Widom method

Figure B3.3.8. Insertion probability for hard spheres of various diameters (indieated on the right) in the hard sphere fluid, as a fiinetion of paeking fraetion p, predieted using sealed partiele theory. The dashed line is a guide to the lowest aeeeptable value for ehemieal potential estimation by the simple Widom method.

A variant of the Rosenbluth-Rosenbluth method tailored to overcome the test chain insertion problem in the Widom method " (Eq. [1.8]) has been developed by Frenkel et and is known as configurational bias... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

B. Widom, Structure and Thermodynamics of Interfaces, in Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York, 1977, pp. 33-71. 

A simple method of improvmg the effieieney of test partiele insertion [106. 107. 108 and 109] involves dividing die simulation box into small eubie regions, and identifying those whieh would make a negligible eontribution to the Widom fonnula, due to overlap with one or more atoms. These eubes are exeluded from the sampling, and a eorreetion applied afterwards for the eonsequent bias. 

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. 

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... 

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. 

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... 

Most free energy and phase-equilibrium calculations by simulation up to the late 1980s were performed with the Widom test particle method [7]. The method is still appealing in its simplicity and generality - for example, it can be applied directly to MD calculations without disturbing the time evolution of a system. The potential distribution theorem on which the test particle method is based as well as its applications are discussed in Chap. 9. 

The NPT + test particle method [8, 9] aims to determine phase coexistence points based on calculations of the chemical potentials for a number of state points. A phase coexistence point is determined at the intersection of the vapor and liquid branches of the chemical potential versus pressure diagram. The Widom test particle method [7] of the previous paragraph or any other suitable method [10] can be used to obtain the chemical potentials. Corrections to the chemical potential of the liquid and vapor phases can be made, using standard thermodynamic relationships, for deviations... 

Widom test-particle method. Provides the chemical potential in various ensembles. Relatively easy to implement and can be used as an additional measurement in standard MC ensembles (and also MD). Computational overhead is small. Yields good accuracy in simple systems, although less reliable in very dense or complex systems (i.e., chain molecules). 

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... 

Local concentrations can be determined accurately, if the excess chemical potential at the point is known. A good method to determine the chemical potential was developed by Widom [30] and it was later improved by Svensson et al. [31]. This technique has been used in MC simulations in order to determine PC011 and Pi0n. 

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... 

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