Simulation of model biaxial
particles
A.J. Masters
School of Chemical Engineering and Analytical Science , University of Manchester, UK
Thermodynamics 2013 Manchester, 3-6 September
https://www.meeting.co.uk/confercare/
thermodynamics2013/
Abstract deadline: 1 April, 2013
Invited speakers:
Keith Gubbins; Daan Frenkel; Ross Taylor;Carol
Hall;Alejandro Gil-Villegas; Peter Monson; Aline
Miller; Paola Carbone; Geoff Maitland
2
Rods and discs
• Most liquid-crystal forming particles (molecules or
colloidal particles) can be regarded as either rods or discs
• Axially symmetric rods:
– Isotropic - nematic (+) - smectic A - crystal
• Axially symmetric discs:
– Isotropic - nematic (-) - columnar - crystal
3
What else can you get?
• Look at some other possibilities that might arise
from simple shapes
• Studies are of hard particles - no attractive
forces, no flexibility
• A quick scamper through V-shapes and fused
hexagons!
4
V-shapes and the Biaxial Nematic
Phase
• A liquid crystal phase characterized by molecular
alignment along three orthogonal axes while maintaining
random positional behaviour
• Gives rise to three distinct optical axes
• Uniaxial nematic  biaxial nematic transition can lead
to novel ferroelectric & optical devices
• The theoretical possibility of such a phase was first
discussed in 1970*.
5
* Ref: M. J. Freiser, Phys. Rev. Lett., 1970, 24, 104
Experimental state-of-play
• Observed by Yu and Saupe (1980) in a lyotropic system
(potassium laurate/1-decanol/water)
• Observed in colloidal suspension of goethite - a board-like
particle (van den Pol et al, 2009)
6
Bent-Core Molecules and Biaxiality
• Biaxial nematic phase requires molecules without
cylindrical symmetry for alignment along multiple axes
• “V-shaped” or bent-core molecules have correct symmetry to
allow biaxial nematic phase, and have experimentally viable
shapes
7
B.R. Acharya, A. Primak and S. Kumar, Phys. Rev. Lett., 92, 145506 (2004)
Idealized Phase Diagram
Disc-like nematic
Rod-like nematic
Density
Biaxial nematic
Isotropic
Narrowing internal angle
8
P.I.C. Teixeira, A.J. Masters and B.M. Mulder, Mol. Cryst. Liq. Cryst. 167, 323 (1998)
G.R. Luckhurst, Thin Solid Films, 393, 40 (2001)
Onsager limit?
•I think (!) that in the Onsager limit (large L/D),
that all virials higher than second order can be
neglected
•Tested against 3rd virial calculations but
nothing higher
•Biaxial phase is truly stable in this limit
•Can we see it in a simulation/experiment?
9
Simulation studies on bent-core
models
• Phase transitions for a bent-core model
• Dependence on bond angle and arm length
• Binary mixtures of bent-cores
• House rules - we will obtain all phases by compressing
the isotropic. We are not allowed to start from an
assumed crystal structure!
Y. Lansac et al, Phys. Rev. E, 67, 011703 10
(2003)
A. Dewar and P.J. Camp, Phys. Rev. E, 70, 011704 (2004)
Molecular model
•
Multi-site model of soft, repulsive Weeks-Chandler-Andersen potentials
   12   6 
4        
U (rij)    rij 
 rij  


 0
•
•
•
rij  6 2
rij  6 2
Rigidly linked particles at separation σ – no bond flexibility
Two arms of equal length, with a shared atom at apex
Bend angle defined as θ=180° linear, θ=90° perpendicular
•  = kT
θ=170°
θ=140°
θ=110°
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Initial Methodology
•
Parameter space of bond angle and pressure, N=512 molecules
•
Starting at low-density isotropic liquid phase, perform a series of
time-stepped NPT-MD (constant pressure, constant temperature)
simulations at incremental pressure steps
•
Use order parameters, configurational energy,
pair correlation function g(rij) and snapshots to examine
phase behaviour and transitions
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N=512, n=11 potentials, θ=140°
P*=0.5
P*=1
P*=2.5
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N=512, n=11 potentials, θ<135°
θ=130°, P*=2.5
θ=110°, P*=2.5
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Help needed!
•Can someone who understands simulation
please either tell us what to do or do it for
us?
15
N=4096, n=11, θ=150°, P*=1.4
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n=7 potentials
• Theoretical limit is for bent-core molecules
in Onsager limit (L >> D)
• Examine the effects of the length of the
molecule arms on phase behaviour
• Equivalent parameter space sweep as for n=11 potentials
17
Arm length dependence: n=7 @ P* = 4
θ=170°
θ=160°
θ=140°
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Give up - how about binary mixtures?
•Incommensurate particles will not sit
comfortably together in a smectic layer
•Maybe we can find a uniaxial/biaxial nematic
transition on compression.
•Just consider 50:50 mixtures in terms of
particle numbers
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Binary mixture: n=7 & n=11
θ=150°, P*=4.2
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Binary mixture: n=5 & n=11
θ=160°, P*=1.7
θ=160°, P*=1.9
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Summary of binary mixtures
•On increasing the pressure, a smectic precipitates
out.
•We never see a biaxial nematic
•Similar issues for binary mixtures where we vary
bend angle
•Effect of higher polydispersity? Christine Stokes on
Friday!
22
Other molecule shapes (in progress)
“Symmetric”
“Asymmetric”
“Y-Shaped”
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Boards and hexagons
• Resembles mono-disperse goethite (van der Pol)
• If C/B ~ B/A - self-dual point, biaxial nematic
predicted (and, indeed, found!)
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The Model
Disc
• Discotic nematic
• Columnar
• Crystal
Rod
•Rodlike nematic
•Smectic
•Crystal
Also checked
large discs
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System Parameters
• Rigid shapes
• Spheres interact via the repulsive WeeksChandler-Andersen potential.
• NpT molecular dynamics. N dependence checked.
• Some NsT simulations.
   12   6 
4        
U (r )    r 
 r  


 0
r  6 2
r  6 2
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Particle structures
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Disc system
• Small discs should give no liquid crystal phases.
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Columnar Phase
(no nematic seen)
P*=1.6
T*=1
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5 Disc system
• As
, should be rod-like.
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5 Disc systems
First phase transition is from isotropic to
nematic.
P*=0.8
Coloured according to primary director
Coloured according to secondary director
Centres of mass are random
Uniaxial nematic
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Smectic A Phase
Layered nematic
alignment to the
primary director only
Coloured to the normal
director
P*=0.2
T*=1
P/T=0.2
Random positioning of
centres of mass within
layers.
32
Smectic C phase
Same as before
but with a tilt.
P*=0.4
Getting close
to a crystal
phase
33
Crystalline Phase
Coloured to
the primary
director
Rigid system with no free
movement in the layers
P*=2
T*=2
P/T=1
Coloured to the
secondary director
Perfectly fixed and
packed system
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Conclusion
5 disc system shows all rod phases.
Expected as
.
35
2 Disc system
,
Expect disc like
behaviour
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2 disc systems
Overall isotropic behaviour at low
pressure.
P*=1.3
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Discotic nematic
Coloured normal to
face
Coloured to the
secondary director
Global discotic
phase
alignment.
Random
alignment of
molecules
Coloured to primary
director
P*=1.8
Centres of
mass are
random
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Discotic smectic phase
P*=2
T*=1
P/T=2
Coloured to the primary director
Coloured to the secondary director
Discotic alignment, layers are one molecule thick.
No columns found.
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2 Disc conclusion
Isotropic
Uniaxial discotic nematic
Uniaxial layered smectic A
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3 Disc system
• Near the dual point, A/B ~ B/C
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Discotic nematic systems
Coloured along the primary
axis
Coloured along the normal
axis
P*=1
Shows a Discotic
nematic phase
(possibly biaxial?)
Centres of mass are
random
42
Weakly biaxial smectic
P*=0.8
Ordering along long
axis
Ordering along
the normal.
Ordering along
third axis
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3 Disc conclusion
Phase sequence appears to be:
Isotropic
Discotic nematic (maybe weak biaxial? )
Biaxial layered smectic
(weakly biaxial but significant in our humble
opinions - )
44
4 Disc system
• Now
, should be rod like
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4 Disc system
Uniaxial alignment along the
primary director
P*=1.3
Centres of mass
are still randomly
arranged
4 Discs are large enough to
show uniaxial nematic
phase preference over
discotic.
46
Biaxial nematic phase
P*=1.5
Strong uniaxial
nematic alignment
with the primary
director.
Good discotic
nematic
alignment with
the secondary
director.
Centre of
mass is still
random
47
Biaxial smectic phase
P*=1.6
Alignment of
primary
director.
Looks like a biaxial
smectic phase.
Alignment of
secondary
director.
Smectic
arrangement of
center of mass.
48
4 Disc order parameters
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4 Disc conclusion
• Phase sequence is:
Isotropic
Rod-like uniaxial nematic (we think)
Biaxial nematic
Biaxial rod-like smectic
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Theory - M. P. Taylor and J. Herzfeld,
Phys. Rev. A, 44, 3742 (1991)
QuickTime™ and a
decompressor
are needed to see this picture.
51
Comparison
•We only see a columnar phase for large
hexagonal discs
•Everything else we have seen somewhere …
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Summary
• Bent-core model
• I think the biaxial nematic is there in the Onsager limit
• We cannot see it in our simulations for one component
systems
• We cannot simulate the interesting bend-angles!
• Binary mixtures - we see smectic demixing
• Hexagons/boards
• Interesting collection of phases, but little columnar
• Layered smectic phase rather unexpected.
53
Acknowledgements
• Prof. Mark Wilson (Durham)
• Robert Sargant (Manchester)
• Adam Rigby (Manchester)
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