Model Edwards Charge
=======================

.. warning:
  
  This model is in beta. It isn't recommended that you use it unless you know what you're doing.


Arguments:

  * `type` (string) - Name of the model to use (`MeltChiAB`)
  * `Nref` (float) - Reference degree of polymerization 
  * `bref` (float) - Reference statistical segment length
  * `initfields` (json object) - Describes how to initialize fields within model. See :ref:`label_init_model_fields`
  * `charge_ref` (float) - Reference charge density
  * `B` or `u0` (float) - Excluded volume parameter :math:`B = u0 N_{ref}^2 / R_g^3`. Only one can be specified.
  * `C` or `rho0` (float) - System density :math:`C = \rho_0 R_g^3 / N_{ref}`. Only one can be specified.
  * `E` or `lB` (float) - Electrostatic interaction strength. Only one can be specified.

Example (python)::

  import openfts
  fts = openfts.OpenFTS()
  ...
  fts.model(Nref=1.0,bref=1.0,charge_ref=1.0, C=6.0,B=0.3,E=64000.0,type='EdwardsCharge')
  ...

Example (json) ::

  "model": {
    "B": 0.3,
    "C": 6.0,
    "E": 64000.0,
    "Nref": 1.0,
    "bref": 1.0,
    "charge_ref": 1.0,
    "initfields": {
      "mu": {
        "mean": 1.0, "stdev": 0.1, "type": "random"
      },  
      "phi": {
        "mean": 1.0, "stdev": 0.1, "type": "random"
      }   
    },  
    "type": "EdwardsCharge"
  },

Model Edwards Charge Formalism
-------------------------------

References: Delaney2017, Riggleman2012, Lee2008

The model

.. math::
  \beta U(\pmb{r}) = \beta U_{bond}(\pmb{r}) + \beta U_{int}

where 

.. math ::
  \beta U_{int} = \frac{\nu}{2} \int d\pmb{r} \bar{\rho}(\pmb{r})^2 + \frac{\ell_B}{2} \int d\pmb{r} \int d\pmb{r'} \frac{\bar{\rho}_e(\pmb{r}) \bar{\rho}_e(\pmb{r'})}{|\pmb{r} - \pmb{r'}|} + self interaction

where 

* :math:`\hat{\rho}` is the microscopic  density defined as :math:`\hat{\rho} = \sum_{i=1}^n \int ds \delta(\pmb{r} - \pmb{r'}_(s))`
* :math:`\hat{\rho}_e` is the microscopic charge density defined as :math:`\hat{\rho}_e = \sum_{i=1}^n \int ds \sigma_i(s) \delta(\pmb{r} - \pmb{r'}_(s))`
* :math:`\sigma_i(s)` is the charge valence at position `s` of polymer `i`
* :math:`\Gamma(\pmb{r}) = \frac{1}{(2\pi a^2)^{3/2}} \exp{\left(-\frac{\lvert\pmb{r}\rvert}{2 a^2} \right)}` is the smearing function 
* :math:`\bar{\rho}` is the smeared density defined as :math:`\bar{\rho} = \Gamma * \hat{\rho}(\pmb{r}) = \int d\pmb{r} \Gamma (\pmb{r} - \pmb{r}') \hat{\rho}(\pmb{r}')`. :math:`\bar{\rho}_e` is defined similarly.

The field theory derived for this model is 

.. math:: 
  \mathcal{Z} = \frac{Z_0}{Z_w Z_\varphi} \int \mathcal{D} w  \int \mathcal{D} \varphi \exp{\left(-H[w,\varphi]\right)}

where 

.. math::
  H[w] = \frac{1}{2 \nu} \int d \pmb{r}  [w(\pmb{r})]^2 - \sum_p^P n_p \ln Q_p[i \Gamma * w]. FIXME missing charge term

By introducing the following variables :math:`B=\frac{\nu N^2}{R_{g0}^3}`, :math:`C=\frac{\rho_0 R_{g0}}{N}`, :math:`E=\frac{4\pi\ell_B\sigma^2N^2}{R_g}`,  :math:`\tilde{V} = \frac{V}{R_{g0}^3}`, :math:`\pmb{x} = \frac{\pmb{r}}{R_{g0}}`, :math:`n_p = \frac{C \tilde{V} \phi_p}{\alpha_p}`, and :math:`\mu = i w(\pmb{r}) N` TODO: rescale phi field, (take a look at Duchs2014 if theres variables that were not defined), the field theory becomes

TODO: lots of these rescaling conventions are recycled from model to model. Can I combine into one place?

.. math::
  H[\mu] = \frac{1}{2 B} \int d \pmb{x}  [\mu(\pmb{x})]^2 + \frac{1}{2 E} \int d \pmb{x}  | \nabla \varphi(\pmb{x})| ^2  - C \tilde{V} \sum_p^P \frac{\phi_p}{\alpha_p} \ln Q_p[\varphi, \mu; a]