Topological Defects 18.S995 - L23 Order Parameters, Broken - - PowerPoint PPT Presentation

Topological Defects

18.S995 - L23

Order Parameters, Broken Symmetry, and Topology

James P. Sethna

Laboratory of Applied Physics, Technical University of Denmark, DK-2800 Lyngby, DENMARK, and NORDITA, DK-2100 Copenhagen Ø, DENMARK and Laboratory of Atomic and Solid State Physics (LASSP), Clark Hall, Cornell University, Ithaca, NY 14853-2501, USA (Dated: May 27, 2003, 10:27 pm)

dunkel@mit.edu

Topological defects are discontinuities in

• rder-parameter fields
• optical effects
• work hardening, etc

"umbilic defects" in a nematic liquid crystal

• rder = symmetry = invariance

(under certain group actions )

symmetry groups can be discrete, continuous, Lie-groups, ….

More or less symmetric ?

http://www.doitpoms.ac.uk/tlplib/crystallography3/printall.php

Mg2Al4Si5O18

More or less symmetric ?

More or less symmetric ?

broken continuous translation/rotation symmetry (invariance)

Order parameters: 2D crystal

4

⃗ u ≡ ⃗ u + aˆ x = ⃗ u + maˆ x + naˆ y.

E =

• dx (κ/2)(du/dx)2.

Order parameters: magnets

tant.

Order parameters: nematic liquid crystals

“projective plane” = half-sphere with opposite points on equator identified

Topological defects

Work hardening

Dislocations

edge screw

Dislocations

Disclination pair

Dislocation-mediated growth of bacterial cell walls

Ariel Amir and David R. Nelson1

PNAS ∣ June 19, 2012 ∣

• vol. 109

∣

• no. 25

∣ 9833–9838

Dunkel et al PRL 2013

Bacterial vortices

PIV

+1

• 1
• 1

+1

Microtubule asters

+1

• 1

Blower et al (2005) Cell

mitotic spindle organization

Active nematics

Dogic lab (Brandeis) Nature 2012

Active nematics

Giomi et al PRL 2012

Defects in nematics

winding number

Defects in nematics

winding number

s bilized ). taneously geometric l- col- p fabrication loids re- l- ecision micro- a- ce d s. ic id

DOI: 10.1126/science.1129660 , 954 (2006); 313 Science et al. Igor Musevic Topological Defects Two-Dimensional Nematic Colloidal Crystals Self-Assembled by

DOI: 10.1126/science.1129660 , 954 (2006); 313 Science et al. Igor Musevic Topological Defects Two-Dimensional Nematic Colloidal Crystals Self-Assembled by

DOI: 10.1126/science.1205705 , 62 (2011); 333 Science et al. Uros Tkalec Reconfigurable Knots and Links in Chiral Nematic Colloids

efect ts s. g nt rs a he

• n

e

• ri-
• Defect

dimer, re nknot. e topologi-

• loops.

d- ns numerical- u-de el l- nd r- the fu- are by using a program for representing knots (33) to show the relaxation

c

Substrate Film

a

R pi pe

• Experiment

Pedro Reis Denis Terwagne

PDMS Oxid layer Relaxation

Yin et al (2014) Sci Rep

5 orders of magnitude in length … similar patterns & transitions

Curvature / stress-induced wrinkling transitions

Breid & Crosby (2013) Soft Matter

f e d

1cm

D Terwagne, M Brojan and P M Reis "Smart Morphable Surfaces for Aerodynamic Drag Control" Adv. Mater. 26:38, 6608 (2014)

Generalized Swift-Hohenberg theory

4ψ = rαrαψ = aγδψ,γδ aγδΓλ

γδψ,λ

(aαβ) = ✓ (R sin θ2)2 R2 ◆ Small deformations of a sphere

@tu = 01u212uaubu2 cu3 +1

⇥(ru)2 +2u1u ⇤+2 ⇥u(ru)2 +u21u ⇤

Air channel

b c

Substrate Film

a

R pi pe h

• u

R

Nature Materials 2015 (joint work with Reis lab MIT MechE) Norbert Stoop

γ0 = κ2 3 − 1 6 p η4/3 + 24(1 + ν)κ2 + 16κ4 a = η4/3 12 + 6(1 + ν) − η2/3 3 κ2 + κ4 3 + ˜ a2Σe b = 3(1 + ν)κ3 c = 2(1 + ν)η2/3 3 c1 + (1 + ν)κ4 Γ1 = 1 + ν 2 κ Γ2 = 1 + ν 2 κ2 ˜ a2 = −η4/3(c + 3|γ0|Γ2) 48γ2 TABLE I: List of parameters for Eq. (1) in units h = 1, with η = 3Es/Ef, γ2 = 1/12, Σe = (σ/σc) − 1 and κ = h/R. The

• nly remaining fitting parameter of the model is c1.

Nature Materials 2015 (joint work with Reis lab MIT MechE)

Generalized Swift-Hohenberg theory

@tu = 01u212uaubu2 cu3 +1

⇥(ru)2 +2u1u ⇤+2 ⇥u(ru)2 +u21u ⇤

Norbert Stoop

Air channel

b c

Substrate Film

a

R pi pe h

• u

R

Theory correctly predicts

Nature Materials 2015 (joint work with Reis lab MIT MechE)

morphology

phase transitions

c f b e a d

Experiment Theory Increasing e ective radius R/h R/h=50 R/h=75 R/h=200 umin/h u/h umax/h

• 10

20 50 100 200 500 1,000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Labyrinth phase Bistable phase Hexagonal phase Unwrinkled Hysteresis Effective radius R/h Excess stress e

Arbitrarily curved surfaces

µ∂tu =

γ04u γ242u au bu2 cu3 + (35) h 2

• (ν 1)

⇥ bαβrαurβu + 2urβ

• bαβrαu

⇤ + 2ν ⇥ H(ru)2 2r · (Huru) ⇤ + h 2 ⇥ (1 ν)urβ

• ucαβrαu
• νRu(ru)2+

νr · (Ru2ru) ⇤

a b c

20 hexagons 12 pentagons Leonard Euler

a

Topological defects

• κ

κ

• κ

J Lidmar, L Mirny and DR Nelson (2003) PRE

Why interesting ?

topological defects nucleate size-induced shape transition in viral capsids bending of graphene introduces defects and changes electronic properties

A Cortijo and MAH Vozmediano (2007) EPL

Surface crystallography

Irvine et al (2010) Nature

Statics Nucleation

Meng, Paulose, Nelson & Manoharan (2014) Science

Statics: surface crystallography

PRL 2016 (joint work with Reis lab MIT MechE)

(f)

• π

π

• π

π

θ [rad]

(e) (f) φ [rad]

• π

π

• π

π

θ [rad]

Z = 5 Z = 6 Z = 7

Film

umax umin

u

(c)

Core

(a) (b)

y x z

Dynamics: Kibble-Zurek mechanism (KZM)

Kibble & Zurek (1970s): System driven through a 2nd order phase transition

• exhibits critical slowing-down
• dynamics cannot follow changes of external system parameters
• density of topological defects after quench reveals information about the quench dynamics
• bserved topological structures in the universe provide a window into early evolution

KZ predictions:

Elastic surface crystals as testbed for KZM in curved geometries?

• 10

20 50 100 200 500 1,000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Labyrinth phase Bistable phase Hexagonal phase Unwrinkled Hysteresis Effective radius R/h Excess stress e

Time

tc=0

quench rate µ Temperature

tfreeze

Order parameter

Tc δT

Dynamics of phase transition

a b

Excess stress UH Subcritical hexagonal phase Unwrinkled Hexagonal phase −0.10 −0.05 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 Amplitude UH δ

e

δΣ

Adiabatic/equilibrium

bifurcation from flat state u=0 to hexagonal pattern at Σe =0

Stoop & Dunkel

arXiv:1703.03540

Freeze-out time follows KZ scaling

a b

Excess stress UH Subcritical hexagonal phase Unwrinkled Hexagonal phase −0.10 −0.05 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 Amplitude UH δ

e

δΣ

bifurcation from flat state u=0 to hexagonal pattern at Σe=0

Stoop & Dunkel

arXiv:1703.03540

Adiabatic/equilibrium Linear quench

Σe(t) = µt

v(x) 1 2

Σe

0.1 0.2 0.3 0.4 0.5

⟨u2⟩ D

10-8 10-6 10-4 µ 10-2 10-1 100 Σf

e

max

E

∼ µ1/2

Σf

e ∼ µ1/2

Freeze-out time follows KZ scaling

Linear quench

Σe(t) = µt

v(x) 1 2

Σe

0.1 0.2 0.3 0.4 0.5

⟨u2⟩ D

10-8 10-6 10-4 µ 10-2 10-1 100 Σf

e

max

E

∼ µ1/2

Σf

e ∼ µ1/2

Stoop & Dunkel

arXiv:1703.03540

Defect density follows KZ predictions !?

Stoop & Dunkel (2016) Preprint

Voronoi tessellation at freeze-out Σef

µ 10-8 10-6 10-4 ρ − ρ0 10-2 10-1

J

µ

∼ µ1/2

ρ − ρ∞

(ND-ND∞)/area Defect density

Time t=0 t=xx t=xx B

u

F G H I

µ = 5 · 10−8 µ = 2 · 10−5

µ = 10−4

Z ≤ 5 Z = 6 Z ≥ 7

Time t=0 t=xx t=xx B

max

u C t

Nucleation dynamics

Meng, Paulose, Nelson & Manoharan (2014) Science

t=xx t=xx t=xx E D

Time t=0 t=xx t=xx B

max

u C t

Nucleation dynamics

explains ‘geodesic wrapping’

10-8 10-6 10-4 10-2 10-1 100

∼ µ1/2

10−8 10−6 10−4

Quench rate µ

10-8 10-6 10-4 10-2 10-1

ρ − ρ0

10−8 10−6 10−4

0.5 1 1.5

Σe

0.2 0.4 0.6 0.8

⟨u2⟩

Order parameter ⟨u2⟩

A E F B C

umax umin

u D

∼ µ1/2 µ=

5 · 10−8 2 · 10−7 5 · 10−7 2 · 10−6 5 · 10−6 1 · 10−5 2 · 10−5

π π/2 5 10 5 ϕ

Nd

Z ≤ 5 Z = 6 Z ≥ 7 Z ≤ 5 Z ≥ 7

µ = 5 ·10 −8 µ = 2 ·10 −5 n=14 n=14 n=14 n=14 Film stress Σe Freeze-out stress Quench rate µ Number of defects

∆Σ

f

Excess defect density ∆ρf ϕ

Stoop & Dunkel

arXiv:1703.03540