Particle insertion methods

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. 

Other variations on these basic free energy methods have been published, although for various reasons they have not yet been widely adopted. These methods include MD/MC methods,38 the acceptance ratio method,39, 40 the weighted histogram method,41 the particle insertion method,42 43 and the energy distribution method.39 The reader is referred to the original publications for additional discussion of these approaches. 

This last decade, the chemical potential has been the subject of intensive efforts in the IETs field, while it can be easily obtained from simulation through the test particle insertion method [73, 74],... 

This relation is the potential distribution theorem [73, 74], which gives a physical interpretation of the cavity function in terms of the chemical potential, and the excess interaction generated by the test particle, Y j>2 u(rv)> yia the ensemble average of its Boltzmann factor. In numerical simulation, the use of such a test-particle insertion method is of prime importance in calculating the cavity function at small distances and particularly at zero separation. Note that if the particle labeled 1 approaches the particle labeled 2, a dumbbell particle [41] is created with a bond length L = r2 n corresponding to a dimer at infinite... 

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... 

In the grand equilibrium method, a simulation of the condensed phase is done to calculate the excess chemical potentials, /x,ex, and the partial molar volumes, V,-, of all components. One may use the test-particle insertion method [59] to calculate the excess chemical potentials and the partial molar volumes as... 

Figure 17-3. A schematic distribution of A.U value in the particle insertion method...

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. 

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... 

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