Simulation of model biaxial particles

A.J. Masters

School of Chemical Engineering and Analytical Science , University of Manchester, UK

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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

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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

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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!

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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*.

* Ref: M. J. Freiser, Phys. Rev. Lett., 1970, 24, 104

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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)

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Bent-Core Molecules and Biaxiality

• Biaxial nematic phase requires molecules withoutcylindrical 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

B.R. Acharya, A. Primak and S. Kumar, Phys. Rev. Lett., 92, 145506 (2004)

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Densi

ty

Narrowing internal angle

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)

Isotropic

Rod-like nematicDisc-like nematic

Biaxial nematic

Idealized Phase Diagram

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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?

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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 (2003)A. Dewar and P.J. Camp, Phys. Rev. E, 70, 011704 (2004)

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Molecular model• Multi-site model of soft, repulsive Weeks-Chandler-Andersen potentials

• 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=110°

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6

612

20

24)(

r

rrrrU

θ=140°

ij ij ijij

ij

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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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P*=0.5 P*=1 P*=2.5

N=512, n=11 potentials, θ=140°

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θ=130°, P*=2.5 θ=110°, P*=2.5

N=512, n=11 potentials, θ<135°

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Help needed!

•Can someone who understands simulation please either tell us what to do or do it for us?

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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

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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!

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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 Weeks-

Chandler-Andersen potential.• NpT molecular dynamics. N dependence checked.• Some NsT simulations.

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6

612

20

24)(

r

rrrrU

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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*=1.6T*=1

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5 Disc system

• As , should be rod-like.

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5 Disc systemsFirst phase transition is from isotropic to nematic.

Coloured according to primary director

Coloured according to secondary director

Centres of mass are random

Uniaxial nematic

P*=0.8

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Smectic A PhaseLayered nematic alignment to the primary director only

Random positioning of centres of mass within layers.

Coloured to the normal director

P*=0.2T*=1P/T=0.2

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Smectic C phase

Same as before but with a tilt.

Getting close to a crystal phase

P*=0.4

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Crystalline Phase

Rigid system with no free movement in the layers

Perfectly fixed and packed system

Coloured to the secondary director

Coloured to the primary director

P*=2T*=2P/T=1

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Conclusion

5 disc system shows all rod phases.

Expected as .

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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 to primary director

Coloured normal to face

Global discotic phase alignment.

Centres of mass are random

Coloured to the secondary director

Random alignment of molecules

P*=1.8

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Discotic smectic phase

Discotic alignment, layers are one molecule thick.No columns found.

P*=2T*=1P/T=2

Coloured to the primary director Coloured to the secondary director

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2 Disc conclusion

IsotropicUniaxial discotic nematicUniaxial layered smectic A

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3 Disc system

• Near the dual point, A/B ~ B/C

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Discotic nematic systemsColoured along the primary axis

Coloured along the normal axis

Centres of mass are random

Shows a Discotic nematic phase (possibly biaxial?)

P*=1

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Weakly biaxial smectic

Ordering along long axis

Ordering along the normal.

Ordering along third axis

P*=0.8

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3 Disc conclusion

Phase sequence appears to be:

IsotropicDiscotic nematic (maybe weak biaxial? )Biaxial layered smectic

(weakly biaxial but significant in our humble opinions - )

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4 Disc system

• Now , should be rod like

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4 Disc systemUniaxial alignment along the primary director

Centres of mass are still randomly arranged

4 Discs are large enough to show uniaxial nematic phase preference over discotic.

P*=1.3

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Strong uniaxial nematic alignment with the primary director.

Good discotic nematic alignment with the secondary director.

Centre of mass is still random

Biaxial nematic phase

P*=1.5

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Biaxial smectic phase

Alignment of primary director.

Alignment of secondary director.

Smectic arrangement of center of mass.Looks like a biaxial

smectic phase.

P*=1.6

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4 Disc order parameters

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4 Disc conclusion

• Phase sequence is:

IsotropicRod-like uniaxial nematic (we think)Biaxial nematicBiaxial 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.

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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.

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Acknowledgements

• Prof. Mark Wilson (Durham)

• Robert Sargant (Manchester)

• Adam Rigby (Manchester)