.. _polymer-excl-volume:

polymer_excl_volume
=======================================================

Polymer Excluded Volume model

========== ================== ======= =============
Parameter  Description        Units   Default value
========== ================== ======= =============
scale      Source intensity   None                1
background Source background  |cm^-1|         0.001
rg         Radius of Gyration |Ang|              60
porod_exp  Porod exponent     None                3
========== ================== ======= =============

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


This model describes the scattering from polymer chains subject to excluded
volume effects and has been used as a template for describing mass fractals.

**Definition**

The form factor was originally presented in the following integral form
(Benoit, 1957)

.. math::

    P(Q)=2\int_0^{1}dx(1-x)exp\left[-\frac{Q^2a^2}{6}n^{2v}x^{2v}\right]

where $\nu$ is the excluded volume parameter
(which is related to the Porod exponent $m$ as $\nu=1/m$ ),
$a$ is the statistical segment length of the polymer chain,
and $n$ is the degree of polymerization.
This integral was later put into an almost analytical form as follows
(Hammouda, 1993)

.. math::

    P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) -
    \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right)

where $\gamma(x,U)$ is the incomplete gamma function

.. math::

    \gamma(x,U)=\int_0^{U}dt\ exp(-t)t^{x-1}

and the variable $U$ is given in terms of the scattering vector $Q$ as

.. math::

    U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6}

The square of the radius-of-gyration is defined as

.. math::

    R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)}

Note that this model applies only in the mass fractal range (ie, $5/3<=m<=3$ )
and **does not apply** to surface fractals ( $3<m<=4$ ).
It also does not reproduce the rigid rod limit (m=1) because it assumes chain
flexibility from the outset. It may cover a portion of the semi-flexible chain
range ( $1<m<5/3$ ).

A low-Q expansion yields the Guinier form and a high-Q expansion yields the
Porod form which is given by

.. math::

    P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left(
    \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left(
    \frac{1}{\nu}\right)

Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function.

The asymptotic limit is dominated by the first term

.. math::

    P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) =
    \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2}
    \Gamma (m/2)

The special case when $\nu=0.5$ (or $m=1/\nu=2$ ) corresponds to Gaussian chains for
which the form factor is given by the familiar Debye function.

.. math::

    P(Q) = \frac{2}{Q^4R_{g}^4} \left[exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right]

For 2D data: The 2D scattering intensity is calculated in the same way as 1D,
where the $q$ vector is defined as

.. math::

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



.. figure:: img/polymer_excl_volume_autogenfig.png

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

**References**

H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247

B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview,
Advances in Polym. Sci.* 106(1993) 87-133


