.. _polymer-excl-volume:

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

Polymer Excluded Volume model

========== =============================== ======= =============
Parameter  Description                     Units   Default value
========== =============================== ======= =============
scale      Scale factor or Volume fraction 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 put into an almost analytical form as follows
(Hammouda, 1993)

.. math::

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

and later recast as (for example, Hore, 2013; Hammouda & Kim, 2017)

.. 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 two analytic forms are equivalent. In the 1993 paper

.. math::

    \frac{1}{\nu U^{1/2\nu}}

has been factored out.

**SasView implements the 1993 expression**.

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::
    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.


**Source**

:download:`polymer_excl_volume.py <src/polymer_excl_volume.py>`

**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
#. M Hore et al, *Co-Nonsolvency of Poly(N-isopropylacrylamide) in Deuterated
   Water/Ethanol Mixtures*, *Macromolecules* 46 (2013) 7894-7901
#. B Hammouda & M-H Kim, *The empirical core-chain model*,
   *Journal of Molecular Liquids* 247 (2017) 434-440

**Authorship and Verification**

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