FRAGMENT-MNP¶

FRAGMENT-MNP is a mechanistic model of Micro and NanoPlastic FRAGMentation in the ENvironmenT.

• Project webpage
• Model code repository
• Model documentation

This app is a simple demonstration of the latest version of the model. The model is still under active development and so except frequent changes, including additions to the model conceptualisation.

Theory¶

FRAGMENT-MNP models the time evolution of micro and nanoplastic mass and particle number concentrations by splitting the size distribution of plastic particles into a number of size classes (bins) and numerically solving the following differential equation for each size class:

$$ \frac{dc_k}{dt} = -k_{\text{frag},k} c_k + \Sigma_i f_{i,k} k_{\text{frag},i} c_i - k_\text{diss} c_k $$

Here, $c_k$ is the mass number concentration in size class $k$, $t$ is time, $k_{\text{frag},k}$ and $k_{\text{diss},k}$ are the fragmentation and dissolution rates of size class $k$, and $f_{i,k}$ is the fraction of daughter fragments produced from a fragmenting particle of size $i$ that are of size $k$. Note that there is no source term - the only input of plastic particles is the initial value at $t=0$.

$k_\text{frag}$ and $k_\text{diss}$ are implicitly dependent on the phys-chem properties and the environment the polymer is in (which governs the degradation and mechanical stresses it encounters). For $k_\text{frag}$, we express this using energy dissipation rate $\epsilon$ and surface energy $\sigma$, along with empirical constants $\Phi$ and $\theta$:

$$ k_\text{frag} = \Phi_1 \sigma^{-\theta_1} \Phi_2 \epsilon^{\theta_2} $$

The surface energy is inversely proportional to the square of the diameter of the particle, $\sigma \propto d_p^{-2}$, and so for an energy dissipation rate that is constant at each length scale, $k_\text{frag}$ varies as $d_p^{2\theta_1}$. We use this to vary $k_\text{frag}$ across size classes using a given average $k_\text{frag}$ and value for $\theta_1$. We also use this to scale $k_\text{diss}$ as proportional to the surface-to-volume ratio of the particles $s$, and use another proportionality constant $\gamma$ to do so: $k_\text{diss} \propto s^\gamma$. To convert particle number to mass concentrations, we assume spherical particles.