Flexible Cylinder Elliptical

This model calculates the form factor for a flexible cylinder with an elliptical cross section and a uniform scattering length density. The non-negligible diameter of the cylinder is included by accounting for excluded volume interactions within the walk of a single cylinder. **Inter-cylinder interactions are NOT provided for.**

The form factor is normalized by the particle volume such that

$$ P(q) = \text{scale} \left<F^2\right>/V + \text{background}
$$
where the averaging $\left<\ldots\right>$ is over all possible orientations of the flexible cylinder.

The 2D scattering intensity is the same as 1D, regardless of the orientation of the q vector which is defined as

$$ q = \sqrt{q_x^2 + q_y^2}
$$

Definitions

The function is calculated in a similar way to that for the `flexible-cylinder` model in reference [1] below using the author's "Method 3 With Excluded Volume".

The model is a parameterization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudo-continuous limit. See equations (13, 26-27) in the original reference for the details.

.. note::

There are several typos in the original reference that have been corrected by WRC [2]. Details of the corrections are in the reference below. Most notably

- Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$

- Equations (23) and (24) are incorrect; WRC has entered these into Mathematica and solved analytically. The results were then converted to code.

- Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of $max(a3*b(Rg^2)^{1/2},3)$

- The scattering function is negative for a range of parameter values and q-values that are experimentally accessible. A correction function has been added to give the proper behavior.

The chain of contour length, $L$, (the total length) can be described as a chain of some number of locally stiff segments of length $l_p$, the persistence length (the length along the cylinder over which the flexible cylinder can be considered a rigid rod). The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.

The cross section of the cylinder is elliptical, with minor radius $a$ . The major radius is larger, so of course, **the axis_ratio must be greater than one.** Simple constraints should be applied during curve fitting to maintain this inequality.

In the parameters, the $sld$ and $sld\_solvent$ represent the SLD of the chain/cylinder and solvent respectively. The *scale*, and the contrast are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting.

**This is a model with complex behaviour depending on the ratio of** $L/b$ **and the reader is strongly encouraged to read reference [1] before use.**

References

J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612

Correction of the formula can be found in

W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548

Authorship and Verification

**Author:**
**Last Modified by:** Richard Heenan **Date:** December, 2016
**Last Reviewed by:** Steve King **Date:** March 26, 2019