Mixtures

The mixture module provides tools to represent and manipulate mixtures of spherical particles with various size distributions.

Creating a Mixture

To create a mixture, you can use the Mixture class directly or use one of the convenience classes provided for specific distributions. Here are some examples:

Binary Mixture

Manually create a 50:50 binary mixture of two species with radii 100 and 250, respectively:

from mixscatter.mixture import Mixture

binary_mixture = Mixture(radius=[100, 250], number_fraction=[0.5, 0.5])

Monodisperse System

Create a pseudo-mixture representing a monodisperse system with a radius of 500:

from mixscatter.mixture import SingleComponent

monodisperse_system = SingleComponent(radius=500)

Flory-Schulz Distribution

Specifically, a Gamma distribution (often termed Schulz/Flory/Zimm-distribution depending on the scientific community), with probability density function

\[ P(R) = \frac{1}{\Gamma(Z+1)} \left( \frac{Z+1}{\langle R \rangle} \right)^{Z+1} R^Z \exp\left( -\frac{Z+1}{\langle R \rangle} R \right), \]

with mean radius \(\langle R \rangle\) and variance \(\langle R^2 \rangle - \langle R \rangle^2 =\langle R \rangle^2/(Z+1)\). \(Z\) is also called shape parameter.

Create a mixture approximating a Schulz-Flory distribution with a mean radius of 100 and a shape parameter of 99:

from mixscatter.mixture import FlorySchulzMixture

flory_mixture = FlorySchulzMixture(
    number_of_components=16, mean_radius=100, shape_parameter=99)

The approximation is based on generalized Gauss-Laguerre quadrature.

Gaussian Distribution

Gaussian or normal distribution with probability density function

\[ P(R) = \frac{1}{\sigma\sqrt{(2\pi)}} \exp \left[ - \dfrac{1}{2} \left(\frac{R-\langle R \rangle} {\sigma}\right) ^2 \right], \]

with mean radius \(\langle R \rangle\) and standard deviation \(\sigma\).

Create a mixture approximating a Gaussian distribution with a mean radius of 100 and a standard deviation of 10:

from mixscatter.mixture import GaussianMixture

gaussian_mixture = GaussianMixture(
    number_of_components=16, mean_radius=100, standard_deviation=10)

The approximation is based on Gauss-Hermite quadrature.

Uniform Distribution

Continuous uniform distribution with probability density function

\[ P(R) = \left\{ \begin{array}{ll} \dfrac{1}{b-a} & a \leq R \leq b \\ 0 & \, \textrm{otherwise} \\ \end{array} \right. \]

with lower bound \(a\) and upper bound \(b\).

Create a mixture approximating a uniform distribution between radii 50 and 150:

from mixscatter.mixture import UniformMixture

uniform_mixture = UniformMixture(
    number_of_components=16, lower_bound=50, upper_bound=150)

The approximation is based on Gauss-Legendre quadrature.

Accessing Mixture Properties

You can access various properties of a mixture, such as the mean radius and polydispersity:

print(f"Mean radius: {flory_mixture.mean}")
print(f"Polydispersity: {flory_mixture.polydispersity}")

See the API Reference for a full list of available attributes and methods.