# flory
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`flory` is a Python package for analyzing field theories of multicomponent mixtures.
In particular, the package provides routines to determine coexisting states numerically, which is a challenging problem since the thermodynamic coexistence conditions are many coupled non-linear equations.
`flory` supports finding coexisting phases with an arbitrary number of components.
The associated average free energy density of the system reads
$$\bar{f}({N_\mathrm{P}}, \{J_p\}, \{\phi_{p,i}\}) = \sum_{p=1}^{{N_\mathrm{P}}} J_p f(\{\phi_{p,i}\}) \; ,$$
where $N_\mathrm{C}$ is the number of components, $N_\mathrm{P}$ is the number of phases, $J_p$ is the fraction of the system volume occupied by phase $p$, and $\phi_{p,i}$ is the volume fraction of component $i$ in phase $p$.
`flory` supports different forms of interaction, entropy, ensemble, and constraints to describe the free energy of phases.
For example, with the commonly used Flory-Huggins free energy, the free energy density of each homogeneous phase reads
$$f(\{\phi_i\}) = \frac{1}{2}\sum_{i,j=1}^{N_\mathrm{C}} \chi_{ij} \phi_i \phi_j + \sum_{i=1}^{N_\mathrm{C}} \frac{\phi_i}{l_i} \ln \phi_i \; ,$$
where $\chi_{ij}$ is the Flory-Huggins interaction parameter between component $i$ and $j$, and $l_i$ is the relative molecular size of component $i$.
Given an interaction matrix $\chi_{ij}$, average volume fractions of all components across the system $\bar{\phi}_i$, and the relative molecule sizes $l_i$, `flory` provides tools to find the coexisting phases in equilibrium.
Installation
------------
`flory` is available on `pypi`, so you should be able to install it through `pip`:
```bash
pip install flory
```
As an alternative, you can install `flory` through [conda](https://docs.conda.io/en/latest/) using the [conda-forge](https://conda-forge.org/) channel:
```bash
conda install -c conda-forge flory
```
Usage
-----
The following example determines the coexisting phases of a binary mixture with Flory-Huggins free energy:
```python
import flory
num_comp = 2 # Set number of components
chis = [[0, 4.0], [4.0, 0]] # Set the \chi matrix
phi_means = [0.5, 0.5] # Set the average volume fractions
# obtain coexisting phases
phases = flory.find_coexisting_phases(num_comp, chis, phi_means)
```
It is equivalent to a more advanced example:
```python
import flory
num_comp = 2 # Set number of components
chis = [[0, 4.0], [4.0, 0]] # Set the \chi matrix
phi_means = [0.5, 0.5] # Set the average volume fractions
# create a free energy
fh = flory.FloryHuggins(num_comp, chis)
# create a ensemble
ensemble = flory.CanonicalEnsemble(num_comp, phi_means)
# construct a finder from interaction, entropy and ensemble
finder = flory.CoexistingPhasesFinder(fh.interaction, fh.entropy, ensemble)
# obtain phases by clustering compartments
phases = finder.run().get_clusters()
```
The free energy instance provides more tools for analysis, such as:
```python
# calculate the chemical potentials of the coexisting phases
mus = fh.chemical_potentials(phases.fractions)
```
More information
----------------
* See examples in [examples folder](https://github.com/qiangyicheng/flory/tree/main/examples)
* [Full documentation on readthedocs](https://flory.readthedocs.io/)
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