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Small Angle Neutron Scattering – SANS Mu-Ping Nieh CNBC 2009 Summer School (June 16)

Small Angle Neutron Scattering – SANS Mu-Ping Nieh CNBC 2009 Summer School (June 16). CNBC Summer School, June 16, 2009. Outline . Common Features of SANS Scattering Principle SANS Instrument & Data Reduction Contrast Dilute Particulate Systems – Form Factor Data analysis

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Small Angle Neutron Scattering – SANS Mu-Ping Nieh CNBC 2009 Summer School (June 16)

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  1. Small Angle Neutron Scattering – SANS Mu-Ping Nieh CNBC 2009 Summer School (June 16)

  2. CNBC Summer School, June 16, 2009 Outline • Common Features of SANS • Scattering Principle • SANS Instrument & Data Reduction • Contrast • Dilute Particulate Systems – Form Factor • Data analysis • Examples • Concentrated Particulate Systems – Structure Factor • Data analysis • Examples • Dilute Polymer Solutions • Analysis for Gaussian Chain – Debye Function • More Generalized Analysis – Zimm Plot • Small Angle Diffraction Analysis • Model Independent Scaling Method Analysis • Summary

  3. CNBC Summer School, June 16, 2009 Common Features & Advantages of SANS • The q range of SANS : between 0.002 and 0.6 Å-1 • – achievable long wavelengths of neutrons and low detecting angles are important • A powerful technique for in-situ study on the global structures of isotropic samples • Easy to play contrast with isotope substitution • Data analysis: • (A) Model dependent: well-defined particulate systems (possible non-unique solutions) • (B) Model independent: general feature of the structure

  4. CNBC Summer School, June 16, 2009 d2s d2s dWdE dWdE f dW y [time-1] ds dW [time-1area-1] Differential cross section Neutron Scattering Principle - I General Equation The number of neutrons scattered per second into a solid angle dW with the final energy between E and E+dE = neutron flux of the incident beam• dWdE For Elastic Scattering We do not analyze the energy but only count the number of the scattered neutrons The number of neutrons scattered per second into a solid angle dW = dE = neutron flux of the incident beam• dW [area]

  5. CNBC Summer School, June 16, 2009 4p q q = ko – ki = sin l 2 Wo The measured intensity at q (or q), Im(q) can be expressed as Beam area on the sample ( )v,sam = ( ) v,std ( )v·A ·t Im(q) = IF · Wo ·e ·T · Path length Flux Solid angle Differential cross-section Detector efficiency Sample transmission ds ds ds ds dW dW dW dW Im,sam(q) · Astd · tstd · Tstd Im,std(q) · Asam · tsam · Tsam Neutron Scattering Principle - II Scattering vector, q ko q ki q ki

  6. CNBC Summer School, June 16, 2009 4p q sin q = l 2 circularly averaged into 1-D Typical SANS Instrument 2-D Detector Neutron Guide Sample table Velocity Selector Neutron Beam from Reactor For l = 8 Å and qmin~ 0.25o, qmin ~ 0.003 Å-1. Max. attainable length scale = 2p/qmin ~ 2000 Å.

  7. CNBC Summer School, June 16, 2009 Current CNBC SANS Setup M.-P. Nieh Y. Yamani, N. Kučerka and J. KatsarasRev. Sci. Instrum. (2008) 79, 095102 qmin ~ 0.006 Å-1: Max. attainable length = 2p/qmin ~ 1000 Å.

  8. CNBC Summer School, June 16, 2009 = (rp – ro) · Vp·P( q ) S( q ) ( )v r, Å-2 ds dW Contrast – A Key Parameter N 2 2 V Scattering intensity is proportional to “the square of the scattering length density difference between studied materials and medium” (known as “contrast factor”). low contrast 2 ways of increasing contrast Ideally, we would like to increase the contrast yet without changing the chemical properties of the system. This is one of the greatest advantages using neutron scattering, since the neutron scattering lengths of isotopes can be very different.

  9. CNBC Summer School, June 16, 2009 Contrast Variation J. Pencer, V. N. P. Anghel, N. Kučerka and J. Katsaras, J Appl Cryst40 (2007) 513-525

  10. CNBC Summer School, June 16, 2009 = (rp – ro) · Vp·P( q ) S( q ) N  r(r1)r(r2) <e-iq· (r1- r2) > dr1dr2 = V vp 2 N  r( r )·<e-iq · r > dr = vp V r1 and r2 are vectors of scattering center from one particle ( )v ( )v Particle Form Factor N 1 S < |Ak(q) |2> = Amplitude of the Form Factor V ds ds k=1  dW dW r( r )·<e-iq · r > dr Ak(q) = particle k SANS Analysis – Dilute Particulate Systems V ro Vp rp orientational average r1 r2 N particles O N 2 2 = (rp – ro) · Vp·P( q ) V The form factor is determined by the structure of the particle.

  11. CNBC Summer School, June 16, 2006 Since spheres are isotropic, there is no need to do orientational average ( )v ds dW SANS Analysis – Dilute Particulate Systems Simulation of a particular system - Spheres 2 1  r( r )·e-iq · r dr P(q) = 2 Vsphere vsphere j 2 q = p j= 2p r= R q 1    e-iqr·cosqr2 sinq dj dq dr = r 2 Vsphere q= 0 j= 0 r= 0 9 = [sin(qR) - qR·cos(qR)]2 6 (qR) N 2 2 R= 100 Å Dr = 1x10-6 Å-2 f= 0.1 = (rsphere – ro) · Vsphere·P( q ) V 9 2 [sin(qR) - qR·cos(qR)]2 =fsphereVsphereDr 6 (qR)

  12. CNBC Summer School, June 16, 2009 9 2 [sin(qR) - qR·cos(qR)]2 =Asph(qR) 6 (qR) [R13·Asph(qR1)– R23·Asph(qR2)]2 (R13 – R23)2 1 1 2   Asph[q a2 cos2(px/2) + b2sin2(px/2)(1-y2)1 + c2y2 ] dx dy 0 0 sin2(qLx/2)  4 dx (qLx/2)2 J12[qR 1-x2 ] 0 [qR 1-x2 ]2 2 - J1(2qR)/qR q2R2 qL sin2(qL/2) sin(t) 2  dt - qL t (qL/2)2 0 SANS Analysis – Dilute Particulate Systems Common form factors of particulate systems P(q) Morphologies Morphologies Spheres (radius :R) Spherical shells (outer radius: R1 inner radius: R2) Triaxial ellipsoids (semiaxes: a,b,c) 1 Cylinders (radius: R length: L) J1(x) is the first kind Bessel function of order 1 Thin disk (radius: R) By setting L = 0 Long rod (length: L) By setting R = 0 “Structure Analysis by Small Angle X-Ray and Neutron Scattering” L. A. Feigen and D. I. Svergun

  13. CNBC Summer School, June 16, 2009 SANS Analysis – Dilute Particulate Systems Procedure of Data Analysis Fit the experimental data using the selected model (Fix the values of any “known” physical parameters as many as possible) Select a model for possible structure of the aggregates Change the model Is it a good fit ? A Possible Structure Yes No

  14. CNBC Summer School, June 16, 2009 In these cases, the known (or constrained) physical parameters are SLDs of lipid and D2O and the bilayer thickness. Thus, there are only 2 parameters to be determined through fitting in each case. Oblate shells: a=180 Å, b=62 Å Spherical shells: R=133 Å, p= 0.15 Bilayer disks: R=156 Å, L= 45 Å Examples of A Dilute Particulate System Sample: A mixture of dimyristoyl and dihexanoyl phosphatidylcholine (DMPC, DHPC), and dimyristoyl phosphatidylglycerol (DMPG) in D2O; total lipid conc. = 0.1 wt.% Demo of SANS data analysis in three afternoons! Path

  15. CNBC Summer School, June 16, 2009 = (rp – ro) · Vp·P( q ) S( q ) N  r(r1)r(r2) <e-iq· (r1- r2) > dr1dr2 = V Vp ( )v ( )v ds ds Structure Factor dW dW In dilute solution, S( q ) = 1 SANS Analysis – Concentrated Particulate Systems V ro Vp rp N 1 R2 S < |Ak(q) |2> = R1 V r1 k=1 r2 N N N particles 1 S S e-iq· (Rk- Rj) Ak(q)Aj*(q) O + V jk k=1 N N 1 S S ~ e-iq· (Rk- Rj) < |Ak(q) |2> {1+ } V jk k=1 N 2 2 = (rp – ro) · Vp·P( q ) S( q ) V

  16. CNBC Summer School, June 16, 2009 + + + + + + + + S(q=0) = kT(N/p) S(0)>1 more compressible S(0)<1 less compressible Osmotic compressibility SANS Analysis – Concentrated Particulate Systems Models for S(q) u(rjk) Coulomb Repulsion Q = 20, csalt=0.01M Hard Sphere R= 50 Å Conc. = 5% 2R Sqaure Well Attraction -3kT 2.4R

  17. CNBC Summer School, June 16, 2009 Fixed, due to P(q) SANS Analysis – Concentrated Particulate Systems Simulation Result using Coulomb Repulsive Model R= 50 Å Q = 50 Csalt = 0.001M

  18. CNBC Summer School, June 16, 2009 Examples of Concentrated Particulate Systems “Bicelles” composed of DMPC/DHPC and small amount of Tm3+

  19. CNBC Summer School, June 16, 2009 RN R1 R2 R3 R4 N a2 = S |Rn - Rn-1|2 N n=1 The distribution vector between any two “steps” m, n, i.e., (Rn - Rm) is Gaussian 3 3(Rn - Rm)2 F(Rn - Rm, n-m) =[ ]3/2 exp[ - ] 2pa2|n-m| 2a2|n-m| This model is adequate for describing long polymer chains in a theta solvent, where the segment-segment and segment-solvent interaction and the excluded volume effect are canceled out. Such polymer chains are also known as “phantom” chains. “The Theory of Polymer Dynamics” M. Doi and S. F. Edwards “Gaussian Chain” “Gaussian Chain” Model : The effective bond length, a, of one step (composed of several segments of the chain) has a Gaussian distribution.

  20. CNBC Summer School, June 16, 2009 Debye Function 2 = Npfp(rp – ro)2VM(e-q2RG2 – 1 + q2RG2) q4RG4 ( )v 1 Radius of Gyration, RG2=S <(Rn – Rm)2> 2N2 m,n RG = 80 Å ds q-2 dW SANS Analysis – Dilute “Gaussian Chain” Solutions Scattering from a Gaussian Chain VM: molecular volume of monomer Np : degree of polymerization

  21. CNBC Summer School, June 16, 2009 fp Np RG (Å) 111 1% 39 74 32 2% 42 27 4% Example – Polymer Solutions Debye function is not always adequate to describe the polymer conformation! An example is that as polymers are in a good solvent, the intensity at high q regime will decay with q-5/3 instead of q-2. Polylactidein CD2Cl2 4% 2% 1%

  22. CNBC Summer School, June 16, 2009 VmfpDr2 VmfpDr2 VmfpDr2 I(q) I(q) I(q) Zimm Plot For dilute polymer solutions, the scattering intensity at low q regime (i.e., q·RG < 1) can be expressed as a function of RG, MW (molecular weight), A2 (second virial coefficient) and fp (volume fraction of the polymer) f2 f1 f4 f3 fp=0 1 = (1 + q2RG2/3 + …)( + 2fpA2 + …) Np q3 q2 Vm is the molecular volume of the monomer Np is the degree of polymerization q1 1/Np RG, Np and A2 can be obtained by Zimm plot, where q=0 q2 + cfp is plotted against (q2 +cfp) and two extrapolation lines for q=0 and fp=0 are also obtained (c is an arbitrary number.). The slope of the extrapolation line for fp=0 is RG2/3Np. The slope of the extrapolation line for q=0 is 2A2/c. The intercept of both extrapolation lines is 1/Np.

  23. CNBC Summer School, June 16, 2009 fp = 0 fp Np RG (Å) 111 1% 39 q = 0 74 32 2% 42 27 4% 1/Np Application of Zimm Plot Polylactidein CD2Cl2 Np = 387 RG = 79 Å A2= 88.6 Compared to the result obtained from Debye fitting It requires at least two concentrations to make up a Zimm plot.

  24. CNBC Summer School, June 16, 2009 ( )v ( )v ( )v ln[ ] = ln(Io) – q2RG2/3 ln[ ] ds ds ds N 2 2 dW dW dW = (rp – ro) ·Vp· [1- (qRG)2/3 + …] V SANS Analysis – Model Independent Scaling Method Guinier regime – low-q scattering ~ fp(rp – ro)2Vpe-q2RG2/3 -RG2/3 q2 “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit

  25. CNBC Summer School, June 16, 2009 9 2 [sin(qR) - qR·cos(qR)]2 =fsphereVsphereDr 6 (qR) ( )v ( )v 2 2 NsphereVsphereDr ds 2pDr2ST q4( )v = ds ds dW V dW dW SANS Analysis – Model Independent Scaling Method Porod regime – high-q (with respect to the length scale) scattering The form factor of a sphere (radius of R) For qR>>1, i.e, focusing at smooth interface between two bulks 3-dimensionally 2pDr2ST 9 total surface area = ~ 4 V Vq4 2(qR) log I -4 log q “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit

  26. CNBC Summer School, June 16, 2009 n particles • Long Rigid rod 1 • Smooth 2-D Objects 2 • Linear Gaussian Chain 2 • Chain with Excluded Volume 5/3 • Interfacial Scattering from 3-D Objects • with Smooth Surface (Porod regime) 4 • with fractal Surface 3 ~ 4 SANS Analysis – Model Independent Scaling Method If Im(q) scales with q-n over a wide range of q. The “possible” structure can be obtained with some knowledge of the systems. “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit

  27. CNBC Summer School, June 16, 2009 Small Angle Diffraction DOPC -CD3 -CH3

  28. CNBC Summer School, June 16, 2009 Example: Orientation of cholesterol in Biomembrane Harroun, T. A. et al. Biochemistry45 (2006) 1227-1233.

  29. CNBC Summer School, June 16, 2009 Take Home Messages • SANS is a powerful tool to study the global structures of isotropic systems in length scales ranging from 10 to 1000 Å. • Isotope replacement could enhance/vary the contrast without changing the chemistry of the systems. • The scattering function is proportional to the product of contrast factor, form factor (intraparticle scattering function) and structure factor (interparticle interaction). • Structural parameters can be obtained through model dependent approaches, while model independent scaling law can also reveal possible morphology of the studied system.

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