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

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

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!

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 = 1 2 3 𝑎 𝑖 𝑎 𝑗 − δ 𝑖𝑗 = 𝑆 2 3 𝑛 𝑖 𝑛 𝑗 − δ 𝑖𝑗

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)

Simulations of liquid crystals Static terms Hydrodynamics 𝑓= 1 2 𝐴 𝑇𝑟 𝑄 2 + 1 3 𝐵 𝑇𝑟 𝑄 3 + 1 4 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).

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 0 + 9 2 𝐶 𝑆 2 )

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)

Defect-mediated interactions

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)

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)

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

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)

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

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

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=1+0+ 1 2 + 1 2 =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)

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)

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)

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

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,...