2D anisotropic fluids:phase behaviour

and defects in small planar cavities

D. de las Heras1, Y. Martínez-Ratón2, S. Varga3

and E. Velasco1

1Universidad Autónoma de Madrid, Spain2Universidad Carlos III de Madrid, Spain

3University of Pannonia, Veszprem, Hungary

MODEL SYSTEM

• Hard particles:

• Hard (excluded volume) interactions

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?

hard models contain essential interactions

to explain many properties

Colloidal non-spherical particles

Metallic nanoparticles

• Motivation

• Phase diagrams

new symmetry: tetratic phase in hard rectangles

• Defects

• Phase separation in mixtures

OUTLINE

Synthesis of metallic nanoparticles of non-spherical shape (nanorods)

building blocks for self-assembly, templates in applications (nanoelectronics)

Wiley et al. Nanolett. 7, 1032 (2007)

Vertically-vibrated quasi-monolayer of granular particles

Experiments on granular matter

Macroscopic realisation of statistical-mechanics of particles?

Observation of liquid-crystal textures in two

dimensions:

• uniaxial nematic

• nematic with strong tetratic correlations

• smectic

smectic state with basmati rice

nematic with strong tetratic correlations in copper cylinders

nematic state with rolling pins

Narayan et al.J. Stat. Mech. 2006

Colloidal discsVibrated monolayer of vertical discs (projecting as rectangles)

Zhao et al. PRE 76, R040401 (2007)

hard rectangle

(HC)

hard

ellipse

(HE)

hard disco-

rectangle(HDR)

3D body2D projection

hard cylinder

hard ellipsoid

hard sphero-cylinder

SOME HARD MODEL PARTICLES

F = U-TS = -TS

Shape, packing and excluded volume determine properties

LIQUID-CRYSTALLINE PHASES (mesophases)

Thermotropic (temperature driven)

Lyotropic (concentration driven)

LIQUID CRYSTALS

DIRECTOR

Phase diagram of HDR (hard disco-rectangles)

Isotropicphase

Phase diagram

Quasi long-range order

Continuous isotropic-nematic phase transition of the KT type

NEMATIC PHASE

Crystalline phase

Bates & Frenkel (2000)

HARD RECTANGLES

Tetratic Nt

CRYSTALLINE

Nematic Nu

Columnar

Smectic

PARTIAL SPATIAL ORDERNEMATIC

nematic phase with two equivalent directors

2D analogue of 3D biaxial and cubatic phases

Possible tetratic phase

?

Scaled-particle theory (SPT) in 2D (Density-functional theory)

Scaled free energy density:

Ideal part:

Excess part:

= v

packing fraction

density

particle area= L

Orientational average of excluded area

Orientational distribution function:

(theory á la Onsager)

Excluded volumes

(HDR)

(HR)

secondary minimumin hard

rectangles

Results from SPT: HR

distribution function of Nu and Nt

HRSPT phase diagram

DISTRIBUTION FUNCTIONS:

Nu: symmetric under rotations of

Nt : symmetric under rotations of / 2

PHASE DIAGRAM:

• Isotropic, Nu and Nt phases

• Nt stability for < 2.62

• Rich phase behaviour

(1st and 2nd order phase

transitions)

Nu

Nt

HDR

Hard rectangles versus hard discorectangles

HDR

• 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

HRHDR

isotropic

unia

xial

ne

mat

ic

Monte Carlo simulation of hard rectangles (Martínez-Ratón et al. JCP 125, 014501, ‘06)

= 3

= 3

I

Nt

K

h()

SPT prediction

isotropic

isotropic

tetratic

tetratic

tetratic

Nt

SPT + B3

SPT

MC

Stability of tetratic phase due to clustering effectsIn the simulations, particle configurations exhibit strong clustering

vibrated monolayer

Monte Carlo simulation

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

2D DEFECTS (topological charge and winding number)

q=1

q=1/2

0d0 2Rd0 0

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

0d

R2d0

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)

2. local order parameter q(r)

3. local tilt angle (r):

The free energy functional is minimised numerically with respect to

variational quantities:

min)(),(),(),( rrqrFrF

PLUS external potential that favours perpendicular

or parallel orientation of molecules V(r,)

)sin,(cosˆ n R

(density)

(tilt)q

(order parameter)

q

PHASE DIAGRAM: Chemical potential vs. cavity radius

first-order phase transition

(discontinuous)

no phase transition

remnant of bulk I-N phase transition in cavity

(pseudo phase transition)

structuraltransition

terminalpoint

inflection point

Structure across pseudo phase transition

pseudo capillary

nematisation

path at fixed radius R and increasing

inflection point

Structure of hedgehog: radial vs. tangential defects

radial

tangential

Nematic elasticityElasticity associated to spatial deformations of the directorFrank elastic energy:

K1 K2 K3

IN 2Dradial

hedgehog defect

tangential hedgehog

defect

DFT calculation of elastic constants

Radius and energy of defect core

nn

n

R

r

el Er

RkEnkrdF

n

logˆ2

11

21

as obtained from inflexion

points

as obtained from DFT

energy density

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

Structure of hedgehog: radial vs. tangential defects

radial

tangential

free-energy density along one radius

r

knrkfel 2ˆ

2

1 121

Parsons-Lee theory

linear regime

Demixing (phase separation) in 2D mixtures

Long-standing issue: does a mixtures of spheres or discs or

different size phase separate?

+

But happens with anisotropic bodies?

Answer seems to be: YES, but one phase is a crystal

Sau et al. Langmuir 21, 2923 (2005)

Experimental verification of

demixing in gold nanospheres and

nanorods

RESULTS:

• no I-I demixing

• there is I-N and N-N separation

THEORY: SPT for mixtures

competition between excluded volume, orientational entropy

and mixing entropy

• discs and rectangles

• rectangles of different size

• discorectangles & rectangles

de las Heras et al. PRE 76, 031704 (2007)

hard squares and discs: L1=1, 1=2=1

hard squares: L1=10, L2=1

HR and HDR: L1=1.5, 1=1, L2=1.70, 2=0.85

hard rectangles: L1=4.0, 4.6, 5.0, L2=2, 1=2=1

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

THE END

2D Defectos en una cavidad circular Núcleos

2R=100D

Hard rectangles in confined geometry

BULK ( = 3):

Isotropic ( I )

coexisting with

Columnar (C)

Theory: FMT in Zwanzig (restricted-orientation) approximation(Cuesta & Martínez-Ratón, PRL 1997)

(Y. Martínez-Ratón, PRE 2007)

similar to two species, the densities of which are defined at every point in space

x y

F [] free-energy functional

(r,) x(r), y(r)

Phenomenology similar to confined (3D) hard spherocylinders where ordered phase is a smectic

(de las Heras, Velasco & Mederos, PRL 2005)

CONFINEMENT: Competition between

capillary ordering and layering transitions

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

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

Sistema semi-infinito

Isótropo/nemático en contactocon una superficie.

Sistemas confinados

Isotrópo/nemático confinado en celdas simétrica o asimétricas

Esméctico confinado en una celda simétrica.

3D

3D: Sistema semi-infinito Modelización de la superficie

Anchoring homogéneo ||

Anchoring homeotrópico ||

Polydispersity and nematic stability

Effect of three-body correlations

For three-dimensional rods In two dimensions, the scaled B3 does not vanish in the hard-needle limit

HARD DISCORECTANGLES VIRIAL COEFFICIENTS (isotropic phase)

To incorporate B3, we construct a SPT-based Padé approximant:

resulting in:

functionals of h()

Excess free energy per particle for isotropic phase:

Extension of SPT including B3 (ISOTROPIC & NEMATIC phases)

Bk

For the nematic phase:

Monte Carlo simulation

vibrated monolayer

Scaled-particle theory (SPT) in 2D (Schlacken et al. Mol. Phys. '98)

Scaled free energy density:

Ideal part:

Excess part:

Averaged excluded area:

Order parameters:

uniaxial

tetratic

= v

packing fraction (fraction of area

occupied by particles)

density

particle area= L

Tetratic phases in 3D hard biaxial parallelepipeds

FMT Phase diagrams

FMT Phase diagrams

NB

Density profilesBiaxial smectic SmB

Biaxial order parameter

Nematic elasticity

Elasticity associated to spatial deformations of the director

K1 K3 K2

IDEAL FREE ENERGY

Fid[ ] is built from an ideal-gas mixture:

Formally identify i-th species with particles at r with orientation :

so that the general ideal-gas functional is:

(minus) translational entropy (minus) rotational entropy

orientational probability

functionmean

density(constant)

ONSAGER theory for Isotropic-Nematic transition

(NEMATIC PHASE)

Free-energy functional: F [ ] = Fid[ ] + Fex[ ]

Fex[ ] is obtained from a truncated virial expansion:

Onsager showed that:

so that the theory is asymptotically exact in the limit D / L

The Bn[ ]'s are the virial coefficients. For instance:

Mayer function:

pair potential

EXCESS FREE ENERGY

Cluster statistics in Monte CarloCriterion for connectedness:

< 10º

r < 1.3

Cluster histogram weakly dependent on crtiterion

size distribution function

Order in two-dimensional nematic phases

Elastic theory predicts absence of true long-range orientational order:

Nematic order parameter vanishes in thermodynamic limit:

Orientational correlation function decays algebraically with distance:

Quasi long-range order

B. J. Wiley et al. Nano Letters 7, 1032 (2007)

SOLAPE

The excluded volume is defined as the

volume integral of the overlap integral:

IT DEPENDS ON THE ANGLES

f o = 0 f o = 1

For hard bodies:

fix one particle at

origin

fix orientations of both

particles

For hard spherocylinders:

Resultados de la teoría (comparar con simulación) y perfilar mejoras

• The interaction (excess) contribution always decreases with S

(excess entropy increases with S)

F = U-TS = -TS

• Beyond some density there arises a minimum for S 0

• The ideal contribution always increases with S (orientational entropy decreases with S)

= 3

including B3

SPT

MC

= 9

including B3

SPT

MC

NEMATIC PHASE

We generalise the first two virial coefficients to oriented phases:

and hence the excess free energy:

B3 is parameterised in terms of a Gaussian function B3(q1,q2)

Construction of DFT for inhomogeneous phases

Excess free-energy:

averaged densities:

FEATURES:• Parallel HR reference system (other choices are possible)• tends to the Onsager limit for low densities• captures high-density limit with FMT-like structure• recovers SPT in the uniform limit

PERFORMANCE:

As a test, we consider parallel hard squares,

comparing with more acurate FMT (Cuesta & Martínez-Ratón, 1997)

• Continuous transition to a square-lattice crystal• Crystal stabilised at slightly lower packing fraction• Lower fraction of vacancies• GOOD overall performance

APPLICATION TO FREELY-ROTATING HRs

• Calculation of spinodal line with respect to spatial order

• Tetratic phase preempted by (possibly) columnar phase

• But our simulations point to stable tetratic phase!

C?

Semiinfinite geometry: wetting phenomena

Young condition for wetting of the WI interface by the C phase:

Isotropic-columnar interface (slab) Wall-columnar interface (slab)

, calculated using the film method

calculated approaching IC coexistence

The condition is met so the C phase wets the WI interface

WI interface

Macroscopic approach

Kelvin equation:

Modified Kelvin equation, including elasticity effects:

free energy columnar slab free energy isotropic slab

elastic contribution

Mixtures of hard rectangles with other bodies

We again use scaled-particle theory for mixtures:

Ideal part:

Excess part:

Excluded volumes:

Order parameters:

Contains mixing entropy

Some conclusions

• Due to reduced dimensionality, rich phase diagrams

including exotic points (critical, tricritical, azeotropic...)

• DFT studies may help explain phenomenology in

vibrated granular monolayer experiments

• confined layers: analogous to 3D system

(bulk 1st-order phase transition)

• Clarify phenomena in terms of depletion forces

(more easily calculated than in 3D)

Some future lines for research

• Nature of crystalline phases (simulations)• Exploration of FMT-like theories• Experiments on vibrated layers

confined layers (capillary effects)

depletion forces (direct measurement)

hard squares: L1=10, L2=1

Some conclusions

• Due to reduced dimensionality, rich phase diagrams

including exotic points (critical, tricritical, azeotropic...)

• DFT studies may help explain phenomenology in

vibrated granular monolayer experiments

• confined layers: analogous to 3D system

(bulk 1st-order phase transition)

• Clarify phenomena in terms of depletion forces

(more easily calculated than in 3D)

Some future lines for research

• Nature of crystalline phases (simulations)• Exploration of FMT-like theories• Experiments on vibrated layers

confined layers (capillary effects)

depletion forces (direct measurement)

In summary: Hard rectangles versus hard discorectangles

HDR

• 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

• SPT underestimates stability of tetratic phase

HRHDR

isotropic

unia

xial

ne

mat

ic

• SPT underestimates stability of tetratic phase

area distribution function of clusters from simulation

Peak at square clusters

Enhanced distribution

angular distribution function of clusters (from theory)

angular distribution function of monomers (from theory)

= 3

= 3

Edwards theoryEdwards & Oakeshott, Physica A 157, 1080 (1989)

Some phenomenology of granular materials can be described using concepts of equilibrium statistical mechanics:

• Static granular configurations are described by a single parameter:

the packing fraction • Static configurations are distributed according to a canonical distribution:

Ciamarra et al., PRL 97, 158001 (2006)

Tconf: configurational temperature

Cálculo de la coexistencia isótropo-nemático

A cada densidad se obtiene la configuración de

equilibrio del fluido y de ahí S

Se obtiene una transición de fase de primer orden con una barrera de energía libre entre la fase desordenada (isótropa) y la ordenada (nemática)

*IN

• calor latente

• fases metaestables

• barrera pequeña: transición débil

• *IN= 4.53

S