-Landau (GL) Functional

2

Introduction to Liquid Crystals (LCs)

3

Effects of Defects in Liquid Crystals

4

The Generalized GL Functional

5

References

Colbert-Kelly Analysis of a G-L Energy

slide-46
SLIDE 46

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

[BPP] P . Bauman, J. Park, and D. Phillips, Analysis of Nematic Liquid Crystals with Disclination Lines. preprint, arXiv:1106.5031. [BBH] F . Bethuel, H. Brezis., F . H´ elein. Ginzburg-Landau Vortices. Birkh¨ auser, Boston, 1994. [LPM ] J-B. Lee, R. A. Pelcovits, Robert A, R. B. Meyer. Role of Electrostatics in the Texture of Islands in Free-Standing Ferroelectric Liquid Crystal Films. Physical Review E, 75(051701):1-5, 2007

Colbert-Kelly Analysis of a G-L Energy

slide-47
SLIDE 47

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

[L] F-H. Lin. Static and Moving Vortices in Ginzburg-Landau Theories. Progress in Nonlinear Differential Equations and Their Applications, 29:71-111, Verlag Basel/Switzerland, 1997. [Sa] E. Sandier Lower Bounds for the Energy of Unit Vector Fields and Applications Journal of Functional Analysis, 152:379-403, 1998 [St] M. Struwe On the Asymptotic Behavior of Minimizers of the Ginzburg-Landau Model in 2 Dimensions Differential and Integral Equations, 7(6):1613-1624, 1994

Colbert-Kelly Analysis of a G-L Energy

slide-48
SLIDE 48

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

[V] E. Virga. Variational Theories for Liquid Crystals. Chapman & Hall, London, 1994.

Colbert-Kelly Analysis of a G-L Energy

slide-49
SLIDE 49

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References

Thank you!

Colbert-Kelly Analysis of a G-L Energy