# Flexible Cylinder Elliptical

## Description:

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. 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 calculated in a similar way to that for the flexible_cylinder model from the reference given 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. 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 $q0 = max(a3/sqrt(RgSquare),3)$ instead of $max(a3*b/sqrt(RgSquare),3)$

- 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 (parameter 5) must be greater than one.** Simple constraints should be applied during curve fitting to maintain this inequality.

The returned value is in units of $cm^{-1}$, on absolute scale.

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.

**No inter-cylinder interference effects are included in this calculation.**

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

## Details:

 Created By sasview Uploaded Sept. 7, 2017, 3:56 p.m. Category Cylinder Score 0 Verified Verified by SasView Team on 07 Sep 2017 In Library This model is included in the SasView library by default Files flexible_cylinder_elliptical.py flexible_cylinder_elliptical.c