.. _be-polyelectrolyte:

be_polyelectrolyte
=======================================================

Polyelectrolyte with the RPA expression derived by Borue and Erukhimovich

===================== ================================ =========== =============
Parameter             Description                      Units       Default value
===================== ================================ =========== =============
scale                 Source intensity                 None                    1
background            Source background                |cm^-1|             0.001
contrast_factor       Contrast factor of the polymer   barns                  10
bjerrum_length        Bjerrum length                   |Ang|                 7.1
virial_param          Virial parameter                 |Ang^3|/mol            12
monomer_length        Monomer length                   |Ang|                  10
salt_concentration    Concentration of monovalent salt mol/L                   0
ionization_degree     Degree of ionization             None                 0.05
polymer_concentration Polymer molar concentration      mol/L                 0.7
===================== ================================ =========== =============

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


**Definition**
This model calculates the structure factor of a polyelectrolyte solution with
the RPA expression derived by Borue and Erukhimovich\ [#Borue]_.  Note however
that the fitting procedure here does not follow the notation in that reference
as 's' and 't' are **not** decoupled. Instead the scattering intensity $I(q)$
is calculated as

.. math::

    I(q) = K\frac{q^2+k^2}{4\pi L_b\alpha ^2}
    \frac{1}{1+r_{0}^2(q^2+k^2)(q^2-12hC_a/b^2)} + background

    k^2 = 4\pi L_b(2C_s + \alpha C_a)

    r_{0}^2 = \frac{1}{\alpha \sqrt{C_a} \left( b/\sqrt{48\pi L_b}\right)}

where 

$K$ is the contrast factor for the polymer which is defined differently than in
other models and is given in barns where $1 barn = 10^{-24} cm^2$.  $K$ is
defined as:

.. math::

    K = a^2

    a = b_p - (v_p/v_s) b_s
    
where $b_p$ and $b_s$ are sum of the scattering lengths of the atoms
constituting the monomer of the polymer and the sum of the scattering lengths
of the atoms constituting the solvent molecules respectively, and $v_p$ and
$v_s$ are the partial molar volume of the polymer and the solvent respectively

$L_b$ is the Bjerrum length(|Ang|) - **Note:** This parameter needs to be
kept constant for a given solvent and temperature! 

$h$ is the virial parameter (|Ang^3|/mol) - **Note:** See [#Borue]_ for the
correct interpretation of this parameter.  It incorporates second and third
virial coefficients and can be Negative.

$b$ is the monomer length(|Ang|), $C_s$ is the concentration of monovalent
salt(mol/L), $\alpha$ is the ionization degree (ionization degree : ratio of
charged monomers  to total number of monomers), $C_a$ is the polymer molar
concentration(mol/L), and $background$ is the incoherent background.

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

.. math::

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


.. figure:: img/be_polyelectrolyte_autogenfig.png

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

**References**

.. [#Borue] V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240
.. [#] J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545
.. [#] A Moussaid, F Schosseler, J P Munch, S Candau, *J. Journal de Physique
   II France*, 3 (1993) 573
.. [#] E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Kienzle **Date:** July 24, 2016
* **Last Reviewed by:** Paul Butler and Richard Heenan **Date:** 
  October 07, 2016