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
Fact The elusive biaxial nematic Claims to discovery Recognition of the biaxial nematic
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
Lyotropic Biaxial Nematic (potassium laurate (KL), 1-decanol, D 2 ) L J Yu, A Saupe, Phys.Rev.Lett., 1980, 45, 1000-1003 Phase diagram
Identifying a Biaxial Nematic F P Nicoletta, G Chidichimo, A Golemme, P Picci, Liq.Cryst., 1991, 10, 665-674 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
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
Thermotropic Biaxial Nematics 2004 V-shaped molecules L A Madsen, T J Dingemans, M Nakata, and E T Samulski, Phys. Rev. Lett., 2004, 92, 145505 η=0 η=0.11 Biaxiality in q η=0.11 n.b. for lyotropic biaxial nematics η 0. 7 1. 0 η=0. 22
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, 107802
Tetrapodes: the NMR Evidence
Theory Molecular field approach Landau approach
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 2 0 2 = 3 8 sin 2 β cos 2 γ R 2 2 0 = 3 8 sin 2 β cos 2 2 α R 2 2 2 = 1 2 1 2 1 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
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 2 0 0 R 2 3 0 2 R 2 3 2 0 6 R 2 22
Molecular Field Theory N Boccara, R Medjani, L de Seze, J.Phys., 1977, 38, 149-151 Phase diagram (molecular biaxiality ε = λ 6 ) Geometric mean approximation 1 u 220 = u u 2 200 222 λ= u 222 2 u 200 1 Bounds on ε of 0 and 3 correspond to uniaxial molecules ε = 1is the maximum biaxiality giving the Landau point
Molecular Field Theory A M Sonnet, E G Virga and G E Durand, Phys. Rev., E, 2003, 67, 061701 Phase Diagram I 1/β N U N B u 220 =0 λ ~u 222 /u 200 λ
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
Landau Theory Biaxial phase of biaxial molecules rder parameters Major S= 3 cos 2 β 1/2 = R 2 0 0 Molecular biaxial T = sin 2 β cos 2 γ = 8 3 R 2 0 2 Phase biaxial U= sin 2 β cos 2 α = 8 3 R 2 2 0
Landau Theory Phase / molecular biaxial V = 1 2 1 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
Landau Theory D W, Allender, M A Lee, N Hafiz, Mol. Cryst. Liq. Cryst., 1985, 124, 45-52 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...
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.
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
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
Strategy of Katriel et al. J Katriel, G F Kventsel, G R Luckhurst, T J Sluckin, Liq. Cryst., 1986, 1, 337-355 (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!
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
Strategy of Katriel et al. A=k B T f β ln 2 f β sin βdβ 1 u 200 2 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
Result A= 5 2 k B T T P 2 2 2 5 2 1 k B T P 2 3 425 196 k B T P 2 4... T =u 200 /5 k B
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
Simulation Rod-Disc Mixtures Rod-Disc Dimers Flexibility
The phase diagram Mixtures of rods and discs P Palffy-Muhoray, J R Bruyn, D A Dunmur, J.Chem.Phys., 1985, 82, 5294-5295
Rod Disc Dimers I D Fletcher, G R Luckhurst, Liq.Cryst., 1995, 18, 175-183 J J Hunt, R W Date, B A Timimi, G R Luckhurst, D W Bruce, J.Am.Chem.Soc., 2001, 123, 10115-10116 P H J Kouwer, G H Mehl., Mol.Cryst.Liq.Cryst., 2003, 397, 301-316
The Phase Behaviour of Rod-Disc Dimers M A Bates, G R Luckhurst, PCCP, 2005, 7, 2821-2829 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
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
Rigid Rod-Disc Dimer Phase diagram as function of relative anisotropy ε DD /ε RR T * I N u N u N B
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
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
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)