Small Angle Neutron Scattering. Mark Telling

Transcription

1 Small Angle Neutrn Scattering a tl fr the study f mlecular structure Mark Telling ISIS Facility mark.telling@stfc.ac.uk

2 Lecture 3 : Overview SANS : why and what gemetry f a SANS experiment cntributins t ds/dw cntrast matching the single particle (shape) factr, P(Q) the inter-particle structure factr, S(Q) transmissin cnsideratins Analysis via standard plts Kratky, Prd, Zimm and Guinier Example triblck c-plymers

3 Selected references The SANS Tlbx, B. Hammuda Applicatins f Neutrn Scattering, B J Gabrys (editr) Grdn and Breach Science Publishers, 000, e.g. Chapters 4,5 and 8 Mdern Techniques fr Plymer Characterisatin, R Pethrick (ed) Jhn Wiley, 1999, ISBN , e.g. Chapter 7 Plymers and neutrn scattering, J.S. Higgins and H.C. Benît, Oxfrd University Press, 1994

4 Small Angle Scattering Cllective name given t techniques f: SAXS, SANS, SALS (r LS) Radiatin type affects: sample, length scale but all three are gverned by same eqns. and laws with minr adjustments fr the radiatin type we will fcus n extractin f M w, R g Interested in S ch (Q,w) which prvides inf n: size, shape f scattering particles and distributin f these particles

5 What can SANS tell us Slutin Dispersin Melt Slid State R g : radius f gyratin A : nd virial cefficient (z) : vlume fractin prfile G : amunt f adsrbed plymer R g : radius f gyratin M w : weight averaged mlecular weight M w : weight averaged mlecular weight s : nd mment f the vlume fractin prfile DG mix : Gibbs free energy D f : fractal dimensins <p> : average bund fractin See slutin E act : activatin energy v : excluded vlume expnents A B : surface f a bilayer S/V : surface area t vlume rati l p : persistence length t rms : rms thickness f adsrbed layer D : diffusin cefficient M L : mass per unit length t max : max extent f adsrbed layer g : real space crrelatin functin

6 Why d small angle scattering measurements

7 The gemetry f a SANS instrument dx dy dw dxdy L Increasing Q Q Q k s k i 4 sin / 4 r L det sd

8 The gemetry f a SANS instrument dx dw dxdy L Increasing Q Cnsidering R g = 100 Å and assume = 1 Å Q = 0.06 Å -1 = 0.6 degrees!

9 The gemetry f a SANS instrument dx dw dxdy L Flux ds I( Q) I( ) DW( ) TV ( Q) dw Nte: differential crss sectin I = incident flux, = detectr efficiency, T = sample transmissin, V = sample vlume

10 The gemetry f a SANS instrument Curtesy f the ILL

11 The NIST guide hall SANS QENS Frm

12 ds/dw Objective f SANS experiment - extract ds/dw cntains inf re: shape, size and interactins ds ( Q) dw N p V p ( D) P( Q) S( Q) B inc N p = density f scattering bdies V p = vlume f ne scattering bdy P(Q) = single bdy frm r shape factr (i) S(Q) = inter-bdy structure factr (ii) (D) = neutrn scattering length density (aka cntrast ) (iii) B inc = istrpic incherent backgrund signal

13 i) P(Q) single bdy (shape) factr Describes hw ds/dw(q) is mdulated by interference effects between radiatin scattered frm different parts f the same scattering bdy i.e. a mnmer perhaps cnsider the Gaussian cil R g = the rt mean square distance f the bject s parts frm either its centre f gravity r a given axis R g Nl / 6 l = segment (end t end) length Generalisatin: R g ~ N In the s called θ slvent, ν = 1 /, which is the result f simple randm walk. The chain behaves as if an ideal chain.

14 i) P(Q) single bdy (shape) factr Analytical expressin fr P(Q) frm Gaussian cil fr a Gaussian distributin f segment density abut a centre f mass with a R g P( Q) (exp( Q Q R 4 g ) Q R 4 g R g 1) this is knwn as the Debye-functin (Debye, 1947) imprtant expansins fr analysis: Lw Q limit: P(QR g <<1) = 1 Q R g /3 High Q limit: P(QR g >>1) = /Q R g

15 i) P(Q) single bdy (shape) factr General frm f P(Q): develped by Van de Hulst Analytic expressins exist fr cmmn shapes i.e. star, cmb, ring plymers i.e. spheres, cncentric cylinders, hinged rds disc f negligible thickness and radius, R p P( Q) ( QR p ) 1 J 1 (QR QR p p ) sphere f radius, R p P( Q) 3(sin( QR p ) QR ( QR p 3 p) (cs( QR p )))

16 i) P(Q) single bdy (shape) factr Here (Q) 0 S(Q) = 1 Very lw Q Only see pints Relate t M w iii dilute and iii cnc See part f mlecule Fr iii: chain statistics/persistence length Fr iii : crrelatin lengths (chain length between cntact pints) Part f a chain See Gaussian chain if persistence length < Q -1 Rd like behaviur if persistence length > Q -1 Q -1 R g N structure detail nly dimensins Guinier regin Q -1 bnd lengths Lcal structure is prbed Frm Higgins and Benît

17 ii) S(Q) inter-bdy structure factr S(Q) defines the mdulatin f (ds/dw)(q) due t interference effects between neutrns scattered by different scattering centres in the sample S(Q) depends n a) the degree f lcal rder b) interactins between scattering centres S( Q) 1 4 N QV s p 0 g( r) 1 r sin( Q r) dr G(r) a density distributin functin the variatin f atmic density as a functin f the distance frm ne atm

18 ii) S(Q) inter-bdy structure factr Relative psitins f scattering bdies? such infrmatin is gained via the radial distributin functin G( r) 4N V p r g( r) r = radial distance frm any scattering bdy, V = vlume N p = cncentratin f scattering bdies Nte: as N p tends t 0 (i.e. we dilute the system) S(Q) tends t unity micrscpic structure infrmatin is therefre nly btained frm cncentrated and/r strngly interacting systems

19 ii) S(Q) inter-bdy structure factr G(r) : damped, scillatry density distributin functin maxima prbable distance f n-n scattering centres

20 iii) (D) i.e. cntrast matching SANS is frm large mlecules n sense t talk abut single atm scattering lengths Instead we talk abut scattering length density, A b N i m i N i b i = bulk density, m = relative mlar mass N = number density f scattering centres b i = cherent neutrn scattering length fr plymers, ne scattering centre is ne repeat unit r mnmer units f = cm -

21 iii) (D) i.e. cntrast matching ds/dw = cntrast-weighted sum f individual cmpnents but if (D) = 0 then ds/dw = 0 we still get intensity in ur detectr but we cannt differentiate between different particles fr example: (D ) = plymer - slvent Greatly simplifies scattering frm multi-cmpnent systems by cntrast matching nce can greatly simplify multi-cmpnent systems by remving the scattering intensity frm certain cmpnent

22 Experimentally An aqueus dispersin f deuterated PS latex particles in D 0 (frm King)

23 Minimising multiple scattering Sample thickness shuld be ptimised t minimise multiple scattering Ideally we want a sample that transmits 80-90% f beam i.e % scatterer T determine thickness use Beer-Lambert relatin I trans = I incident exp(-nst) n = number density s = ttal scattering crss sectin / frmula unit t = sample thickness

24 Standard plts Main analysis tls t extract parameters are least squared fitting t analytical mdels standard plts Standard plts assess linear regins f ds/dw ver different length scales (i.e. Q regimes) t extract characteristic slpes and intercepts cmmn plts: Guinier, Kratky, Zimm, Prd

25 Guinier plt Plt ln(i) vs. Q plt fcuses n the lw Q regin where Q < 1/R g here slpe apprximates t : ln(i)=ln(i )+Q R g /3 infrmatin extracted : M w, R g R g represents effective size f scattering particle inter particle effects affect R g except in infinite dilutin limit s extraplate t zer cncentratin Frm The SANS Tlbx, Hammuda

26 Guinier plt Frm The SANS Tlbx, Hammuda

27 Zimm plt Plt 1/I vs Q Lw Q departure frm linear behaviur nn-hmgeneity in the sample Fr dilute plymer slutins: extraplatin t zer Q and zer cncentratin gives R g, M w and A Fr blends (single phase) slpe : prprtinal t crrelatin length crrelatin length : prprtinal t Flry-Huggins interactin parameter High Q regime (i.e. high Q expansin f P(Q)) fr plymer slutins we can extract the excluded vlume expnent v = 3/5 fr swllen-, 1/ fr theta-, 1/3 fr cllapsed- chains

28 dry slvated V V M A ) ( 1 ) ( s D D W A g A N A QR M c N Q d d 1 1 V s A ex A V M N A Interactin parameter Degree f slvatin Excluded vlume hs A V M N A 4 hard sphere vlume Zimm plt

29 Kratky plt Plt Q vs. I.Q Emphasizes Gaussian nature f plymer chains at high Q, P(Q) fr Gaussian chain tends t: /R gq plt tends t a hrizntal asymptte 1/Q scaling nt affected by interparticle effects deviatins frm asymptte indicates nn Gaussian characteristics Frm The SANS Tlbx, Hammuda

30 Prd plt Plt ln(i) vs. ln(q) prbes regins smaller than scattering particle i.e. lcal structure infrmatin abut the fractal dimensins f the scattering bject at high Q a slpe f Q n where: n = - : signature f Gaussian chains in dilutin n = -1 suggests rigid rds n = -4 highlights smth interface between dmains in multiphase system Fr plymers: n is related t excluded vlume parameter; n = 1/v Frm The SANS Tlbx, Hammuda

31 Triblck c-plymers Plymer slutins and blends phase separate when heated r cled phase separatin characterised via LCST r UCST mst H 0 sluble mixtures separate n heating (H-H bnds sften) UCST LCST Cncentratin nucleatin and grwth regime

32 Triblck c-plymers Plurnic mlecules PEO:PPO:PEO dn t just separate but frm a rich array f micrstructures tri-blck c-plymers cmpsed f cvalently bnded PEO and PPO EO (hydrphilic - simplest water sluble plymer) PO (hydrphbic - nt water sluble at ambient T (30 C) T>30K lw temperature (T<30 C) plymer slutin cnsisting f simple unimers temperature increases - amphiphilic nature results in frmatin f micelles micelles capable f slubilising hydrphbic materials plurnics cmmercially available, used in csmetics pharmaceutical industry SANS interest is micelle frmatin, and phase transitins that frm them

33 Triblck c-plymers Images frm