SANDrA - FRILLS Models

Transcription

1 This document details the SANS data fitting models available within the FRILLS package in the SANDrA program available on the VMS computer system at ISIS. The models very much reflect the scientific interests of the author and are therefore not designed to be comprehensive. If you are seeking alternative, or perhaps more sophisticated implementations of some of these models, you are strongly encouraged to investigate the FISH program described elsewhere. Though many of these models are relatively straightforward to program, or simulate in a spreadsheet, there is also a nice Java-based online simulation package at Whilst every reasonable care has been exercised in the coding of SANDrA and the models, the author cannot accept responsibility for any loss, damage or other consequences arising from their use. Steve King ISIS September 005 NOTES: The names in brackets in the Model field are the subroutine name (eg, Debye, and the FORTRAN module name (eg, DEBYE.FOR. a Correlation length (in inverse Q dimensions unless otherwise stated B Background intensity (cm - C Constant I(Q Intensity ; the differential scattering cross-section (cm - N Number concentration of scatterers (cm - V Volume of one scatterer (cm δ (Bulk density (g/cm except in NMR models where it is a time! φ Volume fraction (V ρ Scattering length density (cm - unless otherwise stated ρ Contrast ; ( ρ i - ρ j ξ Screening length (in inverse Q dimensions unless otherwise stated s.m.king@rl.ac.uk 5/0/007

2 No. Model Parameters Equations Remarks & References Debye Gaussian Coil Q=0 h Y (Debye Z-average Rg Y h (DEBYE.FOR Background Form incorporating a Schultz polydispersity function. I Q N V ( =..( ρ. 4 Polymer Mw/Mn Y Y h Set M w /M n = for monodispersity. Debye P J Phys Coll Chem (947, 5, 8 Monodisperse Spheres (Sphere (SPHERES.FOR Monodisperse Core with Single Shell (Core_Shell (CSHELL.FOR Constant Radius Background Constant Rho of core Rho of shell 4 Rho of bulk 5 Shell thickness 6 Core radius 7 Background Y = ( QR g ( h and h [ Mw Mn ] = ( / ( QR QRcos( QR ( QR sin(. 6π =. N. P( Q 9 6 [( ρ ρ. R. P( R ] P( Q = shell medium shell shell Note no S(Q! Jacrot, B Rep Prog Phys (976, 9, 9-95 Note no S(Q! Markovic, I; Ottewill, RH; Cebula, DJ; Field, I; Marsh, JF Coll Polym Sci, (984, 6, [ ( ρ ρ ( ρ ρ R R PQR (, PQ (, R ] shell medium core shell core shell core shell 6 [( ρ ρ. R. P( R ] core shell core core sin( P ( Ri = ( QRi QRcos( QRi ( QR i s.m.king@rl.ac.uk 5/0/007

3 4 Polydisperse Core with Single Shell (PolyCore_Shell (POLYSHELL.FOR 5 C x (Q^n C x (Q^m ackground (PolyFit (QTON.FOR 6 I0 term for Second Moment of Adsorbed Layer (Simple (I0_Term (SIGMA.FOR 7 I0 term for Second Moment of Adsorbed Layer (Complex (New_I0_Term (SIGMA.FOR 8 Debye-Bueche Randomly-distributed Two-phase System (DebyeB (DB.FOR Constant Rho of core Rho of shell 4 Rho of bulk 5 Shell thickness 6 Core radius 7 Background 8 % Deviation on core Constant Exponent Constant 4 Exponent 5 Background Constant Second Moment Background Calibration factor Density Poly [g/cm] Contrast Term [/cm] 4 Core radius [Angs] 5 Vol fract/n of cores 6 Ads/d amount [mg/m] 7 Second moment [Angs] 8 Background Debye-Bueche scaling Correlation length Background See Model n I( Q = C. Q C. Q m ( Q B C I 0 ( Q. exp σ Q I 0 ( Q ( ρ l ρm 6π φ p exp Q δ Rp ( Q σ Γ I Q = ( I(0. ( Q a I( 0 = 8. π. φ.( φ.( ρ. a Note no S(Q! Uses a zeroth-order log-normal particle size distribution. NOT VERY ROBUST! Using SANS to study adsorbed layers in colloidal dispersions King, SM; Griffiths, PC; Cosgrove, T Chapter 4 in Applications of Neutrons in Soft Condensed Matter, Gabrys, BJ (editor Gordon & Breach, (000 Using SANS to study adsorbed layers in colloidal dispersions King, SM; Griffiths, PC; Cosgrove, T Chapter 4 in Applications of Neutrons in Soft Condensed Matter, Gabrys, BJ (editor Gordon & Breach, (000 Debye, P; Bueche, AM J Appl Phys (949, 0 (June, Debye, P; Anderson, HR; Brumberger, H J Appl Phys (957, 8(6, s.m.king@rl.ac.uk 5/0/007

4 9 Lorentzian Kolberstein-type D-B Excess Scattering (Lorentz (DBXS.FOR Lorentzian Scaling Excess Scaling Short correl. Length 4 Long correl. Length 5 Background f. I (0. ( Q a = f. I (0. ( Q a ( Koberstein; Picot; Benoit Polymer (985, 6, Benguigui; Boue Eur Phys J (999,, Debye-Bueche Marr-type Exponential Excess Scattering (LorentzB (DBXS.FOR Debye-Bueche Scaling Excess Scaling Short correl. Length 4 Long correl. Length 5 Background I 0 8. π. φ.( φ.( ρ. = f. I(0. ( Q a ( a =??? Q a ( f. I(0.exp( 4 Wignall; Farrar; Morris J Mater Sci (990, 5, Marr Macromol (995, 8, I( 0 a = 8. π. φ.( φ.( ρ. Moritani; Inoue; Motegi; Kawai; Macromol (970, (4, 4-44 Lorentzian for Semi-dilute Solutions (SemiConc (LORENTZ.FOR Benoit Gaussian Star (Benoit (BENOIT.FOR Lorentzian Scaling Screening length Background Pre-factor Number of Arms Rg 4 Background I( 0 / = π. φ.( φ.( ρ. a ξ = 8π. φ.( φ.( ρ. ( Q ξ U f ( exp( U ( exp( U 4 f U Higgins JS; Benoit HC Polymers and Neutron Scattering, Oxford Series on Neutron Scattering in Condensed Matter, Volume 8, Clarendon Press, (994 See page 80 for derivation f is the number of arms (the functionality of the star Benoit H J Polym Sci (95,, 507 Huber K; Burchard W Macromol (989,, U = f f QR gstar, s.m.king@rl.ac.uk 5/0/007

5 Cosgrove-Ito polydisperse core polydisperse shell (Maki (POLYCS.FOR Q=0 Background Polydispersity [%] 4 Peak (outer [Q] 5 Peak (inner [%] 6 Shape (outer [%] 7 Shape (inner [%] 8 Number of intervals See Model Note no S(Q! NOT FULLY TESTED! Calculates the scattering law for a core-shell type of model where the core has a logarithmic polydispersity distribution in size, and the shell has a Gaussian polydispersity distribution in thickness but which is linked to the polydispersity of the core by a proportion. 4 Stejskal-Tanner Single-D Echo Attenuation (vs beta (Stejskal (STEJSKAL.FOR Field Gradient [T/m] D [0^- m^/s] Constant A = C [( γ. G. D β ] Ito, M MSc Thesis University of Bristol, (99 For Ln(attenuation vs β PFG- NMR data Requires β in seconds 5 Karger-Henk Single-D Echo Attenuation (vs q^ (Karger (KARGER.FOR Gradient Length [ms] D [0^- m^/s] Constant β = δ ( δ / A = C [(4. π.. D Q ] For Ln(attenuation vs Q PFG- NMR data Requires that Q be computed with δ in seconds 6 Simplified Murday-Cotts Single-D Echo Attenuation (Murday (MURDAY.FOR Field Gradient [T/m] Radius [nm] D [0^- m^/s] 4 Constant Q = δ / (4. π A = C [( G. γ. R / D δ ] For Ln(attenuation vs δ PFG- NMR data Assumes >δ and that R<0 µm After Pryamitsyn, V. These models are for use with Pulsed Field-Gradient NMR data only! Calpin-Davies, SR, PhD Thesis University of Bristol, (998 s.m.king@rl.ac.uk 5/0/007

6 7 Two component Random Phase Approximation (Rpa (RPA.FOR Contrast term [/cm] Volume fraction A Density A [g/cm] 4 Density B [g/cm] 5 Mol wt A [g/mol] 6 Mol wt B [g/mol] 7 Mw/Mn A 8 Mw/Mn B 9 Z-average Rg A 0 Z-average Rg B Effective chi Background ( ρ P( R = M N A. δ. φ. P( R g, N. δ χ eff M. φ. P( R ( v v A g, h Y Y h = Y Y h g, i B de Gennes PG Scaling Concepts In Polymer Physics, nd edition Cornell University Press, (985 8 Mildner-Hall Surface Fractal (Fractal_Ds (DS.FOR Prefactor Term P(q particle radius Surface Fract. Dim. 4 Cut-off Length 5 Background ( QRg, i Y = and h = [( Mw / Mn ] ( h. P( R. S( Q ( QR QRcos( QR ( QR sin( P ( R = (5 D ( ( = [( Γ(5. s Ds S Q D s ξ.[ ( Q ξ ] 5 / Note that 0 < D s 6. Mildner; Hall, J Phys D Appl Phys (986, 9, See equation ( Triolo et al J Appl Cryst (000,, See equation ( sin[( D s 5.arctan( Qξ ] / Q] s.m.king@rl.ac.uk 5/0/007

7 9 Bale-Schmidt Surface Fractal (Fractal_Ds (DS.FOR 0 Mildner-Hall (Schaefer-Keefer Mass Fractal (Fractal_Dm (DM.FOR Schmidt (Hurd-Schaefer-Martin Mass & Surface Fractal (Fractal_QtoN (QTON.FOR Prefactor Term P(q particle radius Surface Fract. Dim. 4 Background Prefactor Term P(q particle radius Mass Fractal Dimens. 4 Cut-off Length 5 Background Prefactor Term Mass Fractal Dimens. Cluster Rg 4 Surface Fract. Dim. 5 Primary Rg 6 Background. P( R. S( Q ( QR QRcos( QR ( QR sin( P ( R = ( Ds 6 S( Q = Q. Γ(5 Ds.sin[( Ds. π / ]. P( R. S( Q ( QR QRcos( QR ( QR sin( P ( R = ( D ( ( = [( Γ(. m Dms S Q D m ξ.[ ( Q ξ ] sin[( D m /.arctan( Qξ ] / Q]. P( Q Dm P ( Q = {[ ( Q. a] / Note that 0 < D s 6. Mildner; Hall J Phys D Appl Phys (986, 9, See equation ( Bale; Schmidt Phys Rev Lett (984, 5, 596 See equation (8 Note that 0 < D m 6. Mildner; Hall, J Phys D Appl Phys (986, 9, See equation (9 Triolo et al J Appl Cryst (000,, See equation (4 Schmidt J Appl Cryst (99, 4, See equation (8 Note that 0 < D s 6 and 0 < D m 6. Schmidt J Appl Cryst (99, 4, See equation (9 [ a = R g /(. D m (6 D m (. ] s D Q b / / } Hurd; Schaefer; Martin, Phys Rev A (987, 5, 6-64 See equation ( b = r /[.( D 6 D g s m / ] s.m.king@rl.ac.uk 5/0/007

8 Shibayama-Geissler Two-Length Scale Fit for Gels (GelFit (GEISSLER.FOR Lorenztian Scaling Guinier Scaling Short correl. Length 4 Radius of Gyration 5 Scaling Exponent 6 Background I f. I(0. ( [(( D /. Q a ( Q = D / ] ( f. I(0.exp( Q a Sibayama; Tanaka; Han J Chem Phys (99, 97(9, Mallam; Horkay; Hecht; Rennie; Geissler, Macromol (99, 4, 54 D is the scaling exponent a R g Note that this reduces to: = f. I(0. ( f. I (0.exp( Q a ( Q a Guinier Approximation for Spheres (Guinier (GUINIER.FOR Constant Radius Background when D=; ie, when the Flory exponent is 0.5 (theta conditions Q R.exp 5 Note no S(Q! Guinier, A C R Hebd Séance Acad Sci Paris (97, 04, 5 s.m.king@rl.ac.uk 5/0/007

9 4 Chen Fractal Fit for Aggregates (Chen (CHEN.FOR Chen Fractal Scaling Primary Radius [Ang] Aggregation Number 4 Polyd of Cluster,Tau 5 Fractal Dimension Df 6 Background ( ρ = fractal ρ medium. Φ. V Γ( τ [ F ( τ, Qξ ( Q ξ Qξ G( τ, Qξ h D f primary D ( τ / ] f. N agg Note that a small Polydispersity index corresponds to a broad size distribution. Scaling is programmed as: ( ρ.φ Chen, Rouch & Tartaglia Croat. Chem. Acta, 65(, (99, 5-66 Liu, Sheu, Chen & Storm Fuel, 74(9, (995, 5-56 Where V primary = ( 4 /. π. R 0 Fratini, Bonini, Oasmaa, Solantausta, Teixeira & Baglioni Langmuir,, (006, 06- R = ( R / 5. 0 ξ = h (/ D f. R. N agg h = D f ( D f 6 F( a, x = Γ( a Γ( a, u u = h. ( Q Q ξ ξ D f / ( D G( a, x = sin f D f x Γ a,. π. h ( D f s.m.king@rl.ac.uk 5/0/007

10 5 Beaucage Polymeric Mass Fractal Fit (Beaucage (BEAUCAGE.FOR Guinier Scaling, G Rg [Ang] Fractal Dimension Df 4 Kuhn scaling, Gs 5 Kuhn size, Rs [Ang] 6 Mer-unit scaling, Bs 7 Mer-unit exponent,ps 8 Background = G exp( q / B exp R g * P ( q R sub / ( / q G exp( q / s R s Beaucage J. Appl. Cryst., 9, (996, 4-46 B * P ( q s / Bkgd s s Generalised Zimm Function (Zimm (ZIMM.FOR Prefactor Term Fractal Dimension Correl. length 4 Background ( Df ( Qξ I( Q NV ( ρ D f NOT CURRENTLY INCORPORATED INTO THE PROGRAM Equation suspect! s.m.king@rl.ac.uk 5/0/007