1. Topology of complex fluids
Simon Čopar
COST Workshop on Modelling of Flowing Matter,
FMF-UL, Ljubljana
Ljubljana, February 2015

2. Topics Fluids with additional degrees of freedom
Topology of liquid crystals
Numerical methods for nematic liquid crystals
Building blocks of defects
Building blocks of rewiring in colloids (2D,3D)
Transitional dynamics of defect networks
Knots in nematic colloids and emulsions

3. Fluids with degrees of freedom
Hydrodynamic variables: p,v,ρ
Order parameters:
Scalars: concentration in multiphase systems
Vectors: magnetization (ferrofluids), angular momentum (3He)
Director/Tensor: liquid crystals (depending on the mesophase)
Combinations for mixtures and complex fluids
Coupling between flow and order: both directions
Slow dynamics, high viscosity: quasi-static simulation is sufficient; no physical flow still yields dinamics!

4. Topology of liquid crystals
Liquids with orientational (+partial positional order)
Nematics permit point and line defects
Defects compensate frustration
Topological conservation laws
Point defects as charges
Line defects as linked and knotted
string-like components
Director
Nematic order parameter
𝑄 𝑖𝑗 = 𝑎 𝑖 𝑎 𝑗 − δ 𝑖𝑗 = 𝑆 2 3 𝑛 𝑖 𝑛 𝑗 − δ 𝑖𝑗

5. Defect types & classification
𝜋 2 homotopy group
𝜋 1 homotopy group
Point defects
Line defects
Nematics (and most vector fields)
Smectics
Dislocations
Disclinations
Smectics
Cholesterics (?)
Abelian
Nonabelian
Nematic
Biaxial nematic
Cholesteric (?)
Mermin, Rev. Mod. Phys. (1979)
Beller et al, PRX 8 (2014)

6. Simulations of liquid crystals
Static terms
Hydrodynamics
𝑓= 1 2 𝐴 𝑇𝑟 𝑄 𝐵 𝑇𝑟 𝑄 𝐶 𝑇𝑟 𝑄 2 2
Molecular field
𝐻 𝑖𝑗 =− 𝜕𝑓 𝜕 𝑄 𝑖𝑗 [traceless]
+ 1 2 𝐿 𝜕𝑄 𝑖𝑗 𝜕𝑥 𝑘 𝜕𝑄 𝑖𝑗 𝜕𝑥 𝑘 +2𝐿 𝑞 0 ε 𝑖𝑘𝑙 𝑄 𝑖𝑗 𝜕𝑄 𝑙𝑗 𝜕𝑥 𝑘 +[extra terms]
𝑑 𝑄 𝑖𝑗 𝑑𝑡 −S(𝜕𝑢,𝑄)=Г 𝐻 𝑖𝑗
+ 1 2 ε 𝐸 𝑖 𝐸 𝑗 𝑄 𝑖𝑗
ρ 𝑑 𝑢 𝑖 𝑑𝑡 = 𝜕 𝑗 Π 𝑖𝑗 +η 𝜕 𝑗 𝜕 𝑗 𝑢 𝑖 + 𝜕 𝑖 𝑢 𝑗
Distinct elastic constants
Cholesterics, blue phases: extra terms
Coupling with electric field, flexoelectricity
Other phases: smectics, biaxial nematics, TGB phase
Active fluids, swimmers: propelling terms
Frequently, quasi-static is enough
Denniston et al, Phil. Trans. R. Soc. A 362, 1745 (2004).
Orlandini et al, Mol. Cryst. Liq. Cryst. 494, 293 (2008).
Tang et al, Liq. Cryst. 36, 889 (2009).

7. Simulations of liquid crystals
Defects are thin: the simulation length-scale is set by the nematic coherence length.
Limitation of a simulation domain size: typical resolution required ~10nm, typical size limit: a few μm.
Equidistant grid methods: Lattice Boltzmann for hydrodynamics, relaxation methods for order parameter.
Triangulation methods: finite elements, finite volume; issue: grid has to follow the defects!
ξ= 𝐿/(𝐴+𝐵 𝑆 𝐶 𝑆 2 )

8. Liquid crystal colloids
thin cell,
surface anchoring
Škarabot et al, PRE 76 (2007)
Ravnik et al, PRL 99 (2007)
Tkalec et al, Science 333 (2011)
Čopar & Žumer, PRL 106 (2011)
Čopar et al, Soft Matter 8 (2012)
Nych et al, Nature Commun. 4 (2013)
Čopar et al, Soft Matter 9 (2013)
Martinez et al, Nature Mater. 13 (2014)

9. Defect-mediated interactions

10. Dimers: rewiring and linking
Start with dimers: similar, local differences
Optical tweezers for manipulations
Rewiring ~ rotation of a tetrahedron?
Čopar & Žumer, PRL 106 (2011)
Čopar, Phys. Rep. 538 (2014)

11. 2D colloidal crystals Arbitrary knots made by laser manipulaton
Theoretical concepts realized in experiment: knot theory, graph theory, topology of fields
Tkalec et al, Science 333 (2011)
Čopar et al, PNAS 112, 1675 (2015)

12. 2D colloidal crystals, contd...
Classifiable in bulk with Jones polynomials
Medial graph construction
Čopar et al, PNAS 112, 1675 (2015)

13. Building blocks in nematic colloids
Universality of rewiring: all crossings are similar!
Boundary conditions in the cell determine the position and number of rewiring sites
Are there systems where tetrahedra are not enough?
Ravnik et al, PRL 99 (2007)
Čopar & Žumer, PRL 106 (2011)
Čopar et al, Soft Matter 8 (2012)
Tkalec et al, Science 333 (2011)

14. Building in three dimensions
3D crystals: tetrahedra + something else?
Face centered cubic: cube-like voids!
How many possible structures? ?

15. 3D building blocks 7 structures + all rotations!
Simulation confirms the structures
Additional geometric and topological constraints!
Čopar et al, Soft Matter 8 (2012)

16. Topological dynamics on a fibre
Transverse homeotropic fibre: topological playground
Zero-charge and charged loops in the same system
rubbing
-1/2
+1/2
flip
𝑞= =0 (mod 2)
one loop
same-parity
rotations
no self-linking
-1/2
-1/2 +1 -1
+1/2
-1/2 +1
Nikkhou et al, Nature Phys. 11 (2014)

17. Topological dynamics on a fibre
Quenching: coarsening dynamics
Fibre stabilizes defects
Slow annihilation dynamics
+1/2 faster than -1/2
Nikkhou et al, Nature Phys. 11 (2014)

18. Knotting in cholesteric droplets
Cholesteric droplets: intrinsic chiral order
Perpendicular (homeotropic) surface alignment: incompatible with the helix
Frustration lots of metastable states
Knotted defects
Seč et al, Soft Matter 8 (2012)
Seč et al, Nature Commun. 5 (2014)

19. Summary Liquid crystals as complex fluids
Coupling with orientational order
High viscosity: slow dynamics?
Passive or active?
Defects have their own dynamic behaviour

20. Collaboration and affiliations
Soft matter group, FMF, UNI-LJ
F5, IJS
prof. Slobodan Žumer
Miha Ravnik
Tine Porenta
David Seč
M. Čančula, J. Aplinc, Ž. Kos
& others
prof. Igor Muševič
Miha Škarabot
Uroš Tkalec
Rao Jampani
Maryam Nikkhou
Gregor Posnjak
& others
UPenn, Pennsylvania, USA
prof. R. Kamien
D. Beller
R. Mosna
D. Sussman
prof. K. Stebe
& others
Collaborators from other groups:
G. Alexander, R. Mosna, T. Machon, M. Dennis, I. Smalyukh, S. Dhara,...