Test particle insertion

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

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. 

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. 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

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. 

Another trick is applicable to, say, a two-component mixture, in which one of the species. A, is smaller than the other, B. From figure B3.3.8 for hard spheres, we can see that A need not be particularly small in order for the test particle insertion probability to climb to acceptable levels, even when insertion of B would almost always fail. In these circumstances, the chemical potential of A may be determined directly, while that of B is evaluated indirectly, relative to that of A. The related semi-grand ensemble has been discussed in some detail by Kofke and Glandt [110]. 

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... 

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... 

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... 

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... 

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