Widom’s method

In addition to standard thermodynamic integration, perhaps the most widely used idea for simulation of the chemical potential of a simple fluid is embeded in Widom s method, which gives residual contributions to the chemical potential as [46]... 

Widom s method [Eq. (4.2)] represents a single-stage free-energy perturbation method to measure the chemical potential of chain molecules (the... 

Several methods to obtain vapor-liquid equilibria or the Henry s law constant [39] require the accurate calculation of the chemical potential. Widom s test molecule method [207] is the most common approach for that task. Widom s method randomly introduces test molecules of the component for which the chemical potential is evaluated into the simulation volume, to calculate their potential energy i//,. The test molecules are instantly removed after the calculation and do not influence the remaining molecules. Widom s method is applicable to MC as well as to MD simulations [56]. hi the iWT ensemble, the residual chemical potential /ip of component i is calculated by... 

Simulations in the Gibbs ensemble attempt to combine features of Widom s test particle method with the direct simulation of two-phase coexistence in a box. The method of Panagiotopoulos et al [162. 163] uses two fiilly-periodic boxes, I and II. 

Beck, T. L., Quantum path integral extension of Widom s test particle method for chemical potentials with application to isotope effects on hydrogen solubilities in model solids, J. Chem. Phys. 1992, 96, 7175-7177... 

The residual chemical potentials of benzene, p f aI,d P2 p> ar d that of C02 in the fluid phase, p[ are calculated by Widom s test particle insertion method, Eq. (6) [6], which has been embedded in all the simulation programs. 

According to Widom s test particle method for calculating p /feT, the test molecules (benzene) inserted in each simulation do not influence the molecular movements of pure C02 in any sense. Fig. 5 (a)Pore density of C02 and (b)residual chemical Therefore, the stabilization of a test molecule potentials of benzene in fluid and pore phases at... 

Hybrid, or modified real particle method was proposed by Kumar (53) as a method suitable for calculation of chemical potential at high densities. This technique combines Widom s test particle and the so-called real particle methods and calls for factious removal and subsequent reinsertion of particles afready present in the y stem. The hybrid technique can be classified as a nondestructive one since it does not affect the proper time evolution of the system. It was suggested that this method would be particularly advantageous for simulation of macromolecules when a removal of a whole polymer chain is likely to create substantial free spaee and thus facilitate the reinsertion. This technique was tested in the original paper on Lennard-Jones particles and proven to yield good results at densities up to 1.1 and T down to 0.7. 

Another technique to determine the vapor-liquid equilibrium of pure substances or mixtures, which has some similarities with what is described in [190, 204-206], is the grand equilibrium method [192]. It is a two-step procedure, where the coexisting phases are simulated independently and subsequently. In the first step, one NpT simulation of the liquid phase is performed to determine the chemical potentials p] and the partial molar volumes v of all components i. These entropic properties can be determined by Widom s test molecule method [207] or more advanced techniques, such as gradual inserticMi [208-210] (see below). On the basis of the chemical potentials and partial molar volumes at a specified pressure po, first order Taylor expansions can be made for the pressure dependence ... 

Schnabel et al. [252] calculated the Henry s law constant of CO2 in ethanol. They evaluated the chemical potential with Widom s test molecule method [207] cf. (30). In this approach, by simulating the pure solvent, the mole fraction of the solute in the solvent is exactly zero, as required for infinite dilution, because the test molecules are instantly removed after the potential energy calculation. 

Widom s Test Particle Insertion Method. The solubility of small molecules at infinite dilution can be estimated based on the knowledge of the chemical potential of these molecules. Widom s test particle insertion method (483) provides a technique for calculating this quantity. The residual chemical potential (the difference between the chemical potentials of the fluid and the ideal gas at the same temperatin-e and density) of a fluid is given by... 

The test particle method of Widom and related particle insertion schemes reviewed recently by Kofke and Cummings have been used extensively to investigate the hydration of small nonpolar molecules. Widom s approach to this problem involves performing NVT-ensemble simulations of pure water. A variety of solution properties may be investigated by measuring the energy of solute molecules randomly inserted into the pure solvent. These test particles probe the system but do not affect the solvent trajectories. The excess chemical potential of a solute molecule is thus calculated as... 

Another important application of the CBMC algorithm is the calculation of the chemical potential using Widom s test particle method [29,32,96], For the parallel CBMC algorithm described here it is straightforward to show that the excess chemical potential can be calculated using... 

Gas solubiUties can also be determined by molecular dynamics simulations, using the Widom test particle insertion method to calculate the excess chemical potential fiex. or free energy of the penetrant molecules. The solubility can be obtained using Henry s law. If E is the interaction energy of a virtual penetrant molecule with the polymer inserted at random within the sample (the molecule is invisible to the polymer) then... 

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