2d anisotropic fluids: phase behaviour and defects in small planar cavities
DESCRIPTION
2D anisotropic fluids: phase behaviour and defects in small planar cavities. D. de las Heras 1 , Y. Martínez-Ratón 2 , S. Varga 3 and E. Velasco 1 1 Universidad Autónoma de Madrid, Spain 2 Universidad Carlos III de Madrid, Spain 3 University of Pannonia, Veszprem, Hungary. MODEL SYSTEM - PowerPoint PPT PresentationTRANSCRIPT
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