Model Edwards AB
Arguments:
nu0 (float) - Excluded volume between all species \(\nu_0\)
delta_nuAB (float) - Extra excluded volume between A and B species \(\Delta \nu_{AB}\)
rho0 (float) - System density \(\rho_0\)
Example (python)
TODO
Example (json)
"model": {
"type": "EdwardsAB",
"Nref": 1.0,
"bref": 1.0,
"nu0": 40,
"delta_nuAB": 10,
"rho0": 3,
"initfields": {
"mu_minus": {
"center0": [ 0 ],
"height": 1.0,
"ngaussian": 1,
"type": "gaussians",
"width": 0.1
},
"mu_plus": {
"seed": 0,
"type": "random"
}
}
},
Model Edwards AB Formalism
References: Park2020, Delaney2016
This is an extension of the Edwards model to include two species (A and B). This is very similar to the non-specific compressible model derived by Delaney2016 and the model derived by Park2020 in the absence of charges.
The model
where
where \(\bar{\rho}_K\) is the smeared density of species \(K\in{A,B}\) defined as \(\bar{\rho}_K = \Gamma_K * \hat{\rho}_K(\pmb{r}) = \int d\pmb{r} \Gamma_K (\pmb{r} - \pmb{r}') \hat{\rho}_K(\pmb{r}')\), where \(\Gamma_K(\pmb{r}) = \frac{1}{(2\pi a_K^2)^{3/2}} \exp{\left(-\frac{\lvert\pmb{r}\rvert}{2 a_K^2} \right)}\) is the smearing function and \(\hat{\rho}_K\) is the microscopic density.
To derive a field theory, we introduce the \(\bar{\rho}_+ = \bar{\rho}_A + \bar{\rho}_B\) and \(\bar{\rho}_- = \bar{\rho}_A - \bar{\rho}_B\). Following two Hubbard-Stratonovich transforms, the field theory derived for this model is
where
with \(w_A = i w_+ - w_-\) and \(w_B = i w_+ + w_-\). By introducing the following variables \(B_0=\frac{\nu_0 N^2}{R_{g0}^3}\), \(B_{AB}=\frac{\Delta \nu_{AB} N^2}{R_{g0}^3}\), \(C=\frac{\rho_0 R_{g0}}{N}\), \(\tilde{V} = \frac{V}{R_{g0}^3}\), \(\pmb{x} = \frac{\pmb{r}}{R_{g0}}\), \(n_p = \frac{C \tilde{V} \phi_p}{\alpha_p}\), \(\mu_+ = i w_+(\pmb{r}) N\), \(\mu_- = w_-(\pmb{r}) N\) (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?
The density operator of species K \(\varphi_K\) is (for a continuous chain)
The forces are