2D anisotropic fluids: phase behaviour and defects in small planar cavities
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Transcript of 2D anisotropic fluids: phase behaviour and defects in small planar cavities
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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
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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
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• Motivation
• Phase diagrams
new symmetry: tetratic phase in hard rectangles
• Defects
• Phase separation in mixtures
OUTLINE
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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)
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Vertically-vibrated quasi-monolayer of granular particles
Experiments on granular matter
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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
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Colloidal discsVibrated monolayer of vertical discs (projecting as rectangles)
Zhao et al. PRE 76, R040401 (2007)
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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
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LIQUID-CRYSTALLINE PHASES (mesophases)
Thermotropic (temperature driven)
Lyotropic (concentration driven)
LIQUID CRYSTALS
DIRECTOR
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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)
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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
?
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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)
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Excluded volumes
(HDR)
(HR)
secondary minimumin hard
rectangles
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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
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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
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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
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Nt
SPT + B3
SPT
MC
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Stability of tetratic phase due to clustering effectsIn the simulations, particle configurations exhibit strong clustering
vibrated monolayer
Monte Carlo simulation
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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
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2D DEFECTS (topological charge and winding number)
q=1
q=1/2
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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
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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
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(density)
(tilt)q
(order parameter)
q
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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
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Structure across pseudo phase transition
pseudo capillary
nematisation
path at fixed radius R and increasing
inflection point
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Structure of hedgehog: radial vs. tangential defects
radial
tangential
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Nematic elasticityElasticity associated to spatial deformations of the directorFrank elastic energy:
K1 K2 K3
IN 2Dradial
hedgehog defect
tangential hedgehog
defect
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DFT calculation of elastic constants
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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
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Structure of hedgehog: radial vs. tangential defects
radial
tangential
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free-energy density along one radius
r
knrkfel 2ˆ
2
1 121
Parsons-Lee theory
linear regime
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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
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Sau et al. Langmuir 21, 2923 (2005)
Experimental verification of
demixing in gold nanospheres and
nanorods
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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)
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hard squares and discs: L1=1, 1=2=1
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hard squares: L1=10, L2=1
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HR and HDR: L1=1.5, 1=1, L2=1.70, 2=0.85
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hard rectangles: L1=4.0, 4.6, 5.0, L2=2, 1=2=1
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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
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THE END
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2D Defectos en una cavidad circular Núcleos
2R=100D
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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
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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
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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
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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
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3D: Sistema semi-infinito Modelización de la superficie
Anchoring homogéneo ||
Anchoring homeotrópico ||
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Polydispersity and nematic stability
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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)
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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:
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Monte Carlo simulation
vibrated monolayer
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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
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Tetratic phases in 3D hard biaxial parallelepipeds
FMT Phase diagrams
FMT Phase diagrams
NB
Density profilesBiaxial smectic SmB
Biaxial order parameter
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Nematic elasticity
Elasticity associated to spatial deformations of the director
K1 K3 K2
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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[ ]
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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
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Cluster statistics in Monte CarloCriterion for connectedness:
< 10º
r < 1.3
Cluster histogram weakly dependent on crtiterion
size distribution function
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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
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B. J. Wiley et al. Nano Letters 7, 1032 (2007)
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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:
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fix one particle at
origin
fix orientations of both
particles
For hard spherocylinders:
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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)
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= 3
including B3
SPT
MC
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= 9
including B3
SPT
MC
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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)
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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
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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
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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?
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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
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WI interface
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Macroscopic approach
Kelvin equation:
Modified Kelvin equation, including elasticity effects:
free energy columnar slab free energy isotropic slab
elastic contribution
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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
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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)
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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)
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hard squares: L1=10, L2=1
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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)
![Page 79: 2D anisotropic fluids: phase behaviour and defects in small planar cavities](https://reader038.fdocuments.net/reader038/viewer/2022110212/56813c5e550346895da5e067/html5/thumbnails/79.jpg)
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)
![Page 80: 2D anisotropic fluids: phase behaviour and defects in small planar cavities](https://reader038.fdocuments.net/reader038/viewer/2022110212/56813c5e550346895da5e067/html5/thumbnails/80.jpg)
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
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area distribution function of clusters from simulation
Peak at square clusters
Enhanced distribution
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angular distribution function of clusters (from theory)
angular distribution function of monomers (from theory)
= 3
= 3
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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
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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
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