Orientation and Flow in Liquid Crystals

Transcription

1 Orientation and Flow in Liquid Crystals Michael P Allen H H Wills Physics Laboratory, Royal Fort, Tyndall Avenue, Bristol and Department of Physics, University of Warwick, Coventry

2 Outline Introduction Nematic Elasticity Nematic Hydrodynamics

3 Liquid Crystals Timeline discovery Reinitzer, Lehmann characterization Friedel twist cells, Helfrich room-t nematics, Gray discotics, Chandrasekhar Elastic Theory Hydrodynamics Simulations Oseen Frank Frenkel Onsager de Gennes Ericksen Leslie TJ Sluckin, DA Dunmur, H Stegemeyer. Crystals that Flow: Classic papers from the history of liquid crystals. Taylor and Francis (2004). DA Dunmur, TJ Sluckin. Soap, Science and Flat-Screen TVs: A history of liquid crystals. OUP (2011).

4 Liquid Crystal Simulations Molecular Models N Calamitic and Discotic Mesogens C O O O O O O

5 Liquid Crystal Simulations Coarse-Grained Models Discs, Rods, Ellipsoids

6 Liquid Crystal Simulations Interaction Potentials The Gay-Berne Potential φgb(r) r JG Gay, BJ Berne. J Chem Phys, 74, 3316 (1981).

7 Liquid Crystals Orientational Order Short-ranged positional order. Long-ranged orientational order. The director n: a unit vector. Second-rank order: n n. Magnitude of ordering: S. Global reorientation of n costs zero free energy. Quasi-conserved: influences hydrodynamics. Molecular simulation: orientational elasticity; effect on nematic flow.

8 Outline Introduction Nematic Elasticity Nematic Hydrodynamics

9 Orientational Elasticity Spatial deformations of director n incur free energy penalty. Elastic Free Energy F = dr f [ n(r) ] f[n] = 1K ( ) n + 1 }{{} K ( ) n n + 1 }{{} K 2 3 n ( n ) 2 }{{} splay twist bend Description valid at long wavelengths. Elastic constants K 1, K 2, K 3. Fourier transform: ik.

10 Director Fluctuation Modes Splay-Bend and Twist-Bend e 3 k n 1 n n2 splay bend k 3 e 1 e 2 k 1 twist k 3 bend n = (0, 0, 1) δn = ( n 1, n 2, 0 ) k = (k 1, 0, k 3 )

11 Director Fluctuation Modes Free Energy in Fourier Space F = 1 f(k) V ) f(k) = 2( 1 K1 k K 3 k 2 ñ1 3 (k) 2 ) + 2( 1 K2 k K 3 k 2 ñ2 3 (k) 2 }{{}}{{} splay-bend twist-bend k Equipartition of (free) energy: Splay-bend: W 13 ñ 1 (k) 2 1 K1 k K 3k 2 3 Twist-bend: W 23 ñ 2 (k) 2 1 K2 k K 3k 2 3

12 Inverse Orientational Fluctuations N = Molecules A Humpert, MP Allen. Elastic constants and dynamics in nematic liquid crystals. Molec. Phys., DOI: / (2015).

13 Colloidal Platelets Elastic Constant Measurements D van der Beek, P Davidson, HH Wensink, GJ Vroege, HNW Lekkerkerker. Phys Rev E, 77, (2008). I-N Phase Separation Reprinted with permission from Phys Rev E, 77, (2008). (2008) American Physical Society. Alignment dictated by container walls. Magnetic field: Freedericksz transition. Sterically stabilized gibbsite Al(OH) 3 platelets (hexagons). Diameter D 230nm. Thickness H 18nm. H/D 1/13. 20% polydispersity in both dimensions. Gives bend elastic constant K 3 = N.

14 Colloidal Platelets Elastic Constant Measurements AA Verhoeff, RHJ Otten, P van der Schoot, HNW Lekkerkerker. J Phys Chem B, 113, 3704 (2009). Tactoids in Magnetic Field Tactoids droplets. Low magnetic field: radial director structure, hedgehog, splay. High magnetic field: director structure deforms, tactoid elongates. Reprinted with permission from J Phys Chem B, 113, 3704 (2009). (2009) American Chemical Society. Theory balances surface, elastic, and magnetic forces. Gives splay elastic constant K 1 = N.

15 Colloidal Platelets Elastic Constant Simulations Experiment, Theory and Simulation 100 H/D = 0 H/D = 1/20 H/D = 1/15 H/D = 1/10 twist KD/kBT 10 splay bend S S S S PA O Brien, MP Allen, DL Cheung, M Dennison, AJ Masters. Phys Rev E, 78, (2008). PA O Brien, MP Allen, DL Cheung, M Dennison, AJ Masters. Soft Matter, 7, 153 (2010).

16 Outline Introduction Nematic Elasticity Nematic Hydrodynamics

17 Nematodynamics Theory JL Ericksen. Arch. Rat. Mech. Anal., 4, 231 (1960). FM Leslie. Arch. Rat. Mech. Anal., 28, 265 (1968). O Parodi. J. de Physique, 31, 581 (1970). D Forster, TC Lubensky, PC Martin, J Swift, PS Pershan. Phys. Rev. Lett., 26, 1016 (1971). The director is effectively conserved on large length scales. Coupled director ñ(k, t) and velocity ṽ(k, t) fields. Two Goldstone modes associated with director motion.

18 Nematic Hydrodynamics Balance Equations ρ v i = τ ji,j ρ n i = g i + π ji,j i, j = x, y, z f,j f r j We use the Einstein convention. Incompressible fluid. ρ = mass density, ρ = director moment of inertia density. v i (r, t) = velocity field, n i (r, t) = director field. τ ji = stress tensor, π ji = couple stress tensor. g i = intrinsic director body force.

19 Nematic Hydrodynamics Reactive Coefficients Elastic Effects τ R ji = Pδ ji f n k,j n k,i g R i = γn i β j n i,j f n i π R ji = β j n i + f n i,j f = Frank free energy density. P, β j and γ are constants. δ ij is the Kronecker delta.

20 Nematic Hydrodynamics Dissipative Coefficients Viscous Effects τ D ji = α 1 n k n m d km n i n j + α 2 n j N i + α 3 n i N j + α 4 d ji + α 5 n j n k d ki + α 6 n i n k d kj g D i = γ 1 N i γ 2 n j d ji π D ji = 0 α 1... α 6 = Leslie coefficients (viscosities) γ 1 = α 3 α 2, γ 2 = α 6 α 5 = α 3 + α 2. N i = ṅ i w ij n j = co-rotational time flux of the director ) d ij = 2( 1 vi,j + v j,i = symmetric strain rate ) w ij = 2( 1 vi,j v j,i = antisymmetric strain rate, vorticity

21 Nematic Hydrodynamics reactive coefficients τ R ji, π R ji, g R i dissipative coefficients τ D ji, g D i balance equations ρ v i = τ ji,j ρ n i = g i + π ji,j hydrodynamics Drop ρ n i term; ṅ i still appears through N i = ṅ i w ij n j. Adopt same axis system as before. Set n = (0, 0, 1) + δn; drop nonlinear terms in δn. Use incompressibility: v = 0, k ṽ = k 1 ṽ 1 + k 3 ṽ 3 = 0. Define k = k ( sin ϕ, 0, cos ϕ ) and let ṽ = ṽ 1 cos ϕ ṽ 3 sin ϕ.

22 Nematic Hydrodynamics Splay-Bend Fluctuation Dynamics ( γ1 t + Kk2) ñ 1 + ikα ṽ = 0 ikα t ñ1 + ( ρ t + ηk2) ṽ = 0 K = K 1 sin 2 ϕ + K 3 cos 2 ϕ α = α 2 cos 2 ϕ α 3 sin 2 ϕ η = η 1 sin 2 ϕ + η 2 cos 2 ϕ + η 12 sin 2 ϕ cos 2 ϕ

23 Nematic Hydrodynamics Twist-Bend Fluctuation Dynamics ( γ1 t + Kk2) ñ 2 + ikα ṽ 2 = 0 ikα t ñ2 + ( ρ t + ηk2) ṽ 2 = 0 K = K 2 sin 2 ϕ + K 3 cos 2 ϕ α = α 2 cos ϕ η = η 3 sin 2 ϕ + η 2 cos 2 ϕ

24 Linear Response Theory Consider perturbation coupling to A(r, p). Measure response in B(r, p). Define B such that B = 0. The Response Function Hamiltonian Perturbation: H = φ(t)a(r, p) Response: B(t) A = t dt χ BA (t t )φ(t ) Response function: χ BA (t) = 0 for t < 0 Nonequilibrium response: B(t) A. Equilibrium time correlation: C AB (t) AB(t).

25 Nonequilibrium Response Response and Correlation χ BA (t) = 1 k B T AḂ(t) = 1 k B T ĊAB(t), for t > 0 impulsive perturbation: φ(t) = constant δ(t) B(t) χ A BA(t) Ċ AB (t) { constant (t 0) preparation and relaxation: φ(t) = 0 (t > 0) B(t) dt χ A BA (t ) C AB (t) t

26 Nematodynamics Experiment Director fluctuations, and time correlation functions or spectra, are investigated by light scattering experiments. PG de Gennes, J Prost. The Physics of Liquid Crystals. (Clarendon, 1995). For each mode the power spectrum (or frequency distribution) has the form of one single Lorentzian, centred on the incident beam frequency. This means that the behaviour of the fluctuations is purely viscous (no oscillations). MJ Stephen, JP Straley. Rev. Mod. Phys, 46, 617 (1974). The orientation fluctuations of the director are coupled to the fluid velocity by viscous effects, and in fact are overdamped: the modes which the elastic theory... predicts do not propagate.

27 Twist Dynamics ñ ṽ n k k Director and Velocity Fluctuations Hydrodynamic Equations ( ) γ 1 t + K 2 k 2 ñ = 0 (ρ ) t + η 3 k 2 ṽ = 0 TWIST

28 Twist Dynamics Director and Velocity Fluctuations ñ TWIST n k ṽ k Hydrodynamic Equations ( ) γ 1 t + K 2 k 2 ñ = 0 (ρ ) t + η 3 k 2 ṽ = 0 no coupling director rotation elastic constant ñ e (K 2k 2 /γ 1 )t

29 Twist Dynamics Director and Velocity Fluctuations ñ TWIST n k ṽ k Hydrodynamic Equations ( ) γ 1 t + K 2 k 2 ñ = 0 (ρ ) t + η 3 k 2 ṽ = 0 no coupling density shear viscosity ṽ e (η 3k 2 /ρ)t

30 Twist Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) larger k ṽ(k, t) ṽ( k, 0) larger k t t

31 Twist Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) ṽ(k, t) ṽ( k, 0) k 2 t k 2 t

32 Splay Dynamics n ṽ ñ k k Director and Velocity Fluctuations Hydrodynamic Equations SPLAY ( ) γ 1 t + K 1k 2 ñ + ikα 3 ṽ = 0 ( ikα 3 t ñ + ρ t + η 1k )ṽ 2 = 0

33 Splay Dynamics Director and Velocity Fluctuations n ñ SPLAY k ṽ k Hydrodynamic Equations ( ) γ 1 t + K 1k 2 ñ + ikα 3 ṽ = 0 ( ikα 3 t ñ + ρ t + η 1k )ṽ 2 = 0 weak coupling

34 Splay Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) larger k ṽ(k, t) ṽ( k, 0) larger k t t

35 Splay Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) ṽ(k, t) ṽ( k, 0) k 2 t k 2 t

36 Bend Dynamics ñ ṽ n k k Director and Velocity Fluctuations Hydrodynamic Equations BEND ( ) γ 1 t + K 3k 2 ñ + ikα 2 ṽ = 0 ( ikα 2 t ñ + ρ t + η 2k )ṽ 2 = 0

37 Bend Dynamics Director and Velocity Fluctuations ñ n k BEND ṽ k Hydrodynamic Equations ( ) γ 1 t + K 3k 2 ñ + ikα 2 ṽ = 0 ( ikα 2 t ñ + ρ t + η 2k )ṽ 2 = 0 strong coupling

38 Bend Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) larger k ṽ(k, t) ṽ( k, 0) larger k t t

39 Bend Dynamics Simulation Results Director and Velocity Correlation Functions ñ(k, t) ñ( k, 0) ṽ(k, t) ṽ( k, 0) k 2 t k 2 t

40 PG de Gennes Argument Orders of Magnitude ρ 10 3 kg m 3 η α γ Pa s (K/γ 1 )k 2 (η/ρ)k 2 K N µ = ρk γ 1 η densities viscosities elastic constants director decay rate velocity decay rate time scale separation

41 Solving the Hydrodynamics Secular Equations for Bend Fluctuations } ñ(k, t) e λk2 t ṽ(k, t) t λk2 λ 2 ργ 1 + λ ( ) α 2 2 γ 1 η }{{ 2 ρk } 3 + K3 η }{{} 2 = 0 small or large? small Define µ = ρk 3 γ 1 η 2, ξ = 1 α2 2 γ 1 η 2, ξ 2 4µ complex λ

42 What Did We Learn? Simulations agree with hydrodynamics. µ = ρk 3 is small (de Gennes), typically µ γ 1 η 2 α 2 2 γ 1η 2 (de Gennes again!). ξ = 1 α2 2 also small, typically ξ 0.2. γ 1 η 2 { Propagating modes if ξ 2 µ 10 4, ξ µ µ 10 2, ξ 0.2 Experimentally observable for low viscosity liquid crystals? A Humpert, MP Allen. Propagating director bend fluctuations in nematic liquid crystals. Phys. Rev. Lett., 114, (2015). A Humpert, MP Allen. Elastic constants and dynamics in nematic liquid crystals. Molec. Phys., DOI: / (2015).

43 Conclusions liquid crystalline properties molecular structure occasional surprises require a rethink insight from statistical mechanical theories stimulate new experiments

44 Conclusions liquid crystalline properties molecular structure simulations can be useful! occasional surprises require a rethink insight from statistical mechanical theories stimulate new experiments

45 Acknowledgements PhD students: I Anja Humpert I Paul O Brien Collaborators: I David Cheung (Galway) I Andrew Masters (Manchester) Funding: I EPSRC

46 Acknowledgements PhD students: I Anja Humpert I Paul O Brien Collaborators: I David Cheung (Galway) I Andrew Masters (Manchester) Funding: I EPSRC AND THANK YOU FOR YOUR ATTENTION!