Structural characterization Part
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.
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
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
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 ) ( ) ( 4 0 4 0 0 >> = = << = QR QR V Q S Q P QR R Q V Q P ρ ρ ρ ρ π ρ ρ
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
Experimental techniques Limits: λ(nm) Q(Å -1 ) r (nm) Light 400-600 5 10-5 -3 10-3 00-10000 (SALS) X-rays 0.1-0.4 10 - -15 0.05-50 (SAXS) Neutrons 0.1-3 10-3 -15 0.05-500 (SANS) Complementary techniques
Small angle neutron scattering Source: Per Zetterberg
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
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) = CV ( ρ ρ ) 0 S / Q 6 D s Slope between 3 and 4 C is a numerical function of D s. Bale and Schmidt, PRL 53 (1984) 586
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
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
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
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
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
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)
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.
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
Example: Berea sandstone Local density function for different cell size L Local percolation probabilities for different L
Other techniques Nitrogen and water adsorption isotherms Mercury porosimetry Pore size distributions Complementary to scattering techniques: Extended X-ray absorption fine structure (EXAFS)
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: Usually featureless Absorption edge: Steep increase in X-ray absorption coefficient, α(e) Post-edge region: Decreasing α(e) with small oscillations An X-ray photon is absorbed by an atom A photoelectron is ejected and backscattered by neighboring atoms Interference between outgoing and backscattered electron waves
Schematic picture of EXAFS Coordination number Interatomic spacings for nearest and next nearest neighbors Mean square deviations
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
Ex: Glassy (AgI) x (AgPO 3 ) 1-x FSDP F N (Q) 0.10 0.05 0.00-0.05-0.10 F X (Q) 0.4 0. 0.0-0. -0.4-0.15-0.6-0.0-0.5 Neutron -0.8-1.0 X-ray - 0 4 6 8 10 1 14 16 18 0 4 6 Q/Å -1 4 0 4 6 8 10 1 14 16 Q/Å -1 0 F Ag (Q) F I (Q) 0 - Ag K EXAFS 4 6 8 10 Q/Å -1 - -4 I L III EXAFS 4 6 8 Q/Å -1 (From R. McGreevy)
(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)