Structure Refinement of Neutron and X-ray Diffraction Data on Glassy Systems for Liquid and Glassy Systems. Daniel T. Bowron

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1 Structure Refinement of Neutron and X-ray Diffraction Data on Glassy Systems for Liquid and Glassy Systems Daniel T. Bowron ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 QX, United Kingdom

2 Outline (1)Brief summary of diffraction experiments ()Neutrons and X-rays and the structure of silica glass. (3)Checking the model EXAFS and a partial structure view. (4)The structure of a aqueous electrolyte solution (5)Fine tuning an atomic potential (6)From local structure to the mesoscale (7)Conclusions: New possibilities for the structure refinement of disordered systems

3 Schematic of a neutron or X-ray scattering experiment Q=4π/λ sinθ θ Q(Å -1 ) λ F D O (Q)

4 The neutron diffraction experiment Total Structure Factor Atomic concentrations and scattering lengths F( Q) = c c αβ α β b α b β S αβ ( Q) Partial Structure Factors S αβ ( Q) 1 = 4πρ r sin Qr Qr [ () ] ( ) g r 1 dr αβ Atomic density Partial Pair Distribution Functions

5 The X-ray diffraction experiment Total Structure Factor Atomic concentrations and scattering form factors F( Q) = c c αβ α β f α ( Q) f ( Q) S ( Q) β αβ Partial Structure Factors S αβ ( Q) 1 = 4πρ e r sin Qr Qr [ () ] ( ) g r 1 dr αβ Electronic density Partial Pair Distribution Functions

6 F F SiO SiO The structure of silica glass: an under-determined problem [ S ( Q) 1] + c c b b [ S ( Q) 1] + c b [ S ( ) 1] ( Q) = c b Q Si Si SiSi Si O Si O SiO O O OO By neutrons: No isotopic variation in scattering length of silicon or oxygen By X-rays: Silicon and oxygen absorption edges at too low an energy to make feasible anomalous X-ray scattering techniques By neutrons and X-rays ( )[ ( ) ] ( ) ( )[ ( ) ] Q S Q 1 + c c f Q f Q S Q 1 + c f ( Q) [ S ( ) 1] ( Q) = c f Q Si Si SiSi Si O Si O SiO O O OO

7 The measured structure factors for silica Neutron X-ray F(Q). F(Q) Q (Å -1 ) Q (Å -1 )

8 The neutron and X-ray total radial distribution functions for silica Neutron X-ray 3 Si-O 1.6Å 3 Si-O 1.6Å O-O.6Å g(r) Si-Si 3.1Å? g(r) O-O.6Å Si-Si 3.1Å r (Å) r (Å)

. 1.8 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1.4 1. 1..8.6.4.. 4 6 8 1 r(å). 1.8 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1. 1..8.6.4.. 4 6 8 1 r(å) 3..5. 1.5 1..5. 4 6 8 1 r(å) 1. 1..8.6.4.. 4 6 8 1 r(å) 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1.8 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 1. 1..8.6.4.. 4 6 8 1 r(å) 1. 1..8.6.4.. 4 6 8 1 r(å) 1. 1..8.6.4.. 4 6 8 1 r(å) 1.6 1.4 1. 1..8.6.4.. 4 6 8 1 r(å) 16 14 1 1 8 6 4 4 6 8 1 r(å) 1.

5 3..5. 1.5 1..5. 4 6 8 1 r(å) 5 4 3 1 4 6 8 1 r(å) 6 5 4 3 1 4 6 8 1 r(å) Empirical Potential Structure Refinement INPUT EPSR OUTPUT TBA Methyl subs HH S(Q) 14 1 1 8 6 4 - -4 5 Q(Å -1 ) 1 15 1 8 6

9 r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) r(å) Empirical Potential Structure Refinement INPUT EPSR OUTPUT TBA Methyl subs HH S(Q) Q(Å -1 ) Q(Å -1 ) Q(Å -1 ) 1 15 TBA Methyl subs XX S(Q) TBA Methyl subs XH S(Q) g C-C (r) g CC-H (r) g CC-C (r) g CC-M (r) g CC-OW (r) g C-M (r) g CC-O (r) g CC-HW (r) g C-O (r) g M-OW (r) g M-M (r) g C-H (r) g g H-H (r) g O-H (r) OW-OW (r) g C-OW (r) g M-O (r) g M-HW (r) g O-OW (r) g H-OW (r) g OW-HW (r) g C-HW (r) g M-H (r) g O-O (r) g O-HW (r) g H-HW (r) g HW-HW (r) Constraints such as: Density Molecular geometry U N αβ = U U O αβ g N αβ αβ g + kt ln g O r U r αβ D αβ ( ) ( ) αβ D ( r) g ( r) αβ () r () r g CC-CC (r) r(å) A.K.Soper Chem Phys p.95 (1996)

10 Silica simulation box 1 Si atoms O atoms Box side: 35.6Å Density:.664 ats/å 3

11 EPSR model fits to the neutron and X-ray data for silica SiO Neutron S(Q) and EPSR model.8 Experimental Data.6 EPSR model.4 Fit residual Q (Å -1 ) SiO X-ray S(Q) and EPSR model estimate X-ray Data (Mozzi and Warren 1969) EPSR Model Estimate Q (Å -1 )

12 EXAFS measurements: One route to partial structure information Storage Ring Bending Magnet Optics Hutch Experimental Hutch sample Fixed exit Double crystal monochromator I Ion chamber It Ion chamber I t = I e -μ(e)x Where x: thickness of the sample μ: absorption coefficient EXAFS μ(e)x = ln (I /I t ) μ(e) = μ (E) { 1 + χ(k) }

13 X-ray Absorption Spectroscopy E Continuum p p 3/ p 1/ s L III L II L I Absorption L III L II L I K hν 1s K Energy Permitted transitions Selection rules Δl= ± 1

14 EXAFS.3 Absorption (a.u.) χ(k) E (kev) Information about: k(1/å) Type of neighbors Number of neighbors Radial distribution around the absorbing atom Disorder. IFTI. R(Å) 4 6

15 The Gaussian shell model for EXAFS analysis χ ( k) = j N kr ( k ) ' ( k, π ) sin kr + δ ( k) + φ ( k) j R j λ j e f j j j ( ) l j e j σ k. R 1 R 3 Convenient for peak fitting analysis based on a (significant) number of free parameters g(r) σ 1 N 1 N N σ r (Å) R σ 3 R j = Distance of photo-absorber to shell j N j = Number of atoms in shell j exp(-σ j k ) = Debye-Waller disorder for shell j f j (k,π) = Backscattering amplitude of shell j φ j (k) = Backscattering phase function of shell j δ l (k) = Photo-absorber phase function λ j = Mean-free path term for inelastic losses Works well for crystalline solids and well defined molecular structures

16 Breakdown of the EXAFS shell model Liquids and disordered materials are best characterised by the radial distribution function g(r) 15 component glass g(r) g(r) =.g AA (r) +.8g AB (r) 1 g(r) r (Å)

17 Breakdown of the EXAFS shell model Liquids and disordered materials are best characterised by the radial distribution function g(r) 15 component glass g(r) g(r) =.g AA (r) +.8g AB (r) 1 g(r) r (Å)

18 Breakdown of the EXAFS shell model Liquids and disordered materials are best characterised by the radial distribution function g(r) 3 g AA (r) + g AB (r) g AB (r) g AA (r) g(r) r (Å)

19 Reformulation of the EXAFS equation as a configurational average χ χ ( ( k) = γ ) (, i) i γ () is the EXAFS signal associated with a single neighbour atom at a distance r from the photo-absorber. ( ) () ( k = dr4π r ρg r γ ) ( r, k ) χ(k) is the ensemble average of the pair-wise atomic configurations characterised by the radial pair distribution function, g(r), centred on the photo-absorber. This function intrinsically incorporates the static and dynamic disorder in the local environment. A.Filipponi, J. Phys. Condens. Matter, 13, R3 (1)

20 Reformulation of the EXAFS equation χ ( ) () ( k = dr4π r ρg r γ ) ( r, k ) γ () contains the chemically specific phase and backscattering amplitude information Phase term Backscattering amplitude ( r, k) = A( k, r) sin( kr ( k, r) ) () γ +φ Atom, energy and distance dependent loss term ( k, r) f A ( k, r) = exp λ, kr ( r ( k r) ) A.Filipponi, J. Phys. Condens. Matter, 6, 8415 (1994)

21 χ For a fixed atom configuration around a photoabsorber, the EXAFS can be written as: ( ) () ( ) k dr r g r ( r k ) dr dr d r r ( ) g ( r r ) ( 3 = π ρ γ, φ8π sin φ ρ,, φ γ ) ( r, r, φ, k ) + Complementarity between Diffraction and EXAFS A diffraction experiment probes the pair correlation function g(r): χ S ( Q) = 1+ ( g () r 1) r sin( Qr)dr ( ( ) ) ( 3 k γ (, i) + γ ) (, i, j) +... = i To apply this to a real system we need to ensemble average to include structural and dynamic disorder. This is achieved through the inclusion of the pair and higher order correlation functions: 4πρ Q i, j A.Filipponi, J. Phys. Condens. Matter, 13, R3 (1)

EXAFS from an EPSR model Step 1: select a photoabsorbing atom within the simulation box and place at origin Step : identify neighbouring atom coordinates within a radius of 6Å (cluster for potential,

22 EXAFS from an EPSR model Step 1: select a photoabsorbing atom within the simulation box and place at origin Step : identify neighbouring atom coordinates within a radius of 6Å (cluster for potential, phase shift and scattering path calculations) Step 3: Calculate theoretical EXAFS signal using no Debye-waller broadening (e.g. FEFF 8) and all significant paths Step 4: repeat for all photoabsorbing atoms in the box, and average signals

23 EXAFS from an EPSR model Step 1: select a photoabsorbing atom within the simulation box and place at origin Step : identify neighbouring atom coordinates within a radius of 6Å (cluster for potential, phase shift and scattering path calculations) Step 3: Calculate theoretical EXAFS signal using no Debye-waller broadening (e.g. FEFF 8) and all significant paths Step 4: repeat for all photoabsorbing atoms in the box, and average signals

24 EXAFS from an EPSR model Step 1: select a photoabsorbing atom within the simulation box and place at origin Step : identify neighbouring atom coordinates within a radius of 6Å (cluster for potential, phase shift and scattering path calculations) Step 3: Calculate theoretical EXAFS signal using no Debye-waller broadening (e.g. FEFF 8) and all significant paths Step 4: repeat for all photoabsorbing atoms in the box, and average signals

25 Si and O K-edge EXAFS calculated from EPSR refinement of SiO glass 1.5 Experimental Data EPSR Model Estimate SiO EXAFS kχ(k) SiO EXAFS at O K-Edge Stohr et al. Phys. Rev. B 664 (1979) SiO EXAFS at Si K-Edge O-O and O-Si correlations Si-Si and Si-O correlations Greaves et al. Nature (1981) k (Å -1 )

26 EPSR derived partial pair distribution functions Si-Si Si-O O-O r (Å) r (Å) EPSR estimated gsi-si (r) EPSR estimated gsi-o (r) EPSR estimated go-o (r) r (Å)

27 Silicon-silicon pair distribution function 5 Si-Si g(r) Si-Si 3.1Å r (Å) EPSR estimated gsi-si (r) Running Coordination (r) (atoms) NSi-Si

28 Silicon-oxygen pair distribution function Si-O g(r) Si-O 1.6Å r (Å) EPSR estimated gsi-o (r) Running Coordination NSi-O (r) (atoms)

29 Oxygen-oxygen pair distribution function 8 O-O g(r) O-O.61Å r (Å) EPSR estimated go-o (r) Running Coordination (r) (atoms) NO-O

30 Intermediate Range Order in Pure Silica Glass 1..5 F(Q) Å -1 4.Å Q (Å -1 )

31 F SiO Intermediate Range Order in Pure Silica Glass from Faber-Ziman to Bhatia-Thornton Structure Factors [ S ( Q) 1] + c c b b [ S ( Q) 1] + c b [ S ( ) 1] ( Q) = c b Q Si Si SiSi Si O Si O SiO O O OO S NN (Q) reflects correlations between atomic sites irregardless of chemical species S cc (Q) reflects correlations between chemical species and S Nc (Q) represents the cross correlations between number density and chemical species S S S ( ) ( ) Q = c S Q + c S ( Q) + c c S ( Q) NN Si SiSi O OO Nc ( Q) = c c [ c ( S ( Q) S ( Q) ) c ( S ( Q) S ( Q) )] Si O Si SiSi SiO ( Q) = c c [ 1+ c c ( S ( Q) + S ( Q) S ( Q) )] Si O O SiO cc Si O Si O SiSi OO OO SiO SiO A.B.Bhatia and D.E.Thornton, Phys. Rev. B, 34, (197)

32 Intermediate Range Order in Pure Silica Glass 1..5 F(Q) Q (Å -1 )

33 Intermediate Range Order in Pure Silica Glass SiO Faber-Ziman Partial [S(Q)-1] S O-O (Q) S Si-O (Q) S Si-Si (Q) Q (Å -1 )

34 Intermediate Range Order in Pure Silica Glass Intermediate range order correlates with fluctuations general atomic density i.e. it is not correlated with a specific chemical component (Si or O) SiO Bhatia-Thornton Partial S(Q) SiO S NN (Q) Q (Å -1 ) SiO S cc (Q) SiO S Nc (Q)

35 Bulk solution structure of 1m YCl 3 in water Correlation Neutron Weights D O solution Rel. Weight % Y-Y Y-Cl Y-OW Y-HW Cl-Cl Cl-OW Cl-HW OW-OW OW-HW HW-HW H/D substitution to probe the solvent structure D.T.Bowron and S.Diaz-Moreno, J.Phys.Chem B 111, (7)

36 EPSR refinement of 1m YCl 3 in water 1.5 H O 1. Experimental Data EPSR simulation 1. f(r) D O.5 F(Q).5 HDO r (Å). D O Q (Å -1 )

37 EPSR derived pair distribution functions for pure water and water in a 1m YCl 3 in water 6 5 HW-HW OW-OW Range Å Pure Water 4.3 ±.1 1m YCl 3 solution 4.5 ±.1 g(r) 4 3 OW-HW OW-HW ± ±.1 HW-HW ±.1 5. ±.1 1 OW-OW r (Å)

38 The spatial density function of water, and water in a 1m YCl 3 solution

39 Local structure in 1m aqueous solution of YCl 3 and refining an atomic potential EXAFS spectroscopy to investigate the Y 3+ local structure

40 Local structure in 1m aqueous solution of YCl 3 and refining an atomic potential EXAFS spectroscopy to investigate the Y 3+ local structure

41 Y 3+ -OW interaction potential used in the structure refinement U () r = 4ε σ rij 1 σ rij αβ αβ αβ αβ + 6 q α q 4πε r β ij αβ 1 ( εαε β ) 1 ε = σ = ( σ + σ ) αβ α β 1. σ αβ =3.133Å ε αβ =.85kJmol -1-5 U Lennard-Jones (r) [kj/mol].5. U Coulomb (r) [kj/mol] Lennard-Jones potential Y - OW r (Å) -45 Coulomb potential Y 3+ - OW r(å)

42 k χ(k) Y 3+ -OW EXAFS optimised interaction potential Y 3+ -OW =.33Å k(å -1 ) Y 3+ -OW =.39Å Model : σ=3.1å Model 1: σ=3.5å Y 1s3d-5p4d shake-up U LJ+Coulomb (r) [kj/mol] Only one free parameter in fit! ΔE The EXAFS energy scale offset (In this study ΔE = 4eV) Model 1: σ = 3.5Å Model : σ = 3.1Å r (Å) Experimental data: S.Díaz-Moreno, A.Muñoz-Paez and J.Chaboy, J. Phys. Chem. A, ()

43 Local structure of the Y 3+ hydration shell Y 3+ -OW =.33Å Y 3+ -OW mean C.N. 7.4± g Y-OW (r) N Y-OW (r) P(N) r (Å) N (O water ) Well defined second hydration shell at 4.6Å

44 Local geometry of the Y 3+ hydration shell 7º.4 OW-Y-OW angle 14º.3 P(θ) θ Square antiprism

45 Ion-ion interactions in the Y 3+ hydration shell The model tells us that Cl - anions can be found in the first hydration shell 5 g Y-OW (r) g Y-HW (r) g Y-Cl (r) g(r) The mean coordination number of Cl - around Y 3+ is.8± r (Å) Cl - can also be found in the second hydration shell of Y 3+

46 Mesoscale structure in.4 mole fraction tert. butanol water solutions SANS Intensity Q (Å -1 ) Temperature ( C) D.T.Bowron and J.L.Finney, J.Phys.Chem B (7)

47 Mesoscale structure with atomistic resolution F(Q) and fit 5 C (CD 3 ) 3 COD/(CD 3 ) 3 COH in D O/H O (CD 3 ) 3 COH in H O (CD 3 ) 3 COD/(CH 3 ) 3 COH in D O/H O (CH 3 ) 3 COH in H O (CD 3 ) 3 COD in D O -.5 Data Fit Q (Å -1 )

48 Mesoscale structure with atomistic resolution

49 5ºC Mesoscale structure with atomistic resolution

50 8ºC Mesoscale structure with atomistic resolution

51 16ºC Mesoscale structure with atomistic resolution

52 Alcohol-alcohol interactions 5ºC 8ºC 16ºC

53 Alcohol-water interactions 5ºC 8ºC 16ºC

54 Isotopes and absorption edges routes to chemical specificity H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Lr Rf Db Sg Bh Hs Mt La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Elements with isotopes that potentially can be used for NDIS ion solvation studies (Δb c 1fm) J.E.Enderby, Chem. Soc. Revs. 159 (1995)

55 Isotopes and absorption edges routes to chemical specificity H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Lr Rf Db Sg Bh Hs Mt La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Elements with absorption edges suitable for atomic resolution Anomalous X-ray Scattering investigation of local structure in solutions

56 Isotopes and absorption edges routes to chemical specificity H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Lr Rf Db Sg Bh Hs Mt La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Elements with absorption edges suitable for EXAFS investigation of local structure in solutions Concentration limits: Diffraction 1 to 1 atom % Spectroscopy.1 atom %

57 Benefits of the neutron (or X-ray) scattering plus EXAFS, EPSR analysis approach (1) Produces a model that is consistent with both bulk and local structural information. () Brings dilute component sensitivity of the spectroscopic probe to bulk structural models. (3) Circumvents the traditional limitations of direct EXAFS analysis of disordered materials data: (i) No need for peak shape approximations (ii) No need for Debye-Waller factor models (iii) Reduces the number of free parameters in the model to 1 (iv) Provides an unambiguous means to incorporate higher order correlation functions into a disordered materials analysis sensitivity to the local atomic environmental geometry

58 Benefits of the neutron scattering plus EXAFS, EPSR analysis approach (4) Allows pair potential models to be investigated and refined (5) Is cheaper, more sensitive and more versatile than neutron scattering with exotic isotopic substitution methods for the investigation of ionic species (6) Experimentally much simpler and more sensitive than Anomalous X-ray Scattering techniques If the scattering and spectroscopy data exist, the method can be applied.

Solutions and Ions. Pure Substances

Solutions and Ions. Pure Substances Class #4 Solutions and Ions CHEM 107 L.S. Brown Texas A&M University Pure Substances Pure substance: described completely by a single chemical formula Fixed composition 1 Mixtures Combination of 2 or more