Biaxial Nematics Fact, Theory and Simulation

Transcription

1 Soft Matter Mathematical Modelling Cortona 12 th -16 th September 2005 Biaxial Nematics Fact, Theory and Simulation Geoffrey Luckhurst School of Chemistry University of Southampton

2 Fact The elusive biaxial nematic Claims to discovery Recognition of the biaxial nematic

3 Biaxial Nematic N B molecular organisation Macroscopic definition Frank Symmetry of average tensorial property,t, such as the dielectric susceptibility or the quadrupolar splitting in NMR is biaxial. Principal components T XX T YY T ZZ for N B (n.b. T XX = T YY T ZZ for N U ) T The directors, n, l, m are identified with the principal axes of. η=3 T XX T YY / 2 T ZZ T XX T YY

4 Lyotropic Biaxial Nematic (potassium laurate (KL), 1-decanol, D 2 ) L J Yu, A Saupe, Phys.Rev.Lett., 1980, 45, Phase diagram

5 Identifying a Biaxial Nematic F P Nicoletta, G Chidichimo, A Golemme, P Picci, Liq.Cryst., 1991, 10, Discotic (N D ) Another lyotropic biaxial nematic (potassium laurate, decyl ammonium chloride, D 2 ) Cylindric (N C ) The primary 2 H NMR data Biaxial (N Bx ) η=0. 51 Biaxial (N Bx ) η=1 Biaxial (N Bx ) η=0.68

6 Thermotropic Biaxial Nematics? C 12 H 25 CH 2 C 6 H 13 C 6 H 13 C 12 H 25 CH 2 C 2 C 2 C 12 H 25 C 6 H 13 H H C 6 H 13 C 12 H 25 CH 2 C 6 H 13 C 6 H 13 C 2 H 5 H C 10 H 21 Cu C 10 H 21 H 25 C 12 H 25 C 12 C 12 H 25 H C 12 H 25 C 2 H 5 H 25 C 12 C 12 H 25

7 Thermotropic Biaxial Nematics 2004 V-shaped molecules L A Madsen, T J Dingemans, M Nakata, and E T Samulski, Phys. Rev. Lett., 2004, 92, η=0 η=0.11 Biaxiality in q η=0.11 n.b. for lyotropic biaxial nematics η η=0. 22

8 Thermotropic Biaxial Nematics 2005 Tetrapodes: J L Figueirinhas, C Cruz, D Filip, G Feio, A C Ribeiro, Y Frère and T Meyer, G H Mehl Phys. Rev. Lett., 2005, 94,

9 Tetrapodes: the NMR Evidence

10 Theory Molecular field approach Landau approach

11 General Notation Molecular Field Theory Single molecule potential U= u 2 mn 1 δ m2 1 δ n 2 1 δ p 2 R 2 pm R 2 p n Symmetry adapted functions Dependence on Euler angles 2 = 3 cos 2 β 1 /2 R 0 0 R = 3 8 sin 2 β cos 2 γ R = 3 8 sin 2 β cos 2 2 α R = cos 2 β γ cos 2 α cos 2 γ cos β sin 2 α sin 2 rder parameters are averages of R 2 mn Strength parameters u 200 u 220 u 202 u 222

12 rientational rder Parameters Biaxial molecule in biaxial phase Limit major ZZ 2 S zz = <(3l zz 1)/2> 1 molecular biaxial S xx ZZ - S yy ZZ = <3(l xz 2 l yz2 )/2> 0 phase biaxial S zz XX S zz YY = <3(l zx 2 l zy2 )/2> 0 phase biaxial (S xx XX S xx YY ) (S yy XX S yy YY ) = <3{(l xx 2 l xy2 ) - (l yx 2 l yy2 )/ 2}> 3 Axes: molecular xyz and space XYZ 8 8 R R R R 2 22

13 Molecular Field Theory N Boccara, R Medjani, L de Seze, J.Phys., 1977, 38, Phase diagram (molecular biaxiality ε = λ 6 ) Geometric mean approximation 1 u 220 = u u λ= u u Bounds on ε of 0 and 3 correspond to uniaxial molecules ε = 1is the maximum biaxiality giving the Landau point

14 Molecular Field Theory A M Sonnet, E G Virga and G E Durand, Phys. Rev., E, 2003, 67, Phase Diagram I 1/β N U N B u 220 =0 λ ~u 222 /u 200 λ

15 Landau Theory Uniaxial nematic of uniaxial molecules Key order parameter for nematic-isotropic transition: S A=A 0 BS 2 CS 3 DS 4 Free energy expansion of invariants Assumption: B=b T T Dependence on S and magnitude of other coefficients related to experimental quantities

16 Landau Theory Biaxial phase of biaxial molecules rder parameters Major S= 3 cos 2 β 1/2 = R Molecular biaxial T = sin 2 β cos 2 γ = 8 3 R Phase biaxial U= sin 2 β cos 2 α = 8 3 R 2 2 0

17 Landau Theory Phase / molecular biaxial V = cos 2 β cos 2 α cos 2 γ cos β sin 2 α sin 2 γ =2 R 2 2 Uniaxial nematic S 0 T 0 U=0 V =0 Biaxial nematic S 0 T 0 U 0 V 0

18 Landau Theory D W, Allender, M A Lee, N Hafiz, Mol. Cryst. Liq. Cryst., 1985, 124, Free energy A N A I = A S 2 /2 T 2 U 2 V 2 /18 B S 3 /4 3 SU 2 /2 3 ST 2 /2 SV 2 /12 TUV C 1 S 2 /2 T 2 U 2 V 2 /18 2 C 2 VS/2 2TU 2 D 1 S 2 /2 T 2 U 2 V 2 /18 S 3 /4 3 ST 2 /2 3 SU 2 /2 SV 2 /12 TUV D 2 SV /3 2 TU SV 2 /4 TV 2 /2 U 2 V /2 V 3 /108...

19 Landau Theory Free energy (continued)... E 1 S 2 /2 T 2 U 2 V 2 /18 3 E 2 S 2 /2 T 2 U 2 V 2 /18 SV /3 2TU 2 E 3 S 3 /4 ST 2 /2 3 SU 2 /2 TUV S V 2 /12 2 E 4 S 2 V /4 U 2 V /2 T 2 V /2 3 STU V 3 /108 2 E 5... Assumptions A is linear in temperature ~ ( T T * ) B, C 1, C 2, D 1, D 2, E 1, E 2, E 3, E 4, E 5 are temperature independent.

20 Landau Theory for Biaxial Nematics Phase diagram C 1 = 0, C 2 = 1, D 1 = 1, D 2 = 1, E 1 = 1, E 2 = 1, E 3 = 1, E 4 = 1, E 5 = -7

21 Landau Theory for Biaxial Nematics Problems: (b) Large number of unknown parameters (c) Single characteristic temperature, might have expected more (d) Four nematic phases are predicted Phase rder Parameter I 0 - N U1 1 S N U2 N B N B * 2 S, T 2 S, U 4 S, T, U, V

22 Strategy of Katriel et al. J Katriel, G F Kventsel, G R Luckhurst, T J Sluckin, Liq. Cryst., 1986, 1, (1) Free energy A= 1 2 u 200 P 2 2 k B T f β ln 2 f β sin βdβ =U TS (4) rder parameter P 2 = P 2 cos β f β sin βdβ (6) Entropy a functional of distribution function f(β) S=S [ f β ] But A is not yet a function of P!

23 Strategy of Katriel et al. A=k B T f β ln 2 f β sin βdβ 1 u 200 P 2 2 =U TS 2 Maximise entropy term subject to given P (1) f(β) a function of auxiliary parameter η f β = 1 Z η exp ηp cos β 2 (3) Partition function (5) P a function of η Z η = exp P 2 = ln Z η η ηp 2 cos β sin βdβ A is now a function of ηand P

24 Strategy of Katriel et al. A=k B T f β ln 2 f β sin βdβ 1 u P2 2=U TS (1) Invert equation: (3) A was a function of P and η P 2 = ln Z η η Expand P 2 in a power series in η (6) A is now a function of P Invert power series to required order Expand A in power series in P A=A P 2

25 Result A= 5 2 k B T T P k B T P k B T P T =u 200 /5 k B

26 Landau Theory for Biaxial Nematics A molecular field approach Energy U N = 1 2 u 2 mn 1 δ m2 1 δ n 2 1 δ p 2 R 2 pm R 2 p n U N = 1 2 u 200 S 2 2 U 2 4 u 220 ST 2UV 4 u 222 T 2 2 V 2 n.b. expansion coefficients are components of supertensor and not scalars

27 Simulation Rod-Disc Mixtures Rod-Disc Dimers Flexibility

28 The phase diagram Mixtures of rods and discs P Palffy-Muhoray, J R Bruyn, D A Dunmur, J.Chem.Phys., 1985, 82,

29 Rod Disc Dimers I D Fletcher, G R Luckhurst, Liq.Cryst., 1995, 18, J J Hunt, R W Date, B A Timimi, G R Luckhurst, D W Bruce, J.Am.Chem.Soc., 2001, 123, P H J Kouwer, G H Mehl., Mol.Cryst.Liq.Cryst., 2003, 397,

30 The Phase Behaviour of Rod-Disc Dimers M A Bates, G R Luckhurst, PCCP, 2005, 7, The Lebwohl-Lasher lattice Model Anisotropic interactions Torsional potential ε a > 0 symmetry axes orthogonal ε a < 0 symmetry axes parallel d U ij RR = ε RR P 2 r i r j U ij RD =ε RD P 2 r i d j U ij RD =ε RD P 2 d i r j U ij DD = ε DD P 2 d i d j U RD tors =ε a P 2 r d i i r

31 Parameterisation Scaling Parameter ε RR Relative anisotropy e.g. T =k B T /ε RR ε =ε DD /ε RR Controls the molecular biaxiality Geometric mean or Berthelot approximation ε RD = ε DD ε RR ½ Scaled torsional strength ε a =ε a /ε RR

32 Rigid Rod-Disc Dimer Phase diagram as function of relative anisotropy ε DD /ε RR T * I N u N u N B

33 Flexible Rod-Disc Dimers Phase diagram for different torsional strengths ε a ε a /ε RR ε a =0 ε a = 4 ε a = 6 ε a = 7

34 Flexible Rod-Disc Dimers rientational order parameters R Q AA ε a = 7 and D Q AA [ S α XX, S α YY ε =2.0 ε =1. 75, S α ZZ ] α= R or D

35 Acknowledgements Fact Gerd Kothe (Freiburg) Jacques Malthete (Paris) Klaus Praefcke (Berlin) Bakir Timimi (Southampton) Theory Tim Sluckin (Southampton) Ken Thomas (Southampton) Simulation Martin Bates (York) Silvano Romano (Pavia)