Model Edwards Charge¶

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 Initialize Model Fields

charge_ref (float) - Reference charge density

B or u0 (float) - Excluded volume parameter \(B = u0 N_{ref}^2 / R_g^3\). Only one can be specified.

C or rho0 (float) - System density \(C = \rho_0 R_g^3 / N_{ref}\). 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

\[\beta U(\pmb{r}) = \beta U_{bond}(\pmb{r}) + \beta U_{int}\]

where

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

where

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

The field theory derived for this model is

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

where

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

By introducing the following variables \(B=\frac{\nu N^2}{R_{g0}^3}\), \(C=\frac{\rho_0 R_{g0}}{N}\), \(E=\frac{4\pi\ell_B\sigma^2N^2}{R_g}\), \(\tilde{V} = \frac{V}{R_{g0}^3}\), \(\pmb{x} = \frac{\pmb{r}}{R_{g0}}\), \(n_p = \frac{C \tilde{V} \phi_p}{\alpha_p}\), and \(\mu = i w(\pmb{r}) N\) 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?

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