Structural characterization. Part 2

Transcription

1 Structural characterization Part

2 Determining partial pair distribution functions X-ray absorption spectroscopy (XAS). Atoms of different elements have absorption edges at different energies. Structure from interference pattern of scattered electron waves from neighboring atoms. Neutron scattering using different isotopes Different isotopes have different scattering lengths b i Measurements on three samples of different isotopic composition all three partial structure factors can be obtained

3 Extended X-ray Absorption Fine Structure EXAFS is an element specific technique Probes the local structure atound each atom type X-ray absorption spectrum is measured close to an X-ray absorption edge of a particular element Pre-edge region. Absorption edge: Steep increase in X-ray absorption coefficient, µ(e) Post-edge region: Decreasing µ(e) with small oscillations An X-ray photon is absorbed by an atom A photoelectron is ejected and backscattered by neighboring atoms

4 X-ray absorption experiments Performed at beam line of a synchrotron Transmission geometry Source: Aksenov et al. 006

5 Absorption process Direct detection Fluorescence detection Auger electrons Source: Rehr and Albers 000 Source: Aksenov et al. 006

6 Schematic picture of EXAFS Interference pattern extends eV from the edge Absorption of X-rays Photoelectron ejected from central atom Electron waves scattered from neighboring atoms interfere Source: Elliott

7 Theoretical interpretation

8 Theory, continued EXAFS equation can be generalized to represent contributions from N R multiple scattering contributions of path length R. Electrons lose energy as they travel in the material mean free path Limited range of tens of Å in EXAFS measurements Peaks close to nearest and nextnearest neighbor distances After the first peak multiple scattering contributions are of increasing importance Should be taken into account in fitting to experimental data A Fourier transform of χ(k) gives an effective reduced radial distribution function Source: Ravel 005

9 Analysis of EXAFS data Obtain the EXAFS function χ(k) and its Fourier transform χ(r) from fits to the experimental spectrum. Generate a simulated model structure consistent with the experimental data Reverse Monte-Carlo modelling and fitting to data Molecular dynamics simulations of amorphous structure for comparisons Model structures further analyzed What do we learn? Partial pair distribution functions Interatomic spacings for nearest and next nearest neighbors, maybe further out Average coordination numbers Coordination distributions Bond angle distributions Mean square deviations σ (from Debye-Waller factor)

10 Reverse Monte-Carlo Modeling Choose interatomic potential Minimize the energy Minimize difference between experimental data and simulation by varying the atomic configuration Combine data from X-ray, neutron, EXAFS. Gives optimized structural model that is consistent with experiments Not necessarily the true structure Shows important structural features of the material Wide range of applications: Liquids, glasses, polymers, crystals, magnetic materials

11 Example: Amorphous TiO Source: Carlos Triana

12 Neutron diffraction λ i, k r i fixed λ f, k r f θ 3. Detector. Sample Scan as a function of θ

13 Isotope substitution For a two component system the total structure factor S(Q) is made up of 3 different partial structure factors S ij (Q) (~ scattering amplitude) 1 S 11 S S 1 ( S ( Q) 1) + x b ( S ( Q) 1) + x x b b ( S ( ) 1) S( Q) = x b Q 1 S ij (Q) - partial structure factors g ij (r) - partial pair distribution functions

14 Determining partial structure factors Different isotopes have different scattering lengths b i Measurements on three samples of different isotopic composition all three partial structure factors can be obtained Inversion to partial pair distribution functions or partial radial distribution functions Alternative: Combine ordinary and magnetic neutron scattering with X-ray scattering

15 Ex: Amorphous alloy Ni 81 B 19 G ij (r)=4πrn 0 (g,ij (r)-1) Source: Elliott: Physics of Amorphous materials

16 Reverse Monte-Carlo Modeling Choose interatomic potential Minimize the energy Minimize difference between experimental data and simulation by varying the atomic configuration Combine data from X-ray, neutron, EXAFS. Gives optimized structural model that is consistent with experiments Not necessarily the true structure Shows important structural features of the material Wide range of applications: Liquids, glasses, polymers, crystals, magnetic materials

17 Ex: Glassy (AgI) x (AgPO 3 ) 1-x FSDP F N (Q) F X (Q) Neutron X-ray Q/Å Q/Å -1 0 F Ag (Q) F I (Q) 0 - Ag K EXAFS Q/Å I L III EXAFS Q/Å -1 (From R. McGreevy)

18 (AgI) x (AgPO 3 ) 1-x x=0 x=0 AgI pushes apart phosphate chains -> FSDP x=0.5 x=0.5 Ag+I P+O (From R. McGreevy)

19 Small angle scattering The study of structures on larger length scales Composites, particle aggregates Porous materials X-rays, neutrons, light SAXS, SANS, SALS

20 Scattering angle Crystalline materials Bragg s law: Scattering vector Q ~ d -1, where d is interplanar distance Q has dimension [m -1 ], hence large Q (large scattering angles) corresponds to small length scales At large Q we can resolve atomic distances Small Q larger length scales With small scattering angles (small Q) we can study clustering on the nano-scale.

21 Small angle scattering At higher scattering angles/q-values we get scattering from each atom. I 1 r m n m, n iq( rm n ) ( Q) = f f e V At small angles/q-values we have low resolution for individual atoms but see clusters of atoms in a volume V with scattering length density ρ(r). I 1 V iq( rm rn ) ( Q) = ρ( r ) ρ( r ) e V m n dr m dr n

22 Two-component material Consider as particles in a matrix Define the scattering contrast by ρ(r)-ρ 0 Particle form factor Intensity per unit volume f p ( ) ( ) i r r e Q Q = ρ ρ ) dr I( Q) = V p N V ( 0 p ( Q) S( Q) Structure factor S(Q) (assume isotropic particles) f p

23 Spherical particles Define Spheres of radius R asymptotic expressions Radius of gyration often used for other shapes as well as for aggregates Radius of gyration R g =3R /5 for spheres S is the surface area ) ( ) ( Q f Q P p = 1 ) ( ) ( 9 ) ( ) ( 1 5) / (1 ) ( ) ( >> = = << = QR QR V Q S Q P QR R Q V Q P ρ ρ ρ ρ π ρ ρ

24 Pair distribution function Number density n p =N p /V Relations between S(Q) and particle pair distribution function analogous to those for atomic systems Isotropic materials S( Q) g ( r) = 1+ 4πn p 3 = 1+ (8π n r p ) ( g ( r) ) 1 4πQ 1 ((sin Qr) / Qr) dr ( S( Q) 1)((sin Qr) / Qr) dq

25 Experimental techniques Limits: λ(nm) Q(Å -1 ) r (nm) Light (SALS) X-rays (SAXS) Neutrons (SANS) Complementary techniques

26 Small angle neutron scattering Source: Per Zetterberg

27 Limiting expressions Low Q: Guinier approx. High Q: Porod approx. Source: J. Teixeira in On Growth and Form P( Q) = V ( ρ ρ ) P( Q) = π ( ρ ρ ) 0 0 exp( Q S / Q 4 R g / 3) QR QR g g << 1 >> 1 S is the total surface area Influenced by particle shape, size distributions: average of R g Aggregation: correlation length ξ. QR Porod approx. compared to P(Q) for a sphere

28 Fractal surfaces Smooth surface: S~r Fractal surface: S~r D s Porod: P(Q)~(Qr) /Q 6 Fractal surface: Lignite coal D s =.5 P( Q) V ( ρ ρ ) 0 / Q 6 D s Slope between 3 and 4 Proportionality constant is a function of D s. Bale and Schmidt, PRL 53 (1984) 586

29 Volume fractals Pair distribution function g (r)-1~r D f-3 Structure factor ( g ( r) ) S Q) = 1+ 4πn p r 1 ((sin Qr) / Qr) dr ( S(Q)~1 at large Q and I(Q)=n p P(Q) Smaller Q: Fractal region D f 1 S( Q) ~ r ((sin Qr) / Qr) dr S( Q) ~ Q D f Small Q: Guinier type law with correlation length ξ instead of R g y D f sin y dy ~ Q D f

30 Gold colloidal aggregates Model for g (r) SAXS exp. vs model Slope between 1 and 3: Volume fractal D f ~ Source: P. Dimon et al, PRL 57 (1985) 598

31 Examples of porous materials Rocks, sandstones Clays Soils Coals Cement Cellulose, cotton Biomolecules, protein aggregates Food Some porous materials are built up of connected fractal aggregates Fractal surfaces are often present also in cases where the solid is non-fractal Examples of these two cases

32 Volume fractals: Silica aerogel Extremely porous continuous SiO solid network strucutre Combination of light and X-ray scattering data D f =.1 Smooth surfaces Source: Schaefer et al, 1984

33 Greige Cotton SAXS data Guinier type cutoff at low Q D f =.13 Different kinds of cotton have values in the range.1 to.7 Aggregation of cellulose microcrystals Q(nm -1 ) Source: Lin et al, ACS Symp. Ser. 340 (1987) 33

34 Surface fractals: Sandstones Sedimentary rocks Structure and properties interesting for oil industry Toy sandstones : sand, crushed glass Example shows fractal surfaces in sandstones and shales. Small angle neutron scattering (SANS) Source: Po-zen Wong, Phys. Today 41 (1988)

35 Cement: A complex case Calcium-silicate-hydrate (CSH) aggregates Volume fractal D ~ 1.8 to.7 depending on C/S and preparation Ordinary Portland cement during hydration Seems surface fractal Source: Adenot et al. C.R. Acad. Sci. II, 317 (1993) 185. Source: Häussler et al. Phys. Scr. 50 (1994 )10.

36 Local porosity analysis Sintered glass beads Diameter 50 µm Works for both fractal and nonfractal structures! Source: R. Hilfer, Transport and relaxation phenomena in porous media

37 Example: Berea sandstone Local density function for different cell size L Local percolation probabilities for different L

38 Other techniques Nitrogen and water adsorption isotherms Mercury porosimetry Pore size distributions X-ray microtomography for porous structures