Widoms Particle Insertion Method

Some years ago Widom pointed out that the chemical potential of a fluid is given in the thermodynamic limit by 

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In principle it is possible to estimate the chemical potential for a given fluid state by the classic MC methods of the grand-canonical ensemble or the Widom particle-insertion method, and these can be used to locate the coexistence curve. This has proved useful for some systems. However, these methods share a technical problem that means they can often not be used... 

Among the thermodynamic functions easily calculated is of course the chemical potential. One might remark that the difficulty with the traditional MC estimation techniques for chemical potentials, namely the grand-canonical and Widom particle-insertion methods, is that one must attempt insertion or deletion of at least one whole discrete particle. This corresponds to a major disruption of the system, and is accordingly infrequently allowed (or carries negligible weight), at any substantial density, which... 

As in the classical domain, quantum thermal properties are more involved to evaluate. In the canonical ensemble the central quantity is Helmholtz free energy A, from which Gibbs free energy and entropy are straightforward to obtain by making use of E and p. The standard methods to fix thermal properties are thermodynamic integration [22, 23] with respect to a parameter f, and Widom particle insertion method [236]. Thermodynamic integration for Helmholtz A relies on the identity... 

Cavity Formation in Water and n-hexane Using the Widom Particle Insertion Method. 

The solubility coefficient can be calculated via simulations in the canonical ensemble in which the chemical potential is calculated using the Widom particle insertion method. The interaction energy of a gas particle inserted within the accessible free volume of the polymer matrix is calculated and the excess thermodynamic potential Uexcess can be estimated from eqn (1.7) ... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

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. 

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... 

Several methods are available for calculating the excess chemical potential of a penetrant molecule using molecular simulation. Widom particle insertion [156,157] is the most straightforward. The potential energy change that would occur if a penetrant were inserted at a randomly chosen position is calculated, and the excess chemical potential is proportional to the average of the resulting Boltzmann factor. 

This problem has been brilliantly reviewed by Kumar in a recent book and hence we summarize only the most sahent features here. For small molecule systems, sampling of the chemical potential rests on the Widom test particle insertion method ... 

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... 

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

The excess chemiccil potential is thus determined from the average of exp[—lT (r )/fe In ensembles other than the canonical ensemble the expressions for the excess chem potential are slightly different. The ghost particle does not remain in the system and the system is unaffected by the procedure. To achieve statistically significant results m Widom insertion moves may be required. However, practical difficulties are encounte when applying the Widom insertion method to dense fluids and/or to systems contain molecules, because the proportion of insertions that give rise to low values of y f, dramatically. This is because it is difficult to find a hole of the appropriate size and sha... 

A simple method of improving the efficiency of test particle insertion [106, 107, 108 and 109] involves dividing the simulation box into small cubic regions, and identifying those which would make a negligible contribution to the Widom formula, due to overlap with one or more atoms. These cubes are excluded from the sampling, and a correction applied afterwards for the consequent bias. 

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