# Calculation of g(r) Functions from Small-Angle Scattering Data by ...

Calculation of g(r) Functions from Small-Angle Scattering Data by ...

Calculation of g(r) Functions from

Small-Angle Scattering Data by

Means of Indirect Fourier

Transformation

Gerhard FRITZ

Institute of Chemistry

Karl-Franzens University Graz, Austria

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 1/22

Introduction

Scattering generally for particle size, shape, internal

structure. Typically models or model-free description of

size and shape, but always models for interaction effects.

Sometimes, however, interest in interaction effects.

Interaction effects can be described by pair correlation

function g(r): Probability to find a neighbour at distance r

relative mean particle density.

New technique largely model free for g(r) function. Needs

model assumption for particle size and shape.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 2/22

Theory

Intensity product of form factor P(q) and structure factor

S(q)

I(q) = n∆ρ 2 mM 2 P(q)S(q) (1)

S(q) is the Fourier transform of g(r)

S(q) = 1 + 4πn (g(r) − 1)r 2sin(qr)

dr (2)

qr

0

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 3/22

IFT Basics I

g(r) can be represented as a sum of cubic B-splines:

(g(r) − 1)r 2 =

N

ciϕi(r) (3)

ψi(q) = 4π ϕi(r) sin(qr)

dr (4)

qr

0

S(q) = 1 + n

i=1

N

ciψi(q) (5)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 4/22

i=1

IFT Basics II

The intensity I(q) is therefore

I(q) = n∆ρ 2 mM 2 P(q)

1 + n

N

ciψi(q)

i=1

Which is a simple linear equation system with

ξi(q) = P(q)ψi(q). It can be solved for the coefficients

f0 = n∆ρ 2 mM 2 and fi = n 2 ∆ρ 2 mM 2 ci

I(q) = f0P(q) +

(6)

N

fiξi(q) (7)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 5/22

i=1

Forcing g(r) = 0 for Small r-values I

Equation system more stable if g(r) = 0 for r < l.

ζ(r) =

−r 2 if r ≤ l;

0 if r > l.

and defining the splines only in the regime r > l leads to

(g(r) − 1)r 2 = ζ(r) +

(8)

N

ciϕi(r) (9)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 6/22

i=1

Forcing g(r) = 0 for Small r-values II

The equivalent in reciprocal space is

I(q) = f0 (P(q) + nP(q)κ(q)) +

N

fiξi(q) (10)

i=1

With ξ0(q) = (P(q) + nP(q)κ(q)) one gets the linear

equation system

I(q) =

N

fiξi(q) (11)

i=0

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 7/22

Instrumental broadening effects are linear operations. It is

sufficient to apply convolutions with beam profiles to the

Fourier transformed basis functions:

Iexp(q) ≈

Ĩ(q) =

N

i=0

fi ˜ ξi(q) (12)

The equation system can be solved, but a stabilisation is

taken into account.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 8/22

Overview Method

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 9/22

Test on Simulated Data

slit smearing, statistical

noise

Charged spheres, 50

electron charges per

particle in water at

298K, volume

fraction 0.1

Hard spheres, volu-

me fraction 0.01.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 10/22

Resulting g(r) Function

Good estimation for g(r), even if P(q) is not perfect.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 11/22

Effect of g(r) = 0 Condition

Forcing g(r) = 0 necessary for small r, but only minor

effect on large r.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 12/22

Experimental Test

PEO-PPO-PEO triblock copolymer P94. Forms spherical

micelles at 40 ◦ C. Structures can be described by effective

hard sphere model. Effective volume fraction 2.05 times

weight fraction in H2O

SANS experiments on solutions from 0.33% up to 15%

(Corresponding to effective volume fractions of 0.7% up to

31.5%).

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 13/22

Determination of P(q)

P(q) estimated from dilute( 0.33%) solution.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 14/22

Determination of n

n estimated from particle diameter (from P(q)), and

effective volume fraction determined by viscosimetry.

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 15/22

Approximations

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 16/22

Resulting g(r) functions

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 17/22

Resulting g(r) functions

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 17/22

Conclusions

Pair correlation function g(r) can be determined

without model for interactions.

Simulated and experimental data show viability of

approach

No deconvolution of scattering curve and beam profiles

necessary

No extrapolations needed, basis functions are defined

analytically

G. Fritz, J. Chem. Phys. (2006) 124 214707(1-6)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 18/22

Solving the Equation System I

Expansion coefficients:

dν =

Co-variance matrix:

Bµν =

qmax

qmin

qmax

qmin

ψν(q)Iexp(q)

σ2 dq (13)

(q)

ψµ(q)ψν(q)

σ2 dq (14)

(q)

Bf = d (15)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 19/22

Solving the Equation System II

Main condition: Minimise mean deviation MD

qmax Iexp(q) −

MD =

Ĩ(q)

2 σ2 (q)

qmin

dq (16)

Side condition: Minimise difference between neighbouring

splines:

Nf ′ =

N−1

i=1

(fi+1 − fi) 2

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 20/22

(17)

Solving the Equation System III

K =

0 0 0 0 · · · 0 0

0 1 −1 0 · · · 0 0

0 −1 2 −1 · · · 0 0

0 0 −1 2 · · · 0 0

. . . .

. .. . .

0 0 0 0 · · · 2 −1

0 0 0 0 · · · −1 1

(18)

(B + λK)f = d (19)

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 21/22

Determining l

Calculation of g(r) Functions from Small-Angle Scattering Data by Means of Indirect Fourier Transformation – p. 22/22

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