Widom test particle insertion method¶

Brief introduction about the Widom method¶

The excess chemical potential of a guest molecule in a porous host material can be calculated using the Widom test particle insertion method.

\[\mu^{ex} = \frac{1}{\beta} \ln{\frac{\int d\mathbf{s} \langle e^{-\beta \Delta U}\rangle_{host}}{\int d\mathbf{s}}}\]

where $\langle ... \rangle_{host}$ denotes the canonical ensemble average of the guest molecule in the host material and the integration $\int d\mathbf{s}$ runs over all $\mathbf{s}`$ possible configurations of the guest molecule with adsorption energy

\[\Delta U = U_{host + guess} - U_{host} - U_{guess}\]

This integral can be approximated by generating a fixed number of $N$ unbiased random configurations ${\mathbf{s}_i}$, and converting the integral to the sum over this set of configurations

\[\begin{split}\begin{split} \mu^{ex} & \approx -\frac{1}{\beta} \ln\left(\frac{1}{N}\sum_{i=1}^N e^{-\beta \Delta U_i}\right) \\ & \approx \frac{1}{\beta} \ln(\langle e^{-\beta \Delta U}\rangle) \end{split}\end{split}\]

The Henry coefficient $K_H$ at a given temperature $T$ is related to the excess chemical potential:

\[K_H = \rho^{-1} \beta e^{-\beta \mu^{ex}} \approx \rho^{-1} \beta \langle e^{-\beta \Delta U}\rangle\]

where $\rho$ is the density of the material.

The enthalpy of adsorption at zero coverage ($\Delta H^0_{ads}$) can then be calculated as

\[\Delta H_{ads}^0 = \frac{\partial \ln(K_H)}{\partial\beta} \approx \frac{\langle \Delta U e^{-\beta \Delta U}\rangle}{\langle e^{-\beta \Delta U}\rangle} - \frac{1}{\beta}\]