Simulation of model biaxial particles A.J. Masters School of Chemical Engineering and Analytical Science, University of Manchester, UK

2 Thermodynamics Manchester, 3-6 September 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

3 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

4 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!

5 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

6 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)

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

8 Density 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 nematic Disc-like nematic Biaxial nematic Idealized Phase Diagram

9 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?

10 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, (2003) A.Dewar and P.J. Camp, Phys. Rev. E, 70, (2004)

11 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°θ=140° ij

12 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(r ij ) and snapshots to examine phase behaviour and transitions

13 P*=0.5 P*=1P*=2.5 N=512, n=11 potentials, θ=140°

14 θ=130°, P*=2.5 θ=110°, P*=2.5 N=512, n=11 potentials, θ<135°

15 Help needed! Can someone who understands simulation please either tell us what to do or do it for us?

16 N=4096, n=11, θ=150°, P*=1.4

17 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

18 Arm length dependence: P* = 4 θ=170° θ=160° θ=140°

19 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

20 Binary mixture: n=7 & n=11 θ=150°, P*=4.2

21 Binary mixture: n=5 & n=11 θ=160°, P*=1.7 θ=160°, P*=1.9

22 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!

23 Other molecule shapes (in progress) “Symmetric”“Asymmetric” “Y-Shaped”

24 Boards and hexagons Resembles mono-disperse goethite (van der Pol) If C/B ~ B/A - self-dual point, biaxial nematic predicted (and, indeed, found!)

25 The Model Disc Discotic nematic Columnar Crystal Rod Rodlike nematic Smectic Crystal Also checked large discs

26 System Parameters Rigid shapes Spheres interact via the repulsive Weeks- Chandler-Andersen potential. NpT molecular dynamics. N dependence checked. Some NsT simulations.

27 Particle structures

28 Disc system Small discs should give no liquid crystal phases.

29 Columnar Phase (no nematic seen) P*=1.6 T*=1

30 5 Disc system As, should be rod-like.

31 5 Disc systems First 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

32 Smectic A Phase Layered nematic alignment to the primary director only Random positioning of centres of mass within layers. Coloured to the normal director P*=0.2 T*=1 P/T=0.2

33 Smectic C phase Same as before but with a tilt. Getting close to a crystal phase P*=0.4

34 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*=2 T*=2 P/T=1

35 Conclusion 5 disc system shows all rod phases. Expected as.

36 2 Disc system, Expect disc like behaviour

37 2 disc systems Overall isotropic behaviour at low pressure. P*=1.3

38 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

39 Discotic smectic phase Discotic alignment, layers are one molecule thick. No columns found. P*=2 T*=1 P/T=2 Coloured to the primary directorColoured to the secondary director

40 2 Disc conclusion Isotropic Uniaxial discotic nematic Uniaxial layered smectic A

41 3 Disc system Near the dual point, A/B ~ B/C

42 Discotic nematic systems Coloured along the primary axis Coloured along the normal axis Centres of mass are random Shows a Discotic nematic phase (possibly biaxial?) P*=1

43 Weakly biaxial smectic Ordering along long axis Ordering along the normal. Ordering along third axis P*=0.8

44 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 - )

45 4 Disc system Now, should be rod like

46 4 Disc system Uniaxial 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

47 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

48 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

49 4 Disc order parameters

50 4 Disc conclusion Phase sequence is: Isotropic Rod-like uniaxial nematic (we think) Biaxial nematic Biaxial rod-like smectic

51 Theory - M. P. Taylor and J. Herzfeld, Phys. Rev. A, 44, 3742 (1991)

52 Comparison We only see a columnar phase for large hexagonal discs Everything else we have seen somewhere …

53 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.

54 Acknowledgements Prof. Mark Wilson (Durham) Robert Sargant (Manchester) Adam Rigby (Manchester)