Steve Kline

NCNR Summer School

Neutron Small Angle Scattering andReflectometry from Submicron Structures

June 5 - 9 2000

SANS from Concentrated Dispersions

Outline

How dilute is “dilute”?Effect of concentration on Rg determination

The structure factor and the radial distribution functionWhat information is in S(q) and how do I get it?

Data fittingThe real world of polydispersity

ApproximationsExact methods

Example I: Colloidal silicaSize polydispersity

Example II: Surfactant micellesCharge interactions

Summary

SANS from Dilute Systems

np = number density of particlesP(q) = Form factor (Intraparticle structure)

Guinier Approximation:If the scatterers are “sufficiently dilute”:

Linear Plot:

For a uniform sphere:

P(q)nI(q) p=

( )3RqI(0)expI(q) 2g

2−≈

3RqlnI(0)lnI(q) 2g

2−=

22g R

53R =

How dilute is “sufficiently dilute”?

Guinier Plot for Hard SpheresR = 100 Å

-2

-1

0

1

2

3

4

5

6

0 100 2 10-5 4 10-5 6 10-5 8 10-5 1 10-4

ln I(

q)

q2 (Å-2)

φ = 10 −1

φ = 10 −2

φ = 10 −3

φ = 10 −4

How dilute is “sufficiently dilute”?

Guinier Plot for Hard SpheresR = 100 Å

is “dilute” for spheres with hard sphere interactions

3500100.010-4

160099.710-3

75093.310-2

3505.410-1

Average Separation (Å)Rsphere (Å)Volume Fraction

310−≤φ

How dilute is “sufficiently dilute”?

Guinier Plot for Charged Spheres R = 100 Å, Z = 50, [salt] = 10-3, κ-1 = 100 Å

-6

-5

-4

-3

-2

-1

0

1

2

0 100 2 10-5 4 10-5 6 10-5 8 10-5 1 10-4

ln I(

q)

q2 (Å-2)

φ = 10 −3

φ = 10 −4

φ = 10 −5

φ = 10 −6

How dilute is “sufficiently dilute”?

Guinier Plot for Charged Spheres R = 100 Å, Z = 50, [salt] = 10-3, κ-1 = 100 Å

is “dilute” for spheres with screened Coulomb interactions

16000100.010-6

7500 100.010-5

3500 96.2 10-4

160041.210-3

Average Separation (Å)Rsphere (Å)Volume Fraction

310−≤φ

Interparticle Interference Effects

Scattered Intensity:

Rk

Rj

Rk - Rj

( ) ( ) ( ) ( ) ( )∑ ∑∑=

≠=

−⋅

=+=

p pjk

p N

1k

N

kj1j

rrqi*jk

N

1k

2k eqfqf

V1qf

V1q

dΩdΣ

Interparticle Interference Effects

Scattering Amplitude (Intraparticle):

the “Form Factor”

( ) ( )[ ]∫k particle

rqisolvkk rdeρrρqf ⋅−=

( )2k qfP(q) =

The Structure Factor

For monodisperse spheres:

If isotropic, we can average over orientation:

Note:– S(q) is proportional to the number density of particles– S(q) depends on g(r), the pair correlation function

+= ∑ ∑

=≠=

−⋅p p

jk

N

1k

N

kj1j

)rr(qi2p e 1f(q)n)q(

dΩdΣ

dΣdΩ

(q ) = npP(q) ⋅S(q )

S(q ) = S(q) = 1+ 4πn p g(r) − 1[ ]sinqrqr

r2dr0

∞⌠ ⌡

The Pair Correlation Function

npg(r) is a “local” density of particles

Spatial arrangement set by interparticleinteractions and indirect interactions

r/d1

g(r)1

0

2

g(r)1

0

2

g(r)1

0

2

2

S(q) and Statistical Thermodynamics

The form of the interparticle potential has a great effect on the low q value of S(q)

The low q limit is proportional to the osmotic compressibility

Attractive interactions ⇒ more compressibleRepulsive interactions ⇒ less compressible

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

S(q)

q*diameter

Attractive Square Well

Hard Sphere

Coulomb Repulsion

==πnkT0)S(q

∂∂

S(q) Reflected in the Low-q Intensity

The form of the interparticle potential has a great effect on the low q value of S(q) Example of charged spheres:

development of “interaction peak”change in low-q slope and I(0)

Must fit model to dataknow P(q)?calculate S(q)?

P(q)S(q)nI(q) p=

10-5

10-4

10-3

10-2

10-1

100

101

102

103

10-3 10-2 10-1

Inte

nsity

(arb

itrar

y)

q (Å-1)

I(q)

P(q), scaled

S(q)

Calculation of S(q)

Ornstein Zernicke Equation:

c(r) = direct correlation functionIntegral = all indirect interactions

A second relation is necessary to relate c(r) and g(r)Percus-Yevick Closure - an approximation

correct closure gives correct resultsin general a difficult problem

( )∫ xh(x)dxrcnc(r)1g(r)h(r) −+=−=

( ) ( ) ( )[ ]rβue1rgrc −=

S(q ) = S(q) = 1+ 4πn p g(r) − 1[ ]sinqrqr

r2dr0

∞⌠ ⌡

Real Life Complications: Polydispersity

Real colloidal systems are not monodisperse

For p-components:

# particles

radius1 2 3 p...

dΣdΩ

q( ) = nin j( )12 fi q( )f j q( )Sij q( )

j

p∑

i

p∑

Real Life Complications: Polydispersity

Partial structure factors

Set of O-Z equations

Would really like

( ) ( ) ( )[ ]⌡⌠ −+= drr

qrsinqr1rgnn4πδqS 2

ij21

jiijij

( ) ( ) ( ) ( )∫∑ −+=p

kkjikkijij xdxhxrcnrcrh

( ) ( ) ( )qS'qPnqdΩdΣ

p=

Two Approximations:

1. Average Structure Factor

Good for moderate polydispersity 2. “Beta” Decoupling

Good only at low polydispersityUseful for non-spherical particles

( ) ( )qS(q)fnqdΩdΣ 2

p ⋅=

( ) ( )qS(q)fnqdΩdΣ 2

p ′⋅=

( ) ( )( ) ( )[ ] ( ) ( )[ ]1qSqβ11qSqf

qf1qS' 2

2

−+=−

+=

Performance of Polydisperse Approximations

Model calculations for polydisperse hard spheres

Form factor oscillations are damped outBoth approximations work well

10-1

100

101

102

103

10-3 10-2 10-1

I(q)

q (Å-1)

R = 100 Å10% polydispersityφ = 0.10

High Polydispersity, High Concentration

Model calculations for polydisperse hard spheres

Form factor oscillations disappearBoth approximations fail at low q

10-1

100

101

102

103

104

10-3 10-2 10-1

I(q)

q (Å-1)

Average S(q)

Beta Decoupling

R = 100 Å30% polydispersityφ = 0.30

Stretching the Approximation Too Far

Model calculations for polydisperse hard spheres

“Fitting” an approximate model gives incorrect resultsMore exact calculations are necessary for high concentration or highpolydispersity

10-1

100

101

102

103

10-3 10-2 10-1

Beta Decoupling15% polydispersity

I(q)

q (Å-1)

Average S(q)37% polydispersity

R = 100 Å30% polydispersityφ = 0.30

Example I

Determining the Size and Polydispersity of Colloidal Silica

Experimental systemSpherical SiO2 particles in aqueous solvent

Charge stabilized with negative surface charge

+

-+-

+-

+-

+-+-

+

-

+

-

+ -

Example I

For modeling of polydisperse charged spheres:Must know ionic strength to calculate screeningFit the particle charge, ZMust use an approximation for P(q)S(q)

Screen the electrostatic interactions by adding salt

Model with the analytical solution for polydisperse hard spheres

Example I

Polydisperse hard spheres - analytic solution

Known parameters∆ρ = 1.3 x 1010 cm-2

φ = 0.096[NaCl] = 0.1 M (to give hard sphere interactions)

Fitted parametersR = 115 ÅσR / R = 0.17

10-4

10-3

10-2

10-1

100

101

102

10-3 10-2 10-1 100

I(q)

(cm

-1)

q (Å-1)

MonodisperseSpheres

(R = 103 Å)

PolydisperseSpheres

Example II

Aggregation and Charge of Surfactant Micelles

Sodium dodecyl sulfateCH3(CH2)11SO4

- Na+

ρcore

ρshellρsolv

Rcore

Rshell+

w

___

_

+

+

w w

w

w w

w

w w

w

w

w

+

Example II

Micelles form in solution above the CMC of the surfactant

Try to measure the form factor:can’t dilute - only monomeric surfactant upon dilutioncan’t add salt - would change structure as well as interactions

Must fit P(q) and S(q) simultaneouslyP(q) ⇒ Aggregation numberS(q) ⇒ U(r) = fδ, κ(δ), r

Example II

SANS and Modeling Results (q)SP(q)nI(q) ′⋅=

10-2

10-1

100

101

10-2 10-1

I(q)

(cm

-1)

q (Å-1)

SDS = 0.01 M 0.05 M 0.20 M 0.30 M 0.40 M

0.20860.01

0.22990.40

0.25960.30

0.25910.20

0.21860.05

IonizationAgg #[SDS] (M)

Summary

Definition of “dilute” is relativeDetermination of intraparticle structureVery useful, model independent information

When not dilute:Determination of interparticle structureInformation about interparticle interactionsPolydispersity approximationsExact methods

Other Concentrated Systems:Non-spherical particlesRod-like micellesMuch more complex analysisCorrelation between position and orientation