Test particle insertion method

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... 

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... 

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... 

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... 

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. 

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

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. 

Shah and Maginn carried out a Monte Carlo study of CO2 solubility in [C4mim][PF6] to compute the Henry s law constant of CO2 using the test particle insertion free energy perturbation method. The Henry s law constant for solute 2 dissolved in solvent 1, Hi,i, is defined as... 

Particle insertion methods may also be used in spatially inhomogeneous systems. In this case, the spatial variation in the excess chemical potential is directly related to the potential of mean force for the solute molecule. The calculation is more computationally intensive because the test particle insertion energy must be determined as a function of its position. The potential of mean force, w r), is then given by... 

The chemical potential of particles belonging to species a and (3 is measured by using the classical test particle method (as proposed by Fischer and Heinbuch [166]) in parts II and IV of the system i.e., we calculate the average value of (e) = Qxp[—U/kT]), where U denotes the potential energy of the inserted particles. 

Probe methods like particle insertion and test particle methods (29-32) are quite useful for computing chemical potentials of constituent particles in systems with low densities. Test particles are randomly inserted the average Boltzmann factor of the insertion energy yields the free energy. For dense systems these methods work poorly because of the poor statistics obtained. 

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... 

The method of Widom to calculate the excess chemical potential of a system by random insertion of a test particle, particle insertion, is in essence a special case of the thermodynamic perturbation formula. The excess chemical potential (or free energy per particle), is the chemical poten-... 

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... 

This approach may yield a result for that is superior to the simple test particle result if the overlap in the two distribution functions is significant. Contrarily, if the distributions do not overlap significantly, it is difficult to extract a reliable result from either the test particle or overlapping distribution formulae, and an alternative approach like thermodynamic integration or the staged insertion method is favored. 

Most of hydrodynamic methods have focused on increasing the particle back transport from the membrane-liquid interface by increasing the shear rate and the flow instability in the boundary layer. These techniques include secondary flows, spacers and inserts, pulsed flow, high shear rate devices, vibrations, and two-phase flow. The physical methods that are currently been tested to enhance filtration performance of membranes include the application of electric fields and ultrasound. 

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