Nuno Silvestre Nelson Bernardino Margarida Telo da Gama Liquid crystals at surfaces and interfaces: from statics to dynamics

Outline Introduction Nematics at structured surfaces: defects, filling and wetting Cholesteric interface and surface tension Nematic flow in microchannels Nematic droplets on fibres Nematic colloids at surfaces Outlook (open questions)

Simple fluids: surface tension and wetting

Surface tension ϒ Mechanics: Force along line of unit length, parallel to surface Thermodynamics: Isothermal work (free energy) per unit area of the surface (mN/m)

Contact angle: Young’s Equation for structureless surfaces The shape of a liquid-vapor interface is determined by the Young–Laplace (normal stress blalance) equation, with the contact angle playing the role of a boundary condition, via Young's Equation.

Wet states & wetting transitions

Simple fluids at structured surfaces: filling

Structured surfaces: New surface states & phase transitions Filling (unbending) and wetting (unbinding) transitions

Thermodynamics of Wetting & Filling transitions Wetting Wenzel Filling Sinusoidal grating

Nematics: surface tension and wetting

Nematics

Mesoscopic elasticity

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Nematic-Isotropic Interface Homeotropic if k < 0 Planar if k > 0

Structureless surfaces: nematic wetting Homeotropic substrate, planar nematic-isotropic interface

Wetting for homeotropic anchoring Non-wetting for planar

Nematics at structured surfaces

Thermodynamics of nematic wetting: generalised Wenzel’s law Roughness enhances wetting of simple fluids on planar surfaces; wetting follows filling. Not so for nematics when the elastic energy dominates. Defects play a major role: Wetting and filling may be suppressed.

Macroscopic elasticity

Defects: Planar disclination lines Frank 1958

The challenge Surfaces & interfaces where Q varies over the order of ξ Chemical or geometrical surface patterns leading to variations of Q on the scale of µm Wetting layers close to wetting transitions on the scale of mm (macroscopic) Surface phase diagrams require the calculation of the excess free energy F ex =F s -F b with an accuracy at least one order of magnitude better than the difference of F ex for the competing surface states Many competing structures or surface states

Simple fluids at structured surfaces: OverviewNematics at structured surfaces X X X 4

Nematics at rectangular gratings Nematic wetting and filling of crenellated surfaces, N. M. Silvestre, Z. Eskandari, P. Patrício, J. M. Romero Enrique and M. M. Telo da Gama, Physical Review E 86, (2012).

Numerics: FEM with adaptive meshes Grid size Accuracy 1-3% %

Rectangular grating: interfacial states Wet states Filled states u bbsb bb sb NEW Silvestre et al, PRE 86, (2013)

Rectangular grating: Phase diagram Deep groovesShallow grooves Filled states with bent interfaces

Transitions between nematic filled states Interfacial motion in flexo- and order-electric switching between nematic filled states, M. L. Blow and M. M. Telo da Gama, Journal of Physics Condensed Matter 25, (2013).

The fluid density ρ and its velocity u evolve according to the continuity and Navier-Stokes equations Hydrodynamics where And the Q-tensor evolves according to the Beris-Edwards equations  and  are the shear and bulk viscosities,  is a molecular parameter, and  is the mobility (thermodynamic coupling) The hydrodynamics are simulated using a hybrid Lattice-Boltzmann/Finite difference model (Henrich et al, 2010)

For a uniform electric field (as is considered here), these terms only contribute to the surface effects. Acts in the bulk to induce a splay-bend texture The contribution to the molecular field is in the bulk and at the boundary In the system of reduced units we define Flexoelectricity

Homeotropic anchoring – switching between bend- bend and splay-bend states Video 1 here Bend-bend to splay-bend transition Splay-bend to bend-bend

Cholesteric-isotropic interface NR Bernardino, MCF Pereira, NM Silvestre, MM Telo da Gama Structure of the cholesteric-isotropic interface, Soft Matter 10, 9399 (2014).

Cholesterics: nematics with a twist Cholesterics can be though of as layered systems

Challenge: Different length scales Correlation length of 10nm and pitch in the micron scale Accurate calculations of free energy required for surface tension

Most interesting: Perpendicular layers Homeotropic anchoring is not compatible with undistorted layers. Creation of topological defects. Interface undulates

Surface tension Very slow approch to the nematic limit Negative k leads to strong distortions Low pitch leads to double twist and blue phases.

Distortions The amplitude scales as sqrt(pitch) The amplitude of scales with - k

Wetting by a blue phase? Layer of blue phase at the interface. Wetting?

Nematic flow in microchannels Matthew Blow and Vera Batista

Flow of a nematic liquid crystal in microfluidic channels Anupam Sengupta et al., Liquid Crystal Microfuidics for Tunable Flow Shaping, Phys.Rev.Lett., 110, (2013) ; Anupam Sengupta, Tuning Fluidic Resistance via Liquid Crystal Microfluidics, Int. J. Mol. Sci. 14, (2013) Low Flow : very little distortion of the nematic director Medium Flow : distortion of the nematic director ; however, at channel mid-height d the director is not yet aligned with the flow direction Average distortion of the nematic director and viscosity measured for 5CB using pressure-driven flow through microchannels. Sudden drop in the value of viscosity when increasing pressure gradient Low Flow to Medium Flow : Medium Flow to High Flow : director gradually aligns with the flow direction Homeotropic anchoring

The effect of anchoring on nematic flow in channels

Mass flow rate Φ as a function of the pressure gradient G. Different lines correspond to distinct values for the strength of the homeotropic anchoring conditions. Vertical lines represent the value of G above which we observe the Horizontal state. Low flow High flow : Horizontal state Vertical state

 The director θ across the channel for small G (red) and the crossover from ’vertical’ to ’horizontal’ when θ approaches the alignment angle (green to blue) for α = 364. Observed jump in mass flux corresponds to rapid changes in θ (e.g. purple to pink) Vertical Horizontal Changes in θ close to the walls have a greater effect on Φ than changes in θ close to the centre of the channel.

Transition of Vertical to Horizontal state: Director profile as a function of channel height near the Vertical → Horizontal transition. Different curves correspond to distinct α for homeotropic anchoring. Vertical discontinuous grey line indicates channel at mid- height. Weak anchoring, by limiting the initial derivative of θ at the substrate (as seen in figure), impedes rather than assists the Vertical → Horizontal transition (larger G to observe transition).

Nematic droplets on fibers Nuno Silvestre and Vera Batista

Nematic droplets on fibers Schematics of the experimental setup. Cross-polarized image corresponds to nematic droplets constrained on a thin fiber (1.0 mm diameter), suspended in air. Schematic of the director field configuration showing ring defect Experimental study of 5CB beads constrained by fibers under an electric field The radius of the ring increases as the field applied across the cell was ramped from 3.8 to 5.1 V/  m and the ring moves along the fiber axis, from the center of the drop

Lattice Boltzmann simulations: Bead : represented as a sphere Fiber : represented as a cylinder with its major axis along the y-axis Electric field : applied perpendicular to the fiber’s major axis Ring defect Fiber z y Increase electric field Distortion of ring defect Increase electric field temperature  *= 0.67 correlation length :  =1 elastic constants : L1 = 0.08, L2 = 0.16 Snem = 1 density :  = 80 bulk constant : A = 0.13 mobility of nematic order :  = 0.25 dynamic parameter :  = 1.5 5CB Parameters (in simulation units) : Low electric field Director profile Tests were run with DC and AC electric fields over a wide range of frequencies. Systematic comparison between results obtained with DC and AC fields Ring remains static in center of bead. Ongoing work:

Nematic colloids Nuno Silvestre and Zahra Eskandari

M. Skarabot et al, PRE 77, (2008). Quadrupole-quadrupole interaction Saturn-ring colloids

I.I. Smalyukh et al, PRL 95, (2005) Quadrupole-quadrupole interaction with NO repulsion at short distances Boojum colloids

M. Tasinkevych et al, New J. Phys. 14, (2012). M.R. Mozaffari et al, Soft Matter 7, 1107 (2011). Boojum colloids

Z. Eskandari et al, Langmuir 29, (2013). Bonded-boojum colloids

Z. Eskandari et al, Soft Matter 8, (2012) U. Ognysta et al, PRE 83, (2011) Mixing quadrupoles

NMS et al, PRE 69, (2004). F.R. Hung et al, J. Chem Phys. 127, (2007). Nematic Structured surfaces

Z. Eskandari et al, Soft Matter 10, 9681 (2014). For matching anchoring conditions particles assemble at concave patterns. For mismatching anchoring conditions particles assemble at convex patterns. Nematic Structured surfaces

NMS et al, PRL 112, (2014). Nematic Structured surfaces

Summary & Outlook We need to move (Lattice-Boltzmann) We need experiments

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