flexible_cylinder

Flexible cylinder where the form factor is normalized by the volume of the cylinder.

Parameter	Description	Units	Default value
scale	Scale factor or Volume fraction	None	1
background	Source background	cm-1	0.001
length	Length of the flexible cylinder	Å	1000
kuhn_length	Kuhn length of the flexible cylinder	Å	100
radius	Radius of the flexible cylinder	Å	20
sld	Cylinder scattering length density	10-6Å-2	1
sld_solvent	Solvent scattering length density	10-6Å-2	6.3

The returned value is scaled to units of cm^-1 sr^-1, absolute scale.

This model provides the form factor, $P(q)$, for a flexible cylinder where the form factor is normalized by the volume of the cylinder. Inter-cylinder interactions are NOT provided for.

$$P(q) = \text{scale} \left<F^2\right>/V + \text{background}$$

where the averaging $\left<\ldots\right>$ is applied only for the 1D calculation

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 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.

In the parameters, the sld and sld_solvent represent the SLD of the cylinder and solvent respectively.

Our model uses the form factor calculations in reference [1] as implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). This states:

‘Method 3 With Excluded Volume’ is used. The model is a parametrization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous 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 Chen et al (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.

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. In particular, the cylinder form factor used as the limiting case for long narrow rods will not be exactly correct for short and/or wide rods.

Fig. 29 1D plot corresponding to the default parameters of the model.

Source

flexible_cylinder.py $\ \star\ $ flexible_cylinder.c $\ \star\ $ wrc_cyl.c $\ \star\ $ sas_J1.c $\ \star\ $ polevl.c

References

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

2. 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:

• Last Reviewed by: Steve King Date: March 6, 2020