flory

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:

pip install flory

As an alternative, you can install flory through conda using the conda-forge channel:

conda install -c conda-forge flory

Usage

The following example determines the coexisting phases of a binary mixture with Flory-Huggins free energy:

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:

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:

# calculate the chemical potentials of the coexisting phases
mus = fh.chemical_potentials(phases.fractions)

More information

• See examples in examples folder
• Full documentation on readthedocs