Dynamics of fluctuations in smectic membranes

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1 Dynamics of fluctuations in smectic membranes Wim H. de Jeu FOM Institute AMOLF Amsterdam, The Netherlands

2 Contents 1. Introduction to smectic membranes Smectic phases Dimensionality, ordering and fluctuations Smectic membranes studied by x-ray reflectivity. Static x-ray scattering Displacement correlation function Fluctuation profiles 3. Dynamic methods: XPCS and NSE Coherent x rays X-ray photon correlation spectroscopy Neutron spin echo 4. Dynamics of smectic fluctuations Theoretical fluctuation modes Experimental results Conclusions

3 pages

4 Contents 1. Introduction to smectic membranes Smectic phases Dimensionality, ordering and fluctuations Smectic membranes studied by x-ray reflectivity. Static x-ray scattering and conformality 3. Dynamic methods: XPCS and NSE 4. Dynamics of smectic fluctuations Conclusions Issues and outlook

5 Smectic liuid crystals Smectic liuid-crystal phase: - orientationally ordered elongated molecules - stacked liuid layers - distribution centers of mass: f ( z) = n cos( n0 d n= 0 - smectic order parameters: n = cos( πn 0z 0 ) z)

6 Smectogenic molecules K 34 SmA 53.5 N 71.5 I 7AB

7 Landau-De Gennes free energy L d Bend layers: K N Compression layers: B 10 7 N/m Fluctuations destroy layer ordering for large L

8 Types of order G(r) I() constant δ ( 0) η r 1 ( ) n 0 η / e r ξ ξ ( 1 0 ) + 1

9 Caillé lineshapes Full correlation function for a smectic phase: with Asymptotic limits: and γ E = is Euler s constant E 1 (x) is the exponential integral λ = K / B is the penetration length normal to the layers parallel to the layers

10 Smectic liuid crystal layering J. Als-Nielsen et al. Phys. Rev. B, 31 (1980) Further examples: Surfactant membranes C.R. Safinya et al. Phys. Rev. Lett. 57, 718 (1986) Smectic polymers E. Nachaliel et al. Phys. Rev. A 43, 897 (1991) Block copolymers P. Štěpánek et al. Macromol. 35, 787 (00)

11 Capillary waves Width of the liuid-vapour interface σ = σ = σ σ + cw k T B ln πγ max min Short-wavelength cut-off: max = π/a Long- wavelength cut-off due to gravity: min = Δρg /γ In practice the long-wavelength cut-off is not reached, and min is determined d by the resolution of the x-ray set-up Other example: Ordering of Langmuir monolayers Berge et al., Phys. Rev. Lett. 73, 165 (1994)

12 Smectic membranes Typical sizes up to 50 mm mm For neutron work: mm but less control

13 View of smectic membrane 1. Very well oriented (mosaic < 1 up to 10 mdegree).. Centro-symmetric: no substrate! 3. From two to thousands of layers (about 5 nm to 0 μm). 4. Cross-over from bulk behaviour (3D) in thick films to surface-dominated behaviour. 5. In thin films: model for physics of D systems.

14 Experimental situation

15 X-ray reflectivity Simple interface: θ < θ c 0.15 o : total reflection θ > θ c : Fresnel fall-off θ 4 Model calculation of 30 nm membrane.

16 33-layer 7AB film N=33 Troika beamline, ESRF

17 Contents 1. Introduction to smectic membranes. Static x-ray scattering Displacement correlation function Fluctuation profiles 3. Dynamic methods: XPCS and NSE 4. Dynamics of smectic fluctuations Conclusions Issues and outlook

18 Static x-ray scattering Landau-De Gennes-Hołyst theory Competition between surface γ and bulk BK The x-ray structure factor is the Fourier transform of the density-density correlation function: 1 = exp{ i [ u( r) u(0) ]} = exp[ g( r )] G ( r ) = ( 0 0 r

19 Fluctuation profiles - Hydrodynamic fluctuations depending on - Local fluctuations ν = γ BK Calculated profiles for L =.94 nm γ = N/m K= N B = 10 9 N/m ν < N/m ν = N/m ν > 1

20 Experimental profiles N=4 N=0 N=34 Thermal lfluctuation ti profiles for FPP films of various thickness Mol, Shindler, Shalaginov, de Jeu, Phys. Rev. E 54, 536 (1996)

21 Contents 1. Introduction to smectic membranes. Static x-ray scattering 3. Dynamic methods: XPCS and NSE Coherent x-rays X-ray photon ht correlation lti spectroscopy Neutron spin echo 4. Dynamics of smectic fluctuations Conclusions Issues and outlook

22 Young s experiment Zero-order order is coherent by definition

23 Transverse coherence length ξ t s ξ tδθ = λ ξt = 1 λ Δθ As Δθ arises from different points on the source: Δθ=D/R ξ = t λ D / R

24 Longitudinal coherence length ξ l ξ l = Nλ ξ t = Nλ ξ = Nλ = ( N + 1)( λ Δλ ) l ( N + 1) Δλ = λ N N + 1 = λ / Δλ λ ξl = Δ λ

25 Fraunhofer diffraction Troïka beamline ESRF, λ= 1.05 Å. Front pinhole 3.5 μm ; back pinhole before detector 5 μm. Robinson et al., Phys. Rev. B 5, 9917 (1995)

27 Photon correlation spectroscopy Speckle reflects instantaneous position of scatterers. Motion by analyzing the intensity variation I(, t). Motion by analyzing the intensity variation I(, t). Intensity autocorrelation function: + ), ( ), ( t I t I,0) ( ), (,0) ( ), ( = I I I g g (,) g (,) can be related to 1 log() 0 ), ( ), ( ), ( ), 1 ( t S t E t E g + = via the Siegert relation [ ]. ), ( 1 ), ( 1 g g + =

29 Experimental situation Troïka beamline Third/fifth harmonic of set of three undulators. Source size: 98 3 μm (s H s V ) Mono: Pinhole: Coherence lengths: Si(111) at 8 13 kev λ at Å, Δλ/ λ μm at R = 44 m ξ t = λr/(s H ) 10 μm ξ l = λ /(Δλ/ λ) 1.6 μm At the Bragg position θ 1.5 o path length difference: L sinθ = 1.6 μm L max 30 μm

30 Storage ring Bunch structure ESRF: 99 bunches, revolution time.7 μs uniformly distributed in continuous mode:.8 ns spacing Avalanche photo diodes Time resolution: 0.7 ns risetime Baron, Hyperfine Interactions 15, 9 (000) Correlators Standard: ALV5000/E correlator, fast extension down to 1.5 ns. Correlator.com lag time 8 ns.

31 Dynamic methods Length scale [Å] INS Freu uency [Hz] ]Raman Brillouin PCS IXS Spin-Echo NFS XPCS Energy [ev V] Scattering vector [Å -1 ]

32 Neutron Spin Echo For an angular displacement (δ, φ)

33 Neutron Spin-Echo Spectrometer IN15 at the ILL (Grenoble)

34 IN15 at ILL

35 Contents 1. Introduction to smectic membranes. Static x-ray scattering and conformality 3. Dynamic methods: XPCS and NSE 4. Dynamics of smectic fluctuations ti Theoretical fluctuation modes Experimental results Conclusions Issues and outlook

36 Fundamental relaxation time Starting point: Landau-de Gennes-Holyst theory. Calculate the time-dependent correlation function C(, z, z', t) u(, z,0) u(, z', t) =. From the euation of motion involving i viscous forces and restoring elastic force: u ρ t = η t 4 u + ( B z K Δ ) u 0 3 Lowest root solution for << B /( γ L ) γ γ 1 = + 1 η3 K 3 L KB 1 For = 0 we find 1 η3l = γ

37 Exponential relaxations Thick films soft x-rays Price, Sorensen et al., PRL 8, 755 (1999)

38 4O.8 membranes 1. L=0.3 μm (N=105). L=3.1 μm (N=1430) 3. L=5 5.5μm (N=500) Sikharulidze, Dolbnya, Fera, Madsen, Ostrovskii, de Jeu, PRL 88, (00)

39 Full solution including inertia Incompressible membranes (B ): undulations only,0) ( ), ( ), ( * = u t u t C. exp exp ) (,0) ( ), ( ), ( = f s f s B t t L T k u t u t C ρ With values for the fast and slow relaxation: ) ( 0 f s f s L ρ , + L K f s γ η ρ η ρ m. 0 3 c L γ ρ η = For small critical wavelength fluctuations: Shalaginov, Romanov, Phys. Rev. E 48, 1073 (1993)

40 Dispersion curves Oscillations Exponential decay

41 FPP off-specular Sikharulidze, Dolbnya, Fera, Madsen, Ostrovskii, de Jeu, PRL 88, (00)

42 Various regimes , L K f s γ η ρ η ρ m 3 3 L η η 1 8 γ ρ ρ Small : , L f s η γ ρ η ρ m L c γ ρ η 0 3, = transition at <,c >,c 3 = η Complex mode (oscillations): * / + = K L s γ η f s = γ η 3 L s = 3 = K s η

43 Dispersion curves surface dominated complex mode bulk elasticity dominated fast mode slow mode slow mode 1 γ / = η 3 L + K

44 XPCS: 8CB

45 8CB: NSE results Fit to stretched exponential: S(, t) = t exp 0.59

46 8CB: XPCS and NSE NSE λ=0.9 nm NSE λ=1.5 nm XPCS λ=0.09 nm Sikharulidze, Farago, Dolbnya, Madsen, de Jeu, PRL 91, (003)

47 Homodyne/heterodyne detection 4O.8 membrane, thickness 3.8 μm Off-specular: = 3.3 μs homodyne detection scheme Specular: = 6. μs heterodyne detection scheme. De Jeu, Madsen, Sikharulidze, Sprunt, Physica B 357, 39 (005)

48 Crystalline-B membranes Start from the Landau-De Gennes-Hołyst theory to which solid-like elasticity is added. Easy-shear approximation: C 44 associated with shear of the layers is much smaller than the other constants: F uniax = 1 8 d 3 r C 44 [ ] u( r). The effect of C 44 is to renormalize the surface tension, 44 as can be seen from the undulation part of the free energy: F [ Δ + + ] 1 LK( u) (γ LC )( u 1 undul = dxdy 4 44 ) Hence Cr-B membranes fluctuate essentially as Sm-A ones, 1 with an effective damping by γ eff = γ + LC. Shalaginov, Sullivan, Phys. Rev. E 6, 699 (000) eff 8 44

49 Cr-B experiments in 4O o C Cr-B 50.1 o C Sm-A F D lb O ti O t kii d J Fera, Dolbnya, Optiz, Ostrovskii, de Jeu, Phys. Rev. E 63, (001)

50 Conclusions 1. Thanks to the high degree of control, smectic membranes provide excellent model systems to study low-dimensional physics.. XPCS can be done on smectic membranes at energies up to 13 kev and down to time scales of 10 ns. 3. A complex fundamental surface relaxation mode is observed, combining exponential decay with oscillatory behaviour. 4. Transitions from complex behaviour to exponential decay occur as a function of film thickness and of, in agreement with theory. 5. Combination with neutron spin echo shows a transition from surface-tension damped modes to bulk elastic ones. 6. Both heterodyne (at the specular ridge) and homodyne (off-specular) detection schemes have been realized. 7. Thin crystalline-b films fluctuate essentially in the same way as smectic-a membranes, due to the easy shear associated with C 44.

51 THANKS Liesbeth Mol Andrea Fera Irakli Sikharulidze Joe Schindler Ricarda Optitz Daniel Sentenac Boris Ostrovskii Vladimir Kaganer Arcadi Shalaginov Gerhard Grübel/Anders Madsen Igor Dolbnya Bela Farago PhD students post-docs guests ESRF ILL

52 THE END Thanks for your attention Wim H. de Jeu

Dynamics of smectic membranes as studied by X-ray and neutron scattering Sikharulidze, I.

Dynamics of smectic membranes as studied by X-ray and neutron scattering Sikharulidze, I. DOI: 10.6100/IR590907 Published: 01/01/2005 Document Version Publisher s PDF, also known as Version of Record (includes