**Email: **robert.rambo@diamond.ac.uk

**Tel: **+44 (0)1235 56 7675

The *CORFUNC* program performs correlation function analysis of one-dimensional SAXS or SANS patterns, __or__ generates a *model-independent* volume fraction profile from a one-dimensional SANS pattern from an adsorbed layer.

*Contact: Dr Stephen King, s.m.king@stfc.ac.uk, www.isis.stfc.ac.uk/instruments/small-angle-scattering.html, manual*

Only the Java versions of the program currently perform the volume fraction profile analysis.

The correlation function [1] [2] [3] [4] [5] [6] is the Fourier transform of the scattering curve and may be analysed in terms of an ideal lamellar morphology [7] to obtain structural parameters describing the sample. This is illustrated graphically in Figure 1 and Figure 2 below:

Figure 1 (upper) :One-dimensional SAXS data,(lower) :One-dimensional correlation function calculated from the SAXS data usingCORFUNC.

Figure 2:Schematic representation of the one-dimensional correlation function showing the parameters that may be obtained:Long period = Lp

Average hard block thickness = Lc

Average core thickness = D0

Average interface thickness = Dtr

Average soft block thickness = La = Lp - Lc

Local crystallinity = φ1 = Lc / Lp

Bulk crystallinity = φ = Γmin / (Γmin + Γ*)

Polydispersity = Γmin / Γmax

Electron density contrast = (ρc - ρa)² = (Δρ)² = Q Γ*/ (φ (1 - φ))

Specific inner surface = Os = 2φ / Lc

Non-ideality = (Lp - Lp*)² / Lp ²where Q is the Invariant.

The analysis is performed in 3 stages:

(a)

Extrapolationof the experimental scattering curve to q = 0 and q = infinity,

(b)Fourier transformof the extrapolated data to give the 1-D correlation function, and

(c)Interpretationof the 1-D correlation function based on ideal lamellar morphology.

The volume fraction profile is a distribution function that describes how the number density of an adsorbed species (typically a surfactant or polymer) varies with distance from an interface (such as a particle surface) [8] [9]. It may be analysed [10] [11] [12] to obtain parameters describing the extent (thickness) of the adsorbed layer, the amount of material adsorbed, and what proportion is "bound" at the interface. It is obtained by a Hilbert transformation of the scattering data.

This is illustrated graphically in Figure 3 and Figure 4 below:

Figure 3 (upper) :One-dimensional SANS data,(lower) :Volume fraction profile generated from the SANS data usingCORFUNC.

Figure 4:Representative volume fraction profile types showing the parameters that may be obtained:Adsorbed amount = Γ

Bound fraction = <p>

Hydrodynamic layer thickness (or extent) =

Second moment thickness =

Possible other parameters that may be derived include:Surface coverage =

Distance between grafting points =D

The analysis is performed in 3 stages:

(a)

Extrapolationof the experimental scattering curve to q = 0 and q = infinity,

(b)Hilbert transformof the extrapolated data to give a segment density distribution, and

(c)Normalisationof the segment density profile to give the volume fraction profile.

Also see the original article about *CORFUNC* in *Fibre Diffraction Review, (1994) 3, 25-29 or search STFC ePublication Archive for other FDR archive material.*

* *

- Strobl, G. R. and Schneider, M. J.,
*Polym. Sci.*(1980)**18**, 1343-1359. - Balta Calleja, F. J. and Vonk, C. G.,
*X-ray Scattering of Synthetic Polymers*, Elsevier, Amsterdam 1989, 247-257. - Balta Calleja, F. J. and Vonk, C. G.,
*X-ray Scattering of Synthetic Polymers*, Elsevier, Amsterdam 1989, 257-261. - Koberstein, J. and Stein R. J.,
*Polym. Sci. Phys. Ed.*(1983)**21**, 2181-2200. - Press, W. H.
*et al.*,*Numerical Recipes: the Art of Scientific Computing*, Cambridge University Press 1986. - Balta Calleja, F. J. and Vonk, C. G.,
*X-ray Scattering of Synthetic Polymers*, Elsevier, Amsterdam 1989, 261-288. - Glatter, O. and Kratky, O.,
*Small Angle X-ray Scattering*, Academic Press Inc., London Ltd. 1982, 433-466. - Crowley, T. L., D.Phil Thesis, University of Oxford, 1984.
- Cosgrove, T.
*et al.*,*Faraday Symp. of the Chem. Soc., No.16*, Royal Society of Chemistry, London 1981 - Cosgrove, T.,
*J. Chem. Soc. Faraday Trans.*(1990)**86**, 1323-1332 - King, S. M.
*et al., Applications of Neutron Scattering to Soft Condensed Matter*, Gordon & Breach, Amsterdam 2000, 77-105 - King, S. and Flannery, D.,
*Fibre Diffraction Review*(2005)**13**, 19-22

* *

* *

*The ideal lamellar model consists of alternating crystalline and amorphous lamellae that are placed in stacks of dimensions that are large enough not to affect the small angle scattering [7]. The model is assumed to be isotropic i.e. no preferred orientation is accounted for. The correlation function of such a system varies in one direction only, perpendicular to the lamellae. The variation of the correlation function in this direction is described by the one-dimensional correlation function.*

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