Model Edwards

Arguments:

  • type (string) - Name of the model to use (Edwards)

  • 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

  • C or rho0 (float) - System density \(C = \rho_0 R_g^3 / N_{ref}\). Only one can be specified.

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

Example (python):

fts.model(Nref=1.0,bref=1.0,C=4.0,B=2.0,type='Edwards')

Example (json):

"model": {
  "Nref": 1.0,
  "bref": 1.0,
  "C": 4.0,
  "B": 2.0,
  "initfields": {
    "mu": {
      "type": "random",
      "mean": 1.0,
      "stdev": 0.1
    }
  },
  "type": "Edwards"
},


"model": {
  "type": "Edwards",
  "Nref": 1.0,
  "bref": 1.0,
  "nu0": 30,
  "rho0": 10,
  "initfields": {
      "mu": {
      "seed": 0,
      "type": "random"
    }
  }
},

Model Edwards Formalism

References: Delaney2016, Villet2014

The model

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

where

\[\beta U_{excl} = \frac{\nu}{2} \int d\pmb{r} \bar{\rho}(\pmb{r})^2 + self interaction\]

where \(\bar{\rho}\) is the smeared density defined as \(\bar{\rho} = \Gamma * \hat{\rho}(\pmb{r}) = \int d\pmb{r} \Gamma (\pmb{r} - \pmb{r}') \hat{\rho}(\pmb{r}')\), where \(\Gamma(\pmb{r}) = \frac{1}{(2\pi a^2)^{3/2}} \exp{\left(-\frac{\lvert\pmb{r}\rvert}{2 a^2} \right)}\) is the smearing function and \(\hat{\rho}\) is the microscopic density.

The field theory derived for this model is

\[\mathcal{Z} = Z_0 \int \mathcal{D} w \exp{\left(-H[w]\right)}\]

where

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

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

\[H[\mu] = \frac{1}{2 B} \int d \pmb{x} [\mu(\pmb{x})]^2 - C \tilde{V} \sum_p^P \frac{\phi_p}{\alpha_p} \ln Q_p[\Gamma * \mu]\]

The density operator \(\varphi\) is (for a continuous chain)

\[\varphi (\pmb{x} ; [\mu]) = \sum_{p=1}^P \frac {\phi_p}{Q_p \alpha_p} \Gamma * \left( \int_{s} ds q_p(x,s) q^\dagger_p(x,s) \right)\]

The force is

\[\frac{\delta H[\mu]}{\delta \mu(\pmb{x})} = - \frac{1}{B} \mu(\pmb{x}) + \varphi(\pmb{x})\]

For a homogeneous saddle point \(\varphi(\pmb{x}) = C\) and the saddle point is located at

\[\mu(\pmb{x}) = BC\]