Small Angle X-ray Scattering

Transcription

1 XII School on Synchrotron Radiation Small Angle X-ray Scattering Francesco Spinozzi Dipartimento di Scienze della Vita e dell Ambiente (DISVA) Gruppo di Biofisica Molecolare Università Politecnica delle Marche f.spinozzi@univpm.it

2 Scattering experiment Incident beam Transmitted beam X-rays will scatter on each atom of the object (target): - scatter predominantly on electron shells - elastic (= same energy) - in all directions

3 Conceptual diagram of a scattering experiment Detector Slits Transmitted X-Ray Sample SAXS X-Ray beam Beam stop WAXS (=XRD)

4 Sizes of interest = large scale structures = nm or more Mesoporous structures Biological structures (membranes, vesicles, proteins in solution) Gels, Polymers, Liquid crystals Colloids and surfactants (micelles) Magnetic films and nanoparticles Voids and Precipitates

5 Different size and shape Interacting, crowded conditions Oriented, interacting Partially ordered, interacting Not interacting, disordered

6 Scattering analysis of particles

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15 Data reduction

16 expressed in cm 1

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18 ρ(r) δρ(r) ρ 0 r

19 r e (number of electrons)

20 One particle Two particles

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22 Form factor of an oriented homogeneous ellipsoidal particle (linear scale) q y b a c q x P ell φ = sinφ φ cosφ ( q) = 3 3 φ a qx + b q y 2

23 Form factor of an oriented homogeneous ellipsoidal particle (log scale) q y b a c q x P ell φ = sinφ φ cosφ ( q) = 3 3 φ a qx + b q y 2

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25 Different dimensions - Same orientation q y q x

26 General Two-Phase System Samples: independent objects Objects are in a uniform, featureless matrix (e.g. solvent); Assume dilute enough so scattering independently; Assume randomly oriented in all directions; Objects may be proteins, micelles, vesicles in solution, polymer chains in melt/solution, inorganic nano-particles

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29 Ellipsoids with different dimensions and isotropic orientation Radial average

30 Babinet s principle The two structures generate the same scattering: I(q) ( ρ) 2 ρ 1 ρ 2 Structure 1 Structure 2

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32 R B C A H R

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34 Typical 2D SAS pattern - Take radial profile (because isotropic); - (Isotropic) scattering patterns usually show featureless decay

35 Form factor: particle size and shape Size I(q) q Shape

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39 R B C A H R

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41 Both approximations may be easily linearized by suitable plots log I ( q) q 1 I( q) q 0 0 R log I (0) 3 I 1 (0) 2 q 2 g 2 Rg q Guinier Zimm

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45 POLIDISPERSITY EFFECT on the scattering intensity Polydispersed spheres: η=1%, <R>=60 Å

46 POLYDISPERSITY EFFECT on the gyration radius For a polydispersion the radius of gyration is a weighed average wich largely overestimates the contribution of the larger particles 2 R g = R R 8 6 p p dis dis ( R ) dr ( R) dr

47 POLYDISPERSITY EFFECT on the gyration radius R = app R g

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55 q

56 Two groups of particles with distinct dimensions GUINIER POROD intensity a.u. R G =8.2 R G =2.1 slope q 2 h 2 (nm -2 ) h (nm -1 ) q

57 Porod s Invariant Q = 0 I( q) q 2 dq = 2 π 2 η (1 η )( η volume fraction of the red component ρ I 1 (q) I 2 (q) Q 1 = Q 2 ) 2

58 Measure of the specific surface 0 π I( q) q I( q) q 2 4 dq q constant = S V p p S 1 > V 1 S V

59 log(i(q)) q -4 log(q)

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61 Pd cuboctahedric cluster: Average diameter: 5 nm 11 shells; 5083 atoms 1.2 I(q) (a.u a.u.).) I ( q ) = r N N sin( q r ij ) 2 e f i ( q ) f j ( q ) i= 1 j= 1 q rij qh=4πsin(θ)/λ (nm -1 )

62 Pd cuboctahedric cluster: Average diameter: 5 nm 11 shells; 5083 atoms θ = 1.8 (λ=0.154 nm) intensità I(q) (a.u a.u.).) I( q) = r N N sin( ij ) 2 e fi ( q) f j ( q) i= 1 j= 1 q rij q r s q [Å [nm -1 ] -1 ]

63 With low resolution the scattering of the cubeoctaedric cluster is very similar to that of a sphere with radius 2.6 nm which contains the same number of electrons 1000 I(q) intensity ty cuboctahedron cubeoctaeder sphere qh[nm -1 ]

64 In logarithmic scale sphere SFERA CLUSTER cluster inte I(q) ensità (a.u a.u.).) 1E q s [nm [ Å -1 ] -1 ]

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66 Diluted solutions (in one dimension, for simplicity) Real space ρ(x) P(q) Reciprocal space convolution g(x) S(Q) S(q) q convolution theorem = q ρ(x) g(x) I(Q) I(q) q

67 Concentrated solutions (in one dimension, for simplicity) Real space ρ(x) P(q) Reciprocal space q g(x) S(q) q ρ(x) g(x) I(q) q

68 Diffraction from a crystal (in one dimension, for simplicity) Real space Reciprocal space ρ(x) P(q) q g(x) S(q) q ρ(x) g(x) (110) unit cell I(q) (100) (010) q

69 Reciprocity of the Fourier Transform (2D) FT FT FT

70 Radial average (1D) FT rad FT rad FT rad

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75 Hard sphere interaction

76 Many terms interacting potential u(r jk ) -3kT 2R Hard Sphere 2.4R Coulomb Repulsion Q = 20, c salt =0.01M Square Wall Attraction S(q=0) = kt( N/ π) Osmotic compressibility Models for S(q) Structur re Factor q (Å -1 ) R= 50 Å Conc. = 5% 0.20 S(0)>1 more compressible S(0)<1 less compressible

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81 Polarizing glasses S. Polizzi et al, J. Appl. Cryst., 30, 487 (1997); J. Non-Cryst. Solids , 147 (1998) The starting material is photochromic glass containing Ag(Cl,Br) crystallites Heating to 725 C Ag(Cl,Br) droplets Drawing at T>Tsoft cigar-like Ag(Cl,Br) particles Reduction at 430 C in H2 cigar-like Ag particles

82 POLARIZING GLASSES 200nm

83 ULTRA-SAXS (HASYLAB-DESY): Sample-detector distance: 12 m ; λ=0.124nm q

84 2 4 2 I( q) = np ( ρ) D( a, b) πab Pell ( q)dadb 3 2 q y q x q x

85 a b = = η ( R) R η R ( R) R b b a stretching

86 ]} ) / ( exp[ {1 1 ) ( )d ) (, ) (, ( 3 4 ) D( ) ( ) ( lim lim ell m p R R R R R R R R P R R n I + = = η η η η π ρ q q

87 0.014 D(2a) Length distribution length (2a) [nm] Width distribution D(2b) width(2b) [nm]

88 Aggregation of colloidal systems: SULPHATE ZIRCONIA SOL-GEL

89 Fractality An object is called fractal when it shows a scale-invariance in a particular length range Mass fractal R M R D f Df =1, 2, 3 for Euclidean objects 1 Df<3 for fractal objects M=Object Mass R= Object Radius

90 Surface fractal The fractal dimension of a surface Ds comes out to be S R D s R S=Surface Ds=2 for non fractal regular surfaces 2<Ds<3 for fractal surfaces

91 Such trends translate in the reciprocal space so that the small angle scattering fractal dimensions are obtained by I(q) q -D f I(q) q D s-6

92 Silica/siloxane Composite material with different fractal regions

93 Aggregation models Simulated structures resulting from various kinetic growth models Each cluster contains 1000 primary particles. D is the associated fractal dimension

94 Aggregation n 0 identical colloidal particles (monomers) with volume V o I( q) I(0) = = n V o n V o 2 o 2 o ( ρ ) 2 P o ( q) ( ) 2 ρ P ( q) = 1 I( q) I(0) (0) o n =n 0 /k aggregates of k monomers P agg = = n k o kn V = 1 ( kv o 2 o 0 ) 2 ( ρ ) ( ρ ) 2 2 P k agg ( q) mass

95 Aggregation Thus one can measure the aggreates mass I(0) and dimension R g without any assumption on their structure I(0) M R D f Measuring the scattering as a function of time, it is possible to calculate D f and thus determine the growth mechanism

96 Time-resolved measurements of the scattering intensity of the sample open sol q

97 DIFFUSION LIMITED CLUSTER AGGREGATION I(0) 1 slope 1.78(6) 1.78 M R g 0.1 slope 0.98(6) M R g 1 10 R G (nm) P.Riello et al. J.Phys.Chem. 107, 15 (2003) 3390

98 (from PDB), SASMOL

99 QUAFIT The structure of the protein Spinozzi & Beltramini, BJ, 2012 assembly is described through a hierarchical sequence of intermediate oligomeric species, formed by the optimum arrangement of an asymmetric unit (the monomer).

100 (Program: SASHA, MPOLE)

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104 Ortore, Spinozzi et al., Phys. Rev. E, 2011

105 High-pressure SAXS Pressure-assisted cold denaturation of metmyoglobin SAXS data & fitting by GENFIT N N K I K U I U P-T phase diagram (Spinozzi et al., JPC B, 2007)

106 The composition of lysozyme solvation shell is determined by the co-solvent The SAS curves obtained at different conditions (c and ϕ) have been analysed by GENFIT considering an exchange water-cosolvent equilibrium The tetrameric model for CPSso carboxypeptidase, predicted by fold recognition method, is confirmed by SAXS particle shape reconstruction by MPOLE The aggregation state of octopus vulgaris hemocyanin strongly depends on the buffer conditions (SAXS analysis by QUAFIT from functional unit PDB structure) (Spinozzi and Beltramini, Biophys. J, 2012; Spinozzi et al. PLOS one, 2012) hierarchical aggregation from monomers to decamers loose monomer in equilibrium with compact monomers Scaled representations of a solvated lysozyme molecule based on PDB structure:, water in the bulk and in the first solvation layer., glycerol in the bulk in contact with the protein. y x y z z x 50 Å

107 Acknowledgements Politecnic University of Marche - DiSVA, Biophysical Research Group Paolo Mariani (full professor) Maria Grazia Ortore (researcher) Flavio Carsughi (researcher) Enrico Baldassarri (PhD student) Silvia Moscatelli (PhD student) -R. Casadio, University of Bologna -C. M. Bergamini, University of Ferrara -S. F. Funari, DESY -L. Q. Amaral, R. Itri, L. Barbosa, E. M. Sales University of Sao Paulo -A. Paciaroni, University of Perugia -D. Gazzillo, A. Giacometti, University of Venezia -S. Bernstorff, H. Amenitsch, ELETTRA -M. Beltramini, L. Bubacco, I. Micetic, A. Gonnelli University of Padova -C. Ferrero, T. Narayanan, A. Antolinos, ESRF -D. Russo, ILL, Grenoble -M. Maccarini, IBS, Grenoble -C. La Mesa, P. Andreozzi, University of Rome -F. Blasi, C. Bruckmann, IFOM