The Widom insertion method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963.[1] In general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall partition function of the system, which directly yields the system free energy. The second approach, known as the Widom insertion method, instead derives from calculations centering on one molecule. The Widom insertion method directly yields the chemical potential of one component rather than the system free energy. This approach is most widely applied in molecular computer simulations[2] [3] but has also been applied in the development of analytical statistical mechanical models. The Widom insertion method can be understood as an application of the Jarzynski equality since it measures the excess free energy difference via the average work needed to perform, when changing the system from a state with N molecules to a state with N+1 molecules.[4] Therefore it measures the excess chemical potential since
\mu | ||||
|
\DeltaN=1
As originally formulated by Benjamin Widom in 1963,[1] the approach can be summarized by the equation:
B | ||||
|
=\left\langle\exp\left(-
\psii | |
kBT |
\right)\right\rangle
where
Bi
\rhoi
i
ai
i
kB
T
\psi
i
Note that in other ensembles like for example in the semi-grand canonical ensemble the Widom insertion method works with modified formulas.[5]
From the above equation and from the definition of activity, the insertion parameter may be related to the chemical potential by
\mui=-kBTln\left(
Bi | |
\rhoiλ3 |
\right)=\underbrace{kBTln(\rhoi
3)} | |
λ | |
\muid |
\underbrace{-kBTln\left(\left\langle\exp\left(-
\psii | |
kBT |
\right)\right\rangle
\right)} | |
\muex |
=\muid+\muex
The pressure-temperature-density relation, or equation of state of a mixture is related to the insertion parameter via
Z= | P | =1-lnB+ |
\rhokBT |
1 | |
\rho |
\rho | |
\int\limits | |
0 |
lnBd\rho'
where
Z
\rho
lnB
lnB=\sumi{xilnBi}
In the case of a 'hard core' repulsive model in which each molecule or atom consists of a hard core with an infinite repulsive potential, insertions in which two molecules occupy the same space will not contribute to the average. In this case the insertion parameter becomes
Bi=Pins,i\left\langle\exp\left(-
\psii | |
kBT |
\right)\right\rangle
where
Pins,i
i
The above is simplified further via the application of the mean field approximation, which essentially ignores fluctuations and treats all quantities by their average value. Within this framework the insertion factor is given as
Bi=Pins,i\exp\left(-
\left\langle\psii\right\rangle | |
kBT |
\right)