Hidden Symmetry in Disordered Matter

Transcription

1 MAX-PLANCK-INSTITUTE FOR METALS RESEARCH Department LDMM Hidden Symmetry in Disordered Matter P. Wochner T. Demmer V. Bugaev A. Díaz Ortiz H. Dosch C. Gutt T. Autenrieth A. Duri G. Grübel DESY F. Zontone ESRF 4th European XFEL User Meeting, DESY Hamburg, January 27, 2010

2 Outline Motivation Higher Order Correlation Functions and XCCA Proof of Principle Experiment Results Conclusions and Outlook

3 Motivation Liquids and amorphous systems still among the oldest and least understood problems in cond. mat. physics Structure: - only pair-correlations - no directional information Dynamics: - no directional information - time-averaged, long wavelength collective behaviour - ultra-fast, local structural changes of interatomic distances Similar situation: solutions, nano-powders Theoretically: - Glass transition: freezing of density fluctuations g2(r) - dynamical heterogeneities and correlation length treated as fluctuation of g2(r): g4(r) (Parisi, Franz, Donati, Glotzer)

4 Motivation Most mysterious substance worldwide: H 2 O Local order: Tetrahedral vs. rings and chains Ph. Wernet et al., Science (2004) Y. Zubavicus, M. Grunze, Science (2004) T. Head-Gordon, M.E. Johnson: Tetrahedral structure or chains for liquid water, PNAS (2006) C. Huang et al.; "The Inhomogeneous Structure of Water at AmbientConditions", PNAS (2009)

5 N-Point Correlation Function n 0 g 2 (r)dr probability to find particle 2 at distance r from 1 in dr g 2 (r 1,r 2 ) = n 0 2 N N 1 i j i δ(r 1 R i )δ(r 2 R j ) 1 r 2 g2(r) independent of bond angles

6 N-Point Correlation Function n 0 g 2 (r)dr probability to find particle 2 at distance r from 1 in dr g 2 (r 1,r 2 ) = n 0 2 N N 1 i j i δ(r 1 R i )δ(r 2 R j ) g2(r) independent of bond angles 1 r3 " r2 2 analogous: 3-point and n-point distribution function g3(r1, r2, r3) - but depend on angles 3 n 0 g 3 (r 1,r 2,r 3 )dr 3 = (N 2)g 2 (r 1,r 2 ) icosahedra ---- dodecahedra fcc... N-2 different arrangements with same g2(r) Schenk et al., PRL 89, (02)

7 Structure Determination of Disordered Systems Traditional approach: S(Q) = ρ(r)e iq r dr 2 Ensemble or configuration (and time) average <...> g2(r) 2-point (pair) distribution function S(Q) = 1+ (g 2 (r) 1)e iq r dr 2 g 2 (r, r ) = n 0 Major breakthrough: beyond 2-point correlation functions ( ρ(r)ρ( r ) δ(r) )

8 Higher Order Correlation functions Eliminate intrinsic spatial and temporal averaging Coherence Snap shot Construct new correlation function by hand Speckle intensity Speckle width I(Q,t) = ΔQ λ / D b iq r s e ( ) ρ(r,t)ρ(s,t)drds (D b beam size) Intensity-Intensity correlation function (appropriate average) C(Q, Q,t, t ) = I(Q,t)I( Q, t ) iq (r s) i = e Q ( r s ) ρ 4 (r,s,t, r, s, t ) drdsd r d s 4 (r) 4-point correlation function ρ 4 (r,s,t, r, s, t ) = ρ(r,t)ρ(s,t)ρ( r, t )ρ( s, t ) = f (g 2,g 3,g 4 )... to be defined

9 X-ray Cross-Correlation Analysis XCCA... for local orientational correlations (instantaneous t = t ): CQ (Δ) = < I(Q, ϕ )I(Q, ϕ + Δ) >ϕ < I(Q, ϕ ) >ϕ2 < I(Q, ϕ ) >ϕ2 for medium range orientational correlations: Q Q time dependent: t t P. Wochner et al., PNAS 106, (2009)

10 Proof of Principle: Colloidal Glass Beamline ID10A, ESRF" Energy 8.03 kev" Vertical focusing by CRL " Aperture: 10 µm Flux : 3.6e9 ph/s at 56 ma Coherent fraction ~ 30% CCD camera, 22 µm pixel size hard sphere PMMA glass (117 nm) Speckle noise "

11 Experimental Results Fast hard sphere PMMA system (117 nm) Temporal auto-correlation function f (Q,Δt) = < I(Q,t)I(Q,t + Δt) > t < I(Q) > 2 t Structure factor <S(Q)> "

12 Typical angular dependence of CQ( ) Fast hard sphere PMMA system (117 nm) Icosahedral Cluster

13 Time dependence of CQ( ) in fast systems Fast hard sphere PMMA system (117 nm): dynamical heterogeneity Temporal auto-correlation function f (Q,Δt) = < I(Q,t)I(Q,t + Δt) > t < I(Q) > 2 t

14 Time dependence of CQ( ) in fast systems Fast hard sphere PMMA system (117 nm): dynamical heterogeneity H.Shintani, H. Tanaka, Nature Physics 2, 200 (2006)

15 Very simplified Interpretation Single icosahedral cluster - Intensity in Qx-Qy plane Wanted: < I(ϕ)I(ϕ + Δ) > ϕ

16 X-ray cross correlation analysis Single-molecule diffraction

17 X-ray cross correlation analysis Partially coherent diffraction

18 X-ray cross correlation analysis Coherent diffraction Speckle-Size ~ 1 / Beam-Size = Volume of coherently illuminated sample

19 X-ray cross correlation analysis Δ = 2/5 π Δ = 2/5 π

20 X-ray cross correlation analysis Δ = 2/5 π Δ = 2/5 π

21 X-ray cross correlation analysis Δ = 2/5 π Δ = 2/5 π

22 XCCA-averaging Considerations Illuminated volume: - 10 µm x 10 µm x 800 µm ~ 6 x 10 6 PMMA particles max Icosahedra XCCA symmetries: only subset of n-fold axes in beam direction contribute Analogy Powder Diffraction: Angular average selects subset of states lying on Debye-Scherrer Cone

23 Numerical Simulation 8000 random icosahedral cluster on a lattice

24 Numerical Simulation 8000 random icosahedral cluster on a lattice

25 Fresnel Approximation Fresnel Density: add imaginary phase factor ρ F (r) = ρ(r)exp i Ω 2 2 k (r + r 2 ) 0 L 1 L 2 ; Ω = 1+ Δλ / λ

26 MD Simulations CQ( ) with Fraunhofer approximation Q=16.43nm -1 Q=13.14nm -1 Q=18.07nm -1 Q=15.35nm -1 Q=18.44nm -1 Q=18.79nm -1 Q=13.50nm -1 Mono-atomic glass: Dzugutov potential atoms in MD-simulation liquid H2O: particles SPC

27 Origin of Symmetries Hypothesis: Icosahedral clusters (LFS) form factor expansion: in icosahedral harmonics and orthogonal rotator functions e.g. icosahedron: l=0, l=6, l=10, l=12... ρ i (Q) = 4πf sphere (Q) i l g l j l (QR) S lγ γ (Ω Q )U,τ l (ω i ) Conclusion: l,τ Q-Range - form factor gl can select dominant Q-ranges for special symmetry - medium-range correlation length will also influence the Q-dependence γ Angular Symmetry

28 Summary and Outlook XCCA with XFEL will revolutionize studies of liquids (H2O): XCCA with single lasershots (100 fs) XCCA opens a new world for structural analysis of disordered systems Glasses transient complex molecular solutions and reactions in solutions nano-powders Sophisticated Cross-correlators CQ,Q (,t): time-dependent mid-range orientational correlations Q-space Formalism (mode-coupling): Interaction potentials

29 END Thanks to to A. Schofield for samples Thank you for your attention