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

• work hardening, etc • optical effects order-parameter fields are discontinuities in Topological defects "umbilic defects" in a nematic liquid crystal

order = symmetry = invariance (under certain group actions ) symmetry groups can be discrete, continuous, Lie-groups, ….

More or less symmetric ?

More or less symmetric ? Mg 2 Al 4 Si 5 O 18 http://www.doitpoms.ac.uk/tlplib/crystallography3/printall.php

More or less symmetric ? broken continuous translation/rotation symmetry (invariance)

Order parameters: 2D crystal 4 � dx ( κ / 2)( du/dx ) 2 . E = u + a ˆ x = ⃗ u + ma ˆ x + na ˆ ⃗ u ≡ ⃗ y.

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. Nelson 1 PNAS June 19, 2012 vol. 109 no. 25 9833 – 9838 ∣ ∣ ∣ ∣

Bacterial vortices PIV +1 -1 -1 +1 Dunkel et al PRL 2013

Microtubule asters +1 -1 mitotic spindle organization Blower et al (2005) Cell

Active nematics Dogic lab (Brandeis) Nature 2012

Active nematics Giomi et al PRL 2012

Defects in nematics winding number

Defects in nematics winding number

Two-Dimensional Nematic Colloidal Crystals Self-Assembled by Topological Defects Igor Musevic et al. Science 313 , 954 (2006); DOI: 10.1126/science.1129660 s bilized ). taneously geometric l- col- p fabrication loids re- l- ecision micro- a- ce d s. ic id

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

Reconfigurable Knots and Links in Chiral Nematic Colloids Uros Tkalec et al. Science 333 , 62 (2011); DOI: 10.1126/science.1205705 efect ts s. g nt rs a he on e ori- o- Defect dimer, re nknot. e topologi- o 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

Experiment Denis Terwagne Pedro Reis a Film � Substrate R � p i p e c �

Curvature / stress-induced wrinkling transitions d e f Breid & Crosby (2013) Soft Matter 1cm PDMS Yin et al (2014) Sci Rep Oxid layer Relaxation 5 orders of magnitude in length … similar patterns & transitions 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 @ t u = � 0 1 u � � 2 1 2 u � au � bu 2 � cu 3 Norbert Stoop ⇥ ( r u ) 2 + 2 u 1 u ⇥ u ( r u ) 2 + u 2 1 u ⇤ + � 2 ⇤ + � 1 a b h Film � Substrate Small deformations of a sphere R � p i u ✓ ( R sin θ 2 ) 2 ◆ 0 p e ( a αβ ) = R 2 0 c R 4 ψ = r α r α ψ = a γδ ψ , γδ � a γδ Γ λ γδ ψ , λ channel Air � Nature Materials 2015 (joint work with Reis lab MIT MechE)

Generalized Swift-Hohenberg theory @ t u = � 0 1 u � � 2 1 2 u � au � bu 2 � cu 3 Norbert Stoop ⇥ ( r u ) 2 + 2 u 1 u ⇥ u ( r u ) 2 + u 2 1 u ⇤ + � 2 ⇤ + � 1 γ 0 = κ 2 3 − 1 η 4 / 3 + 24(1 + ν ) κ 2 + 16 κ 4 p a b h 6 Film � Substrate = η 4 / 3 12 + 6(1 + ν ) − η 2 / 3 κ 2 + κ 4 a 3 + ˜ a 2 Σ e 3 = 3(1 + ν ) κ 3 b R � p i = 2(1 + ν ) η 2 / 3 u c 1 + (1 + ν ) κ 4 c 3 p e Γ 1 = 1 + ν κ 2 c Γ 2 = 1 + ν R κ 2 2 a 2 = − η 4 / 3 ( c + 3 | γ 0 | Γ 2 ) ˜ channel 48 γ 2 0 Air TABLE I: List of parameters for Eq. (1) in units h = 1, with � η = 3 E s /E f , γ 2 = 1 / 12, Σ e = ( σ / σ c ) − 1 and κ = h/R . The only remaining fitting parameter of the model is c 1 . Nature Materials 2015 (joint work with Reis lab MIT MechE)

Theory correctly predicts phase transitions morphology 3.0 a b c � 2.5 � Theory 2.0 � � Labyrinth � phase Excess stress � e Bistable 1.5 � R/h=50 R/h=75 R/h=200 phase � � d e f � � � � Hexagonal � 1.0 � � � phase � � Experiment � � � � � � 0.5 � � � � � � � � � � � � � � � � � � 0.0 Hysteresis Unwrinkled 10 20 50 100 200 500 1,000 Increasing e ective radius R/h Effective radius R/h u min /h u/h u max /h Nature Materials 2015 (joint work with Reis lab MIT MechE)

Arbitrarily curved surfaces γ 0 4 u � γ 2 4 2 u � au � bu 2 � cu 3 + µ ∂ t u = � (35) h b αβ r α u r β u + 2 u r β b αβ r α u � ⇥ � �⇤ ( ν � 1) + 2 H ( r u ) 2 � 2 r · ( H u r u ) ⇥ ⇤ 2 ν + h uc αβ r α u � ν R u ( r u ) 2 + ⇥ � � (1 � ν ) u r β 2 ν r · ( R u 2 r u ) ⇤ a b c

a Topological defects Leonard Euler 20 hexagons 12 pentagons

Why interesting ? � κ κ bending of graphene topological defects � introduces defects and nucleate size-induced shape transition changes electronic properties in viral capsids κ J Lidmar, L Mirny and DR Nelson (2003) PRE A Cortijo and MAH Vozmediano (2007) EPL

Surface crystallography Statics Nucleation Meng, Paulose, Nelson & Manoharan (2014) Science Irvine et al (2010) Nature

Statics: surface crystallography Film (b) (a) (c) u max Core u z u min x y Z = 5 Z = 6 Z = 7 π π (f) (f) (e) φ [rad] 0 0 - π - π π 0 θ [rad] - π - π π 0 θ [rad] PRL 2016 (joint work with Reis lab MIT MechE)

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 • observed topological structures in the universe provide a window into early evolution 3.0 t freeze Temperature � Order parameter 2.5 � quench rate µ T c 2.0 � � Labyrinth � � phase Excess stress � e δ T Bistable 1.5 � � phase � � � � � � � � Hexagonal � 1.0 � � � phase t c =0 Time � � � � � � � 0.5 � � � � � KZ predictions : � � � � � � � � � � � � � � 0.0 Hysteresis Unwrinkled 10 20 50 100 200 500 1,000 Effective radius R/h Elastic surface crystals as testbed for KZM in curved geometries?

Dynamics of phase transition Adiabatic/equilibrium a b Unwrinkled Subcritical hexagonal U H phase 1.5 Amplitude 1.0 U H δ Hexagonal 0.5 phase 0.0 δΣ e − 0.10 − 0.05 0.00 0.05 0.10 0.15 Excess stress bifurcation from flat state u=0 to hexagonal pattern at Σ e =0 Stoop & Dunkel arXiv:1703.03540

Freeze-out time follows KZ scaling Adiabatic/equilibrium Linear quench Σ e ( t ) = µt a b Unwrinkled Subcritical hexagonal 0.5 v(x) D E U H phase 1.5 10 0 0.4 Amplitude 1.0 0.3 ⟨ u 2 ⟩ ∼ µ 1 / 2 Σ f e U H δ Hexagonal 10 -1 0.2 max 0.5 phase 0.1 0.0 δΣ e 10 -2 0 − 0.10 − 0.05 0.00 0.05 0.10 0.15 0 1 2 10 -8 10 -6 10 -4 Excess stress Σ e µ bifurcation from flat state u= 0 Σ f e ∼ µ 1 / 2 to hexagonal pattern at Σ e = 0 Stoop & Dunkel arXiv:1703.03540

Freeze-out time follows KZ scaling Linear quench Σ e ( t ) = µt 0.5 v(x) D E 10 0 0.4 0.3 ⟨ u 2 ⟩ ∼ µ 1 / 2 Σ f e 10 -1 0.2 max 0.1 10 -2 0 0 1 2 10 -8 10 -6 10 -4 Σ e µ Σ f e ∼ µ 1 / 2 Stoop & Dunkel arXiv:1703.03540

Defect density follows KZ predictions !? µ B J 10 -1 Defect density (N D -N D ∞ )/ area max ρ − ρ ∞ ρ − ρ 0 u ∼ µ 1 / 2 Time t=0 t=xx t=xx 10 -2 µ = 5 · 10 − 8 µ = 2 · 10 − 5 µ = 10 − 4 10 -8 10 -6 10 -4 F G H I µ Z ≤ 5 Z = 6 Z ≥ 7 Voronoi tessellation at freeze-out Σ ef Stoop & Dunkel (2016) Preprint

Nucleation dynamics B C max u Time t t=0 t=xx t=xx Meng, Paulose, Nelson & Manoharan (2014) Science

Nucleation dynamics B C max u Time t t=0 t=xx t=xx D E t=xx t=xx t=xx explains ‘geodesic wrapping’