Scattering Techniques in Soft Condensed Matter

Transcription

1 Scattering Techniques in Soft Condensed Matter Arnout Imhof Soft Condensed Matter, Debye Institute, Utrecht University

2 Scattering of waves Scattering occurs whenever a medium is inhomogeneous Waves scatter from obstacles called scatterers: Water waves ( m): ships, ducks Sound ( mm): sea floor, fish, earth structure, tissue X-rays ( nm): atoms Neutrons ( nm): atomic nuclei Light (0.1 1 µm): water drops (fog), fat globules (milk), interstellar dust Scattering most effective if obstacle size ~ wavelength All forms of scattering are analogous

3 Light scattering Why does light (an EM wave) scatter? EM waves accelerate charges (electrons) in a medium Accelerating charges radiate new EM waves Conversion to heat, fluorescence, Scattering Absorption Absorption + Scattering = Extinction

4 Interference Scattered waves interfere: Intra particle size, shape Inter particle particle ordering (SLS) particle dynamics (DLS)

5 Some other scattering terms Multiple scattering: Single scattering: particle scatters sum of incident and scattered waves particle scatters only incident wave (weak scattering or particles far apart) Elastic scattering: Inelastic scattering: frequency of scattered wave equals frequency of incident wave frequencies differ

6 The scattering geometry incident x k =I 0 π λ E o E o k θ r E s z y E s ˆr scattered I s (, ) θϕ = IF 0 ( θ, ϕ ) kr

7 Measuring scattered light As a function of angle: Scatt. cross section: C sca ( θ, ϕ ) IF 0 Is ( θϕ, ) = kr Ps 1 = = F ( θ, ϕ) sinθdθdϕ I0 k [m ] 4π [J.m -.s -1 ] C abs total scattered power (per unit incident intensity) = P I abs Cext = Csca + Cabs 0

8 Cross sections are obtained from transmission measurements: ρ particles/m 3 I 0 I(z) z dz L I+dI di det. di = ρc I( z) dz t ext I = I ρc L exp 0 ( ext )

9 Scattering our approach Point particle Large particle Ensembles of (large) particles Moving particle Ensemble of (large) moving particles

10 Scattering by one small particle Size << λ (Rayleigh scattering) incident field (, t) Er = E 0 e ikr iωt x induced point dipole p = αe= αe0 i t e ω E 0 E 0 k θ r E s z y E s ˆr mvr radiated field 1 E ˆ ˆ s = ( / ) 4πε r r p ( t r v ) E s k E0 cosθ ikr iω t = α e E 4 s πε mr E 0

11 I s k 3πεr 4 ( θ ) = I0 α ( 1+ cos θ) m 9π m = I V + λ r m + ( 1 cos θ ) p (Rayleigh) ε p ε m α = 3εm ε + ε p m V m 1 np = 3 ε m V p m= m + nm p I perp I par

12 Scattering by a large particle Rayleigh-Gans-Debye approximation: Divide particle into pointlike subunits and sum all wavelets. k 0 r θ k Valid if: each subunit responds only to incident field: wave inside particle suffers no phase lag relative to a wave passing the particle by: m 1 << 1 kd m 1 << 1

13 ( ) φ = k r+ k r = k k r q r 0 0 k 0 k0 r dr k r r θ k k 0 θ q k q = q = 4π sin λ ( θ ) k ikr iωt iqr des = ( m() r 1) E0 e dr π R ( m 1)/( m + ) ( m 1) 3

14 Integrate over the particle: k ikr iωt Es = E0 e f q π R P ( q) ( ) Take the modulus squared: 4 kvp m 1 Is = I0 P + 8π R ( ) = ( () 1) iqr f m e d q r r V ( q)( 1 cos θ ) ( q) ( ) p ( r ) iqr m() 1e dr f Vp = = f 0 m r dr V p ( () 1) Particle form factor Compare with a model.

15 Form factor of a sphere ( ) P q ( qa) qa cos( qa) 3 ( qa) sin = 3 q = q = 4π sin λ ( θ ) nm P(q) qa I s (a.u.) E-3 0 1x10 7 x10 7 q (m -1 ) 64 nm 435 nm

16 Homogeneous sphere ( radius a = λ ) log(i perp ) log(i par )

17 Radius of gyration iqr 1 At small angles or for small particles: e 1 i ( ) P ( q) ( ) ( ) ( ) m 1 q r cos d ( m 1) ( )( ) m 1 iqr dr m 1 qr dr Vp 1 Vp 1 m 1 dr m 1 dr = 1 V p V 1 = 1 qrg+ 3 p V p dr θ r V p ( ) R g = V p V ( m 1) p qr qr + ( m 1) Sphere: R = 35a Rodlike: R = L/ 1 Polymer: R = l n/6 g g r d dr r g

18 Scattering by many particles k0 r k r k 0 r θ k k 0 θ q k ( ( r) ) E ~ E m 1 e dr iqr s 0 V s = 0 if r in medium

19 ( ( r) ) E ~ E m 1 e dr iqr s 0 V s N 0 ( () r 1) iqr j= 1 V = E m e dr j N 0 iqr j ( ( r ) 1) iqr j= 1 V = E e m e dr 0 N j= 1 i j ( ) j = E e qr f q j r = r j + r r j r

20 N j s ~ s = 0 j = 0 j= 1 iqr ( q) ( q) ( q) I E I f e NI P S Identical particles S ( q) N N 1 i k j ( ) = e q r r N j= 1 k= 1 Structure factor ( ) 1 g r 1 e i qr = + ρ dr Radial distribution function

21 Measuring structure factors Dilute sample: I dil ( q) ~ ρ P( q) Concentrated sample: I( q) ~ ρp( q) S( q) dil S( q) = ρdil ρ I I dil ( q) ( q) 31 nm polystyrene spheres mmol/l NaCl Volume fractions: 0.01, 0.04, 0.08, 0.13 (SANS, Ottewill et al.)

22 80 nm polystyrene spheres with SO 3 H surface groups (SLS, Härtl and Versmold, 199) 3.1 µmol/l NaOH 3.1 µmol/l NaCl model fits: Z eff = 503 e Z eff = 474 e κσ = 0.1 κσ = 0.50

23 Inverting structure factors 1) Fourier inversion (limited q-range) ) Fit with model; then Fourier invert the fit curve ρ ρ (80 nm PS spheres, deionized, Härtl and Versmold, 199)

24 Low q scattering ( 0) S q ρ = = kt Π T Equation of state 1 kt Π = 1 + Bρ + ρ T Hard-sphere particles (silica coated with stearyl alcohol, suspended in cyclohexane)

25 Polymer solutions Zimm plot s ( ) = ρ ( ) ( ) I q KM P q S q with: ρ = cn Av M K ρ = + + I q M P q S q M ( ) ( ) ( ) Kc 1 1 NAvB 1+ Rq g c 3 I q + M M ( ) ( Bρ ) 1 Rq 1 3 g Kc I( q) slope: N B M Av intercept: 1 M slope: R g 3M (SLS, PS in toluene, Lechner 1993.)

26 Scattering from crystals SAXS Light scattering

27 Opals and colloidal crystals: The Bragg law: dsinθ = mλ

28 Crystal lattices Crystal: periodic lattice of equivalent points unit cell lattice plane lattice point diameter ~ 300 nm

29 Diffraction Position of particle p in unit cell number m 1, m, m 3 : rj = Rp + m1a1+ ma+ m3a3 a 1 a R p Substitute in structure factor (N=nM 1 M M 3 particles): N n M 1 M 1 M 1 iqr iqr im qa 1 3 j p im1qa 1 imqa e = e e e e j= 1 p= 1 m = 0 m = 0 m = M 1 m= 0 x m = 1 x 1 x M imqa 1 e = i 1 qa e

30 I s = = E s sin ( M1qa 1) sin ( Mqa ) sin M3qa 3 F ( q) sin ( qa 1) sin ( qa ) sin qa 3 MMM ( ) ( ) F ( q) n 1 i p = e qr n p= 1 Structure factor of a unit cell

31 100 The function sin sin ( Mqa ) 1 ( qa ) M = 10 M = 5 0 3π π π 0 π π 3π q a Only diffraction if: qa = π h 1 qa = π k qa = π l 3 h, k, l integers Laue conditions

32 Structure factor of fcc lattice Rp = xa + ya + za 1 3 a a 3 (x,y,z) = (0,0,0) ; (½,½,0) ; (½,0,½) ; (0,½,½) ( ) qr p = π hx + ky + lz F a 1 ( q) n 1 i p = e qr n p= iπ( k+ l) iπ( h+ l) iπ( h+ k) = + e + e + e 4 1 ( 1 ( 1) k+ l ( 1) h+ l ( 1) h+ k = ) 4 4 hkl all even or all odd = 0 otherwise systematic vanishings

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34 Scattering recap N i Es E 0 f j( ) e qr q j= 1 j weak scattering s ( ) ( ) ( )( q ) 0 q q 1+ cos I NI P S θ Form factor: P ( q) ( q) ( ) ( r ) iqr m() 1e dr f Vp = = f 0 m r dr V p ( () 1) Structure factor: S ( q) N N 1 i k j ( ) = e q r r N j= 1 k= 1

35 Dynamic light scattering Brownian particles laser speckle detector N () ( q) j= 1 iqr Es t = f j e (random variable) j () t pinhole

36 How to quantify fluctuating speckle? 150 photon counts (khz) Almost random time (s) intensity autocorrelation function I ( q, τ ) = ( q, ) ( q, + τ ) g I t I t gi (0) = I > g( τ ) = I

37 g I (τ) /τ (s -1 ) x10 14 x x x10 14 q (m - ) τ (s)

38 Brownian motion Einstein s argument: System of a particles with external force: K = Φ In equilibrium Boltzmann says: P ( r ) = P ( Φ kt ) 0 exp / v drift Probability density function ( ~ concentration) On the other hand: drift flux + diffusive flux = 0 (friction factor: γ) J = Pv PK drift γ ( P γ ) Φ D = 0 P 0 kt D0 = = γ kt 6πη a 0

39 Now remove the external force: System will return to equilibrium with flux: J D0 = P Continuity equation (conservation of particles): = t P D 0 P P t = J P(r) ( = 0) = δ ( r r ) P t < r > 0 r = 6D t 0 r 0 r t

40 Dilute dispersions (particles are statistically independent) N g f iq r r r r ( ) 4 ( τ) = exp ( 0) ( 0) ( τ) + ( τ) I j k l m jklm,,, = 1 Statistical independence means: ABCD = A B C D But ( iqrj( τ )) exp = 0 so most terms equal zero!

41 N g f iq r r r r ( ) 4 ( τ) = exp ( 0) ( 0) ( τ) + ( τ) I j k l m jklm,,, = 1 Surviving terms are: i. N terms j=k, l=m these terms give: 1 ii. iii. N N terms j=m, k=l, j k ( i q r j( τ) r j( ) ) i q r k ( τ) r k ( ) ( ) exp + 0 exp + 0 = 0 N N terms j=l, k=m, j k exp( i q r j( τ) r j( 0) ) exp i q r k ( τ) r k ( 0) ( ) particle displacement

42 I 4 4 ( τ) = + ( ) exp( q r ( )) j τ g N f N N f i I = Nf N ( τ) ( τ) ( 0) r = r r j j j gi I i ( ) j ( τ) = 1+ exp q r ( τ) Displacements are described by the diffusion equation: ( r, t) P = t P( r, t = 0) = δ ( r) ( r, ) D0 P t but we only need: exp ( iq r) = P( r, t) exp( iq r) d( r)

43 Fourier transformation of diffusion equation: ( q, t) P = t P( q, t) = 1 So, finally: qdp 0 ( q, t) ( q, ) exp( 0 ) P t = q D t ( ) = 1+ exp( 0 ) gi τ I q D t Stokes-Einstein : D0 kt 6πηR = (for spheres)

44 Determination of particle size 1.0 ( ) = 1+ exp( 0 ) gi τ I q D t g I (τ) /τ (s -1 ) x10 14 x x x10 14 q (m - ) τ (s)

45 Concentrated dispersions (particles are not statistically independent) If scattering volume >> correlation volume then the electric field is still a random variable: Siegert relation: electric field autocorrelation function I ( τ ) = + ( τ ) g I g * ( τ) = ( ) ( + τ) g E t E t E E s s N 1 = I exp q 0 N r r jk, = 1 ( i ( ) ( ) ) j k τ dynamic structure factor

46 Let s again write: ( c ) ( ) = 1+ exp (, ) gi τ I q D q t t Definition of D c (q,t) ( collective diffusion coeff. ) Interpretation: decay of density fluctuations π q

47 Time dependence: no longer pure exponential q Charged silica spheres Volume fraction: 10 % 0.1 % longer times slower motion particles collide

48 q-dependence: no longer pure q D0 Dc ( t = 0) Charged silica spheres S( q) D0 Dc ( t = 0) 1/D c (t=0) looks like S(q)

49 Self-diffusion Diffusion of tracer particles N 1 g I i ( τ) = exp ( 0) ( τ) ( iq r( τ )) ( s ()) = I exp ( q r r ) (Sum over tracers) E j k N jk, = 1 = I exp qd tt Tracers are independent: only terms j=k survive Definition of D s (t) no q-dependence (self-diffusion coefficient)

50 Measuring mean-square displacements short time long time