2D anisotropic fluids: phase behaviour and defects in small planar cavities

1. 2D anisotropic fluids:phase behaviour and defects in small planar cavities D. de las Heras1, Y. Martínez-Ratón2, S. Varga3andE. Velasco1 1Universidad Autónoma de Madrid, Spain 2Universidad Carlos III de Madrid, Spain 3University of Pannonia, Veszprem, Hungary

2. MODEL SYSTEM • Hard particles: • Hard (excluded volume) interactions Colloidal non-spherical particles Metallic nanoparticles hard models contain essential interactions to explain many properties • AIM • Study of self-assembly in monolayers • Orientational transitions in 2D • Frustration effects: defects effect of reduced dimensionality on phases and phase transitions absence of long-range order?

3. OUTLINE • Motivation • Phase diagrams • new symmetry: tetratic phase in hard rectangles • Defects • Phase separation in mixtures

4. building blocks for self-assembly, templates in applications (nanoelectronics) Synthesis of metallic nanoparticles of non-spherical shape (nanorods) Wiley et al. Nanolett. 7, 1032 (2007)

5. Experiments on granular matter Vertically-vibrated quasi-monolayer of granular particles

6. Macroscopic realisation of statistical-mechanics of particles? Observation of liquid-crystal textures in two dimensions: • uniaxial nematic • nematic with strong tetratic correlations • smectic Narayan et al.J. Stat. Mech. 2006 nematic with strong tetratic correlations in copper cylinders nematic state with rolling pins smectic state with basmati rice

7. Colloidal discs Vibrated monolayer of vertical discs (projecting as rectangles) Zhao et al. PRE 76, R040401 (2007)

8. SOME HARD MODEL PARTICLES 2D projection 3D body hard rectangle (HC) hard cylinder hard ellipse (HE) hard ellipsoid hard disco- rectangle (HDR) hard sphero- cylinder F = U-TS = -TS Shape, packing and excluded volume determineproperties

9. LIQUID CRYSTALS Lyotropic(concentration driven) Thermotropic(temperature driven) DIRECTOR LIQUID-CRYSTALLINE PHASES (mesophases)

10. Phase diagram of HDR (hard disco-rectangles) Continuous isotropic-nematic phase transition of the KT type Bates & Frenkel (2000) Phase diagram NEMATIC PHASE Isotropic phase Quasi long-range order Crystalline phase

11. HARD RECTANGLES nematic phase with two equivalent directors 2D analogue of 3D biaxial and cubatic phases Possible tetratic phase Tetratic Nt PARTIAL SPATIAL ORDER NEMATIC CRYSTALLINE Columnar ? Nematic Nu Smectic

12. Scaled-particle theory (SPT) in 2D (Density-functional theory) density h = rv (theory á la Onsager) Scaled free energy density: Orientational distribution function: particle area = L s packing fraction Ideal part: Orientational average of excluded area Excess part:

13. Excluded volumes (HDR) (HR) secondary minimum in hard rectangles

14. Results from SPT: HR DISTRIBUTION FUNCTIONS: Nu: symmetric under rotations ofp Nt : symmetric under rotations of p / 2 HR SPT phase diagram distribution function of Nu and Nt Nu PHASE DIAGRAM: • Isotropic, Nu and Nt phases • Nt stability for k < 2.62 • Rich phase behaviour (1st and 2nd order phase transitions) Nt HDR

15. Hard rectangles versus hard discorectangles • The isotropic-nematic transition for HDRs is always of second order • HRs may be of first or second order • An additional nematic (tetratic) phase exists for HRs of low k HDR uniaxial nematic isotropic HDR HR

16. Monte Carlo simulation of hard rectangles (Martínez-Ratón et al. JCP 125, 014501, ‘06) k=3 tetratic K tetratic Nt I tetratic h(f) isotropic isotropic k= 3 SPT prediction

17. SPT Nt MC SPT + B3

18. Stability of tetratic phase due to clustering effects In the simulations, particle configurations exhibit strong clustering Monte Carlo simulation vibrated monolayer

19. Clustering model: • particles in one cluster are strictly parallel and form a unit • these units are taken as particles in a polydispersed fluid We are led to a polydispersed fluid with a continuous distribution of sizes (species) and where concentration of species is exponential SPT for a polydispersed 2D fluid

20. 2D DEFECTS (topological charge and winding number) q=1 q=1/2

22. DEFECTS IN A SMALL PLANAR CAVITY We confined particles into a circular cavity and impose a strong anchoring surface energy (perpendicular to surface) RADIAL (+1)(hedgehog) POLAR 2x(+1/2) UNIFORMno defects

23. DFT THEORY: Parsons-Lee theory for hard disco-rectangles It is an Onsager-like, second-order theory in two dimensions Basic variational quantities: 1. local density r(r) 2. local order parameter q(r) 3. local tilt angle Y(r): R PLUS external potential that favours perpendicular or parallel orientation of molecules V(r,f) The free energy functional is minimised numerically with respect to variational quantities:

24. h q y q (order parameter) h (density) Y (tilt)

25. PHASE DIAGRAM: Chemical potential vs. cavity radius structural transition no phase transition first-order phase transition (discontinuous) inflection point terminal point remnant of bulk I-N phase transition in cavity (pseudo phase transition)

26. Structure across pseudo phase transition path at fixed radius R and increasing m inflection point pseudo capillary nematisation

27. Structure of hedgehog: radial vs. tangential defects radial tangential

28. Nematic elasticity tangential hedgehog defect Elasticity associated to spatial deformations of the director Frank elastic energy: radial hedgehog defect IN 2D K1 K2 K3

29. DFT calculation of elastic constants

30. Radius and energy of defect core Frank elastic energy for m=+1 radial defect PLUS defect core energy rn and En we obtain by comparing density-functional theory with elastic theory as obtained from inflexion points as obtained from DFT energy density

31. Structure of hedgehog: radial vs. tangential defects radial tangential

32. free-energy density along one radius Parsons-Lee theory linear regime

33. Demixing (phase separation) in 2D mixtures Long-standing issue: does a mixtures of spheres or discs or different size phase separate? + Answer seems to be: YES, but one phase is a crystal But happens with anisotropic bodies?

34. Experimental verification of demixing in gold nanospheres and nanorods Sau et al. Langmuir 21, 2923 (2005)

35. THEORY: SPT for mixtures • competition betweenexcluded volume, orientational entropy and mixing entropy • discs and rectangles • rectangles of different size • discorectangles & rectangles de las Heras et al. PRE 76, 031704 (2007) RESULTS: • no I-I demixing • there is I-N and N-N separation

36. hard squares and discs: L1=1, s1=s2=1

37. hard squares: L1=10, L2=1

38. HR and HDR: L1=1.5, s1=1, L2=1.70, s2=0.85

39. hard rectangles: L1=4.0, 4.6, 5.0, L2=2, s1=s2=1

40. Experiments on vibrated layers of granular objects Plastic inelastic beads confined by two horizontal plates and excited by vertical vibrations. Experiments: one-component: phases, surface phenomena, confinement effects, defects, ... mixtures: "entropic" segregation Future directions: perform full-field tracking of positions and orientations of objects using fast video imaging and obtain correlation functions

41. THE END

42. 2R=100D 2D Defectos en una cavidad circular Núcleos

43. Hard rectangles in confined geometry (Y. Martínez-Ratón, PRE 2007) BULK (k = 3): Isotropic( I ) coexisting with Columnar (C) CONFINEMENT:Competition between capillary ordering and layering transitions Phenomenology similar to confined (3D) hard spherocylinders where ordered phase is a smectic (de las Heras, Velasco & Mederos, PRL 2005) Theory: FMT in Zwanzig (restricted-orientation) approximation (Cuesta & Martínez-Ratón, PRL 1997) similar to two species, the densities of which are defined at every point in space F [r ] free-energy functional r (r,f) rx(r), ry(r) rxry

44. Confined fluid confined Isotropic phase (I) confined Columnar phase with 17 layers (C17) Competition between d and H Strong commensuration effects expected in the C phase

45. Phase diagram of confined fluid • Layering transitions: between columnar phases with different number of layers Cn Cn+1 • Capillary ordering transitions: analogue of capillary condensation • They are related phenomena

47. Sistema semi-infinito Isótropo/nemático en contacto con una superficie. 3D Sistemas confinados Isotrópo/nemático confinado en celdas simétrica o asimétricas Esméctico confinado en una celda simétrica.

48. Anchoring homogéneo || Anchoring homeotrópico | | 3D: Sistema semi-infinito Modelización de la superficie

49. Polydispersity and nematic stability D = 0.8 D = 0.6 D = 0.3 D = 0

50. Effect of three-body correlations In two dimensions, the scaled B3 does not vanish in the hard-needle limit For three-dimensional rods HARD DISCORECTANGLES VIRIAL COEFFICIENTS (isotropic phase)