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

2. 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 CNBC Summer School, June 16, 2009

3. 1.The q range of SANS : between 0.002 and 0.6 Å -1 – achievable long wavelengths of neutrons and low detecting angles are important 2.A powerful technique for in-situ study on the global structures of isotropic samples 3.Easy to play contrast with isotope substitution 4.Data analysis: (A) Model dependent: well-defined particulate systems (possible non-unique solutions) (B) Model independent: general feature of the structure Common Features & Advantages of SANS CNBC Summer School, June 16, 2009

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

5. Neutron Scattering Principle - II Scattering vector, q kiki koko  q kiki q = k o – k i =  sin  2 dd dd ( ) v ·A ·t Beam area on the sample Path length The measured intensity at q (or  ), I m (q) can be expressed as I m (q) = I F ·  o ·  ·  T · dd dd Flux Solid angle Detector efficiency Sample transmission Differential cross-section  dd dd ( ) v,sam = ( ) v,std dd dd I m,sam (q) · A std · t std · T std I m,std (q) · A sam · t sam · T sam CNBC Summer School, June 16, 2009

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

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

8. Contrast – A Key Parameter 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. , Å -2 CNBC Summer School, June 16, 2009 N V 2 2 = (  p –  o ) · V p · P ( q ) S ( q ) dd dd ( ) v

9. Contrast Variation CNBC Summer School, June 16, 2009 J. Pencer, V. N. P. Anghel, N. Kučerka and J. Katsaras, J Appl Cryst 40 (2007) 513-525

10. SANS Analysis – Dilute Particulate Systems dd dd ( ) v = (  p –  o ) · V p · P ( q ) N V 2 2 orientational average The form factor is determined by the structure of the particle. = N V  (r 1 )  (r 2  e -iq· (r 1 - r 2 ) > dr 1 dr 2  vpvp pp oo VpVp V N particles r1r1 r2r2 O r 1 and r 2 are vectors of scattering center from one particle 1   k  q    k=1 V = N Particle Form Factor Amplitude of the Form Factor   ( r )· dr vpvp 2 N V =   ( r )· dr particle k  k  q  CNBC Summer School, June 16, 2009 = (  p –  o ) · V p · P ( q ) S ( q ) dd dd ( ) v

11. Simulation of a particular system - Spheres Since spheres are isotropic, there is no need to do orientational average   ( r )·e -iq · r dr P(q) = 1 V sphere v sphere 2 2 =    e -iqr·cos  r 2 sin  d  d  dr 1 V sphere 2 2 r= 0 r= R  = 0  =   = 0  = 2  = [sin(qR) - qR·cos(qR)] 2 9 (qR) 6 R= 100 Å  = 1x10 -6 Å -2  = 0.1 dd dd ( ) v = (  sphere –  o ) · V sphere · P ( q ) N V 2 2 =  sphere V sphere  2 [sin(qR) - qR·cos(qR)] 2 9 (qR) 6 SANS Analysis – Dilute Particulate Systems r   CNBC Summer School, June 16, 2006

12. Common form factors of particulate systems CNBC Summer School, June 16, 2009 SANS Analysis – Dilute Particulate Systems Cylinders (radius: R length: L) Spherical shells (outer radius: R 1 inner radius: R 2 ) Morphologies P(q) Triaxial ellipsoids (semiaxes: a,b,c) Morphologies Spheres (radius :R) [sin(qR) - qR·cos(qR)] 2 =A sph (qR) 9 (qR) 6 2 (R 1 3 – R 2 3 ) 2 [R 1 3 ·A sph (qR 1 )– R 2 3 ·A sph (qR 2 )] 2    A sph [q a 2 cos 2 (  x/2) + b 2 sin 2 (  x/2)(1-y 2 ) 1 + c 2 y 2 ] dx dy 00 11 2 1  0 J 1 2 [qR 1-x 2 ] 4 [qR 1-x 2 ] 2 dx sin 2 (qLx/2) (qLx/2) 2 Thin disk (radius: R) Long rod (length: L) By setting L = 0 2 - J 1 (2qR)/qR q2R2q2R2 By setting R = 0 qL  0 2 sin(t) t dt - sin 2 (qL/2) (qL/2) 2 “Structure Analysis by Small Angle X-Ray and Neutron Scattering” L. A. Feigen and D. I. Svergun J 1 (x) is the first kind Bessel function of order 1

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

14. Examples of A Dilute Particulate System Sample: A mixture of dimyristoyl and dihexanoyl phosphatidylcholine (DMPC, DHPC), and dimyristoyl phosphatidylglycerol (DMPG) in D 2 O; total lipid conc. = 0.1 wt.% Path In these cases, the known (or constrained) physical parameters are SLDs of lipid and D 2 O 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 CNBC Summer School, June 16, 2009 Demo of SANS data analysis in three afternoons!

15. SANS Analysis – Concentrated Particulate Systems pp oo VpVp V r1r1 r2r2 O N particles R2R2 dd dd ( ) v = N V  (r 1 )  (r 2  e -iq· (r 1 - r 2 ) > dr 1 dr 2  VpVp = (  p –  o ) · V p · P ( q ) S ( q ) N V 2 2 1   k  q    k=1 V = N 1   k  q  j *  q  k=1 V + N  jkjk N e -iq· (R k - R j ) ~ 1  k  q    {1+ } V  k=1 N  jkjk N e -iq· (R k - R j ) Structure Factor In dilute solution, S( q ) = 1 R1R1 CNBC Summer School, June 16, 2009 = (  p –  o ) · V p · P ( q ) S ( q ) dd dd ( ) v

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

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

18. Examples of Concentrated Particulate Systems “Bicelles” composed of DMPC/DHPC and small amount of Tm 3+ CNBC Summer School, June 16, 2009

19. “Gaussian Chain” “Gaussian Chain” Model : The effective bond length, a, of one step (composed of several segments of the chain) has a Gaussian distribution. R1R1 R2R2 R3R3 R4R4 RNRN a 2 =  |R n - R n-1 | 2 n=1 N N The distribution vector between any two “steps” m, n, i.e., ( R n - R m ) is Gaussian  (R n - R m, n-m) =[ ] 3/2 exp[ - ]  a 2 |n-m|   a 2 |n-m|  R n - R m  2 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 CNBC Summer School, June 16, 2009

20. Scattering from a Gaussian Chain dd dd ( ) v = N p  p (  p –  o ) 2 V M (e -q 2 R G 2 – 1 + q 2 R G 2 ) 2 q4RG4q4RG4 Debye Function Radius of Gyration, R G 2 =  2N 2 1 m,n q -2 R G = 80 Å V M : molecular volume of monomer N p : degree of polymerization CNBC Summer School, June 16, 2009 SANS Analysis – Dilute “Gaussian Chain” Solutions

21. 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. 4% 2% 1% Polylactide in CD 2 Cl 2 pp NpNp R G (Å) 1% 2% 4% 111 74 42 39 32 27 CNBC Summer School, June 16, 2009 Example – Polymer Solutions

22. Zimm Plot For dilute polymer solutions, the scattering intensity at low q regime (i.e., q·R G < 1) can be expressed as a function of R G, M W (molecular weight), A 2 (second virial coefficient) and  p (volume fraction of the polymer) I(q) V m  p   = (1 + q 2 R G 2 /3 + …)( + 2  p A 2 + …) NpNp 1 V m is the molecular volume of the monomer N p is the degree of polymerization I(q) V m  p   is plotted against (q 2 +c  p ) and two extrapolation lines The slope of the extrapolation line for  p =0 is R G 2 /3N p. R G, N p and A 2 can be obtained by Zimm plot, where for q=0 and  p =0 are also obtained (c is an arbitrary number.). The slope of the extrapolation line for q=0 is 2A 2 /c. The intercept of both extrapolation lines is 1/N p. q 2 + c  p I(q) V m  p   q=0  p =0 11 22 44 33 q1q1 q2q2 q3q3 1/N p CNBC Summer School, June 16, 2009

23. Application of Zimm Plot N p = 387 R G = 79 Å A 2 = 88.6 Polylactide in CD 2 Cl 2  p = 0 q = 0 1/N p Compared to the result obtained from Debye fitting pp NpNp R G (Å) 1% 2% 4% 111 74 42 39 32 27 It requires at least two concentrations to make up a Zimm plot. CNBC Summer School, June 16, 2009

24. SANS Analysis – Model Independent Scaling Method Guinier regime – low-q scattering ~  p (  p –  o ) 2 V p e -q 2 R G 2 /3 dd dd ( ) v ln[ ] = ln(I o ) – q 2 R G 2 /3 dd dd ( ) v N V 2 2 [1- (qR G ) 2 /3 + …] = (  p –  o ) · V p · dd dd ( ) v ln[ ] q2q2 -R G 2 /3 “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit CNBC Summer School, June 16, 2009

25. Porod regime – high-q (with respect to the length scale) scattering =  sphere V sphere  2 [sin(qR) - qR·cos(qR)] 2 9 (qR) 6 dd dd ( ) v For qR>>1, i.e, focusing at smooth interface between two bulks 3-dimensionally The form factor of a sphere (radius of R) dd dd ( ) v N sphere V sphere  22 ~ V 9 2(qR) 4 2    S T Vq 4 = dd dd q 4 ( ) v = 2    S T V total surface area log I log q -4 “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit CNBC Summer School, June 16, 2009 SANS Analysis – Model Independent Scaling Method

26. If I m (q) scales with q -n over a wide range of q. The “possible” structure can be obtained with some knowledge of the systems. Long Rigid rod1 Smooth 2-D Objects 2 Linear Gaussian Chain2 Chain with Excluded Volume 5/3 Interfacial Scattering from 3-D Objects with Smooth Surface (Porod regime)4 with fractal Surface 3 ~ 4 n particles “Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit CNBC Summer School, June 16, 2009 SANS Analysis – Model Independent Scaling Method

27. -CD 3 -CH 3 DOPC Small Angle Diffraction CNBC Summer School, June 16, 2009

28. Example: Orientation of cholesterol in Biomembrane Harroun, T. A. et al. Biochemistry 45 (2006) 1227-1233. CNBC Summer School, June 16, 2009

29. 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. CNBC Summer School, June 16, 2009