Model Melt Chi AB
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
C or rho0 (float) - System density \(C = \rho_0 R_g^3 / N_{ref}\). Only one can be specified.
chiN or chi (float) - Flory-Huggins \(\chi\) parameter, optionally multiplied Nref. Only one can be specified.
inverse_zetaN or zeta (float) - Helfand compressibility parameter \(\zeta\), optionally multiplied Nref and inverted (i.e. \((\zeta N)^{-1}\)). Only one can be specified. Default: inverse_zetaN=0 (incompressible).
inverse_BC or B or u0 (float) - Excluded volume incompressiblity. \(B =\beta u_0 N^2 / R_g^3\) is the dimensionless excluded volume and \(C = n R_g^3 / V\). See Delaney2016 for details.
Note
Only one compressibility parameter (i.e. u0, B, zeta, inverse_zetaN or inverse_BC) can be specified.
Example (python)
import openfts
fts = openfts.OpenFTS()
...
fts.model(Nref=1.0,bref=1.0,chiN=20.,type='MeltChiAB')
...
Example (json)
"model": {
"type": "MeltChiAB"
"Nref": 1.0,
"bref": 1.0,
"chiN": 20 ,
"initfields": {
... see "Initialize Model Fields" ...
},
},
Model Melt Chi AB Formalism
The form of \(H[w_+,w_-]\) depends on the parameters.
Incompressible model inverse_BC = 0 or inverse_zetaN=0:
where \(w_A = i w_+ - w_-\) and \(w_B = i w_+ + w_-\). where \(\mu_A = \mu_+ - \mu_-\) and \(\mu_B = \mu_+ + \mu_-\).
Helfand compressible model inverse_zetaN != 0:
Excluded volume compressible model inverse_BC != 0:
Model Melt Chi AB Formalism (old)
TODO: add bold math using pmb{}
Fredrickson2006 (FIXME: add citation) shows us how to derive field theories of the form
where for a binany blend consisting of \(n_A\) A homopolymers and \(n_B\) B homopolymers (Model C), \(H[w_+,w_-]\) is given by,
where \(w_A = w_+ - w_-\) and \(w_B=w_+ + w_-\). For a diblock copolymer melt (Model E), \(H[w_+,w_-]\) is given by,
Where
and \(q(r,s)\) is obtained by solving
subject to the initial condition \(q(r,0) = 1\).
To generalize these two models to any number of different polymers, with any block sequence, length or architecture, we follow the procedure described in Duchs2014 (FIXME: add citation). We define our system to consist of a volume \(V\), consisting of \(n\) total molecules consisting of \(P\) distinct molecule types. We define \(n_p\) to be the number of molecules of type \(P\) such that
\(N_p\) is the number of degrees of freedom of molecule type \(p\), which here we take as the degree of polymerization without loss of generality. We define a reference \(N = N_{ref}\) which we use to normalize all chain lengths such that \(N_p = N \alpha_p\). The overall density of monomers is \(\rho_0 = \sum_{p=1}^P \frac{n_p N_p}{V}\) and all statistical segements were defined relative to the same reference volume \(v_0 = 1/\rho_0\). For this system \(H[w_+,w_-]\) is
We now introduce introduce the fields \(\mu_i = \gamma_i w_i N\) where \(i \in {+,-}\), and \(\gamma_i = {1,\sqrt{-1}}\) depending on whether the field is Wick rotated. For this model \(\mu_+ = i w_+ N\) and \(\mu_- = w_- N\). We also rescale lengths in our model \(x=r/R_g\) relative to \(R_g = \sqrt{\frac{b^2N}{6}}\), the radius of gyration of reference polymer chain with statistical segment length \(b = b_{ref}\). By introducing \(C = \frac{\rho_0 R_g^3}{N}\), the dimentionless chain concentration, our field theory is now
where \(\mu_A = \mu_+ - \mu_-\) and \(\mu_B = \mu_+ + \mu_-\). It is convenient to work with polymer volume fractions \(\phi_p = \frac{n_p N_p}{\sum_{i=1}^P n_i N_i}\) instead of explicit molecule numbers \(n_p\). This can be acomplished by first manipulating the expression for \(\phi_p\)
and then solving for \(n_p = \frac{\phi_p \rho_0 V}{\alpha_p N}\). Introducing the dimentionless volume \(\tilde{V} = V/R_g^3\)
We now have the expression used in the code,
With this non-dimentionalization
and
where \(\tilde{\nabla}^2 = R_g^2 \nabla^2\), \(s \in [0,\alpha_p]\), \(q(x,0) = 1\), \(\mu(r,s)\) is the field corresponding to the species at contour position s along molecule p.
The density operator \(\varphi_j\) of species j is
FIXME. It would be more clear to write \(\frac{\delta Q[i\Gamma * \mu]} {\delta \mu_j (\pmb{r})}\) instead of the ds integral. This would be true for any chain model. Also should note that \(\varphi_j = \tilde{\rho_j}/\rho_0\), since most of GHF works with \(\tilde{\rho}\). Also, should I change to \(\tilde{\varphi_j}\) and modify my notation so that anything with a tilde is a field-theoretic operator?
The forces for this model are
The code also has a model with a Helfand compressibility implemented (see Delaney 2016 or Josh’s Notebook #2, p 118) which has the following form
with forces