A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures

First-order phase transitions of binary mixtures at the given pressure (P) and temperature (T) are studied by taking into account the composition fluctuations. Isothermal-isobaric semigrand canonical ensemble is adopted to find the relations among the total number of molecules, the composition fluctuations and Gibbs free energy density. By combining two identical subsystems of mixtures successively, the free energy density is transformed until being stable and its linear segments represent phase transitions. A new method is developed to calculate the phase equilibriums of binary mixtures. The method handles multiple types and number of phase equilibriums at single time and its solutions are physically justified. One example is shown for calculating the phase diagram of binary Lennard-Jones mixture. It demonstrates that the fluctuations of the total number of molecules in mixtures are fundamental behind phase transitions and the van der Waals loops in Gibbs free energy are reasonable.     

Molecular dynamics simulation in the grand canonical ensemble

An extended system Hamiltonian is proposed to perform molecular dynamics (MD) simulation in the grand canonical ensemble. The Hamiltonian is similar to the one proposed by Lynch and Pettitt (Lynch and Pettitt, J Chem Phys 1997, 107, 8594), which consists of the kinetic and potential energies for real and fractional particles as well as the kinetic and potential energy terms for material and heat reservoirs interacting with the system. We perform a nonlinear scaling of the potential energy parameters of the fractional particle, as well as its mass to vary the number of particles dynamically. On the basis of the equations of motion derived from this Hamiltonian, an algorithm has been proposed for MD simulation at constant chemical potential. The algorithm has been tested for the ideal gas, for the Lennard-Jones fluid over a wide range of temperatures and densities, and for water. The results for the low-density Lennard-Jones fluid are compared with the predictions from a truncated virial equation of state. In the case of the dense Lennard-Jones fluid and water our predicted results are compared with the results reported using other available methods for the calculation of the chemical potential. The method is also applied to the case of vapor-liquid coexistence point predictions.     

Size-effects on simulations in the grand canonical ensemble

Results are presented for the effect of periodic boundary conditions on predictions made using the grand canonical ensemble for systems of limited size. A Monte Carlo study of a realistic gas-solid adsorption model and an exact study of the Tonks gas both show that it is necessary for the minor dimension of the replicated system to be 5 σ or greater if errors in the partition function equivalent to δμ > 0.2 kT are to be avoided. The heat capacity C_μ has similar requirements. However, for quantities such as the isosteric heat of adsorption, <N > and <U > even a dimension as small as 3σ does not lead to serious errors. An examination is presented of possible implications for studies of phase changes.     

Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms

We present two efficient iterative Monte Carlo algorithms in the grand canonical ensemble with which the chemical potentials corresponding to prescribed (targeted) partial densities can be determined. The first algorithm works by always using the targeted densities in the kT log(rho(i)) (ideal gas) terms and updating the excess chemical potentials from the previous iteration. The second algorithm extrapolates the chemical potentials in the next iteration from the results of the previous iteration using a first order series expansion of the densities. The coefficients of the series, the derivatives of the densities with respect to the chemical potentials, are obtained from the simulations by fluctuation formulas. The convergence of this procedure is shown for the examples of a homogeneous Lennard-Jones mixture and a NaCl-CaCl(2) electrolyte mixture in the primitive model. The methods are quite robust under the conditions investigated. The first algorithm is less sensitive to initial conditions.     

Dissipative particle dynamics simulations in the grand canonical ensemble: applications to polymer brushes.

We have used the dissipative particle dynamics (DPD) method in the grand canonical ensemble to study the compression of grafted polymer brushes in good solvent conditions. The force-distance profiles calculated from DPD simulations in the grand canonical ensemble are in very good agreement with the self-consistent field (SCF) theoretical models and with experimental results for two polystyrene brush layers grafted onto mica surfaces in toluene.     

Acceleration of Monte Carlo simulations through spatial updating in the grand canonical ensemble

A new grand canonical Monte Carlo algorithm for continuum fluid models is proposed. The method is based on a generalization of sequential Monte Carlo algorithms for lattice gas systems. The elementary moves, particle insertions and removals, are constructed by analogy with those of a lattice gas. The updating is implemented by selecting points in space (spatial updating) either at random or in a definitive order (sequential). The type of move, insertion or removal, is deduced based on the local environment of the selected points. Results on two-dimensional square-well fluids indicate that the sequential version of the proposed algorithm converges faster than standard grand canonical algorithms for continuum fluids. Due to the nature of the updating, additional reduction of simulation time may be achieved by parallel implementation through domain decomposition.     

On interfacial tension calculation from the test-area methodology in the grand canonical ensemble

We propose the extension of the test-area methodology, originally proposed to evaluate the surface tension of planar fluid-fluid interfaces along a computer simulation in the canonical ensemble, to deal with the solid-fluid interfacial tension of systems adsorbed on slitlike pores using the grand canonical ensemble. In order to check the adequacy of the proposed extension, we apply the method for determining the density profiles and interfacial tension of spherical molecules adsorbed in slitlike pore with different pore sizes and solid-fluid dispersive energy parameters along the same simulation. We also calculate the solid-fluid interfacial tension using the original test-area method in the canonical ensemble. Agreement between the results obtained from both methods indicate that both methods are fully equivalent. The advantage of the new methodology is that allows to calculate simultaneously the density profiles and the amount of molecules adsorbed onto a slitlike pore, as well as the solid-fluid interfacial tension. This ensures that the chemical potential at which all properties are evaluated during the simulation is exactly the same since simulations can be performed in the grand canonical ensemble, mimicking the conditions at which the adsorption experiments are most usually carried out in the laboratory.     

On a new self-consistent-field theory for the canonical ensemble

A significant amount of many-body problems of quantum or classical equilibrium statistical mechanics are conveniently treated at fixed temperature and system size. In this paper, we present a new functional integral approach for solving canonical ensemble problems over the entire coupling range, relying on the method of Gaussian equivalent representation of Efimov and Ganbold. We demonstrate its suitability and competitiveness for performing approximate calculations of thermodynamic and structural quantities on the example of a repulsive potential model, widely used in soft matter theory.     

A comparison between two massively parallel algorithms for Monte Carlo computer simulation: An investigation in the grand canonical ensemble

We present a comparison between two different approaches to parallelizing the grand canonical Monte Carlo simulation technique (GCMC) for classical fluids: a spatial decomposition and a time decomposition. The spatial decomposition relies on the fact that for short-ranged fluids, such as the cut and shifted Lennard-Jones potential used in this work, atoms separated by a greater distance than the reach of the potential act independently, and thus different processors can work concurrently in regions of the same system which are sufficiently far apart. The time decomposition is an exactly parallel approach which employs simultaneous (GCMC) simulations, one per processor, identical in every respect except the initial random number seed, with the thermodynamic output variables averaged across all processors. While scaling characteristics for the spatial decomposition are presented for 8–1024 processor systems, the comparison between the two decompositions is limited to the 8–128 processor range due to the warm-up time and memory imitations of the time decomposition. Using a combination of speed and statistical efficiency, the two algorithms are compared at two different state points. While the time decomposition reaches a given value of standard error in the system's potential energy more quickly than the spatial decomposition for both densities, the warm-up time demands of the time decomposition quickly become insurmountable as the system size increases. © 1996 by John Wiley & Sons, Inc.     

Calculation of critical temperature Tc in high temperature superconductivity using grand canonical ensemble

In this paper we use the grand canonical ensemble to calculate the T_c of superconductivity for the compound YBa₂Cu₃O₇ under one dimensional Cu–O chain model. The results are consistent with experiments.     

Specifics of first-order phase transitions in 1,22-docosanediol and 1,44-tetratetracontanediol

A comparative analysis of phase transitions in diols with different chain lengths, (CH₂)₄₄(OH)₂ and (CH₂)₂₂(OH)₂, was performed using differential scanning calorimetry. The use of temperature hysteresis made it possible to reveal a number of new effects associated with the specifics of first-order phase transitions. The parameters of transitions were quantitatively analyzed in terms of the self-consistent field theory for diffuse (A-shaped) first-order transitions.     

A new approach to canonical orthonormalisation

The canonical orthornormalisation procedure is derived through an integral equation with a finite rank kernel. Its optimal properties in connection with the problem of approximate linear dependence are established.     

A simple protocol for the probability weights of the simulated tempering algorithm: applications to first-order phase transitions

The simulated tempering (ST) is an important method to deal with systems whose phase spaces are hard to sample ergodically. However, it uses accepting probabilities weights, which often demand involving and time consuming calculations. Here it is shown that such weights are quite accurately obtained from the largest eigenvalue of the transfer matrix--a quantity straightforward to compute from direct Monte Carlo simulations--thus simplifying the algorithm implementation. As tests, different systems are considered, namely, Ising, Blume-Capel, Blume-Emery-Griffiths, and Bell-Lavis liquid water models. In particular, we address first-order phase transition at low temperatures, a regime notoriously difficulty to simulate because the large free-energy barriers. The good results found (when compared with other well established approaches) suggest that the ST can be a valuable tool to address strong first-order phase transitions, a possibility still not well explored in the literature.     

Solvation of sodium-chloride ion pair in water cluster at atmospheric conditions: grand canonical ensemble Monte Carlo simulation

Open statistical ensemble simulations are used to study the mechanism of nucleation of atmospheric water on sodium-chloride ion pair in a wide range of temperature and relative humidity values. The extended simple point-charge model is used for water molecules. Ions-water nonadditive interactions are taken into account by introducing the mutual polarization of ions and water in the field of each other. Gibbs free-energy variations are calculated from Na+-Cl- pair-correlation function and used as a criterion for determining the possible stable states of the cluster. In this relation, it was found that the dissociation of ion pairs in water clusters occurs even at vapor pressures of only a few millibars. In the conditions under consideration solvent-separated ion-pair states are found to be more probable than contact ion-pair configurations. The susceptibilities of water and ions are found to play an essential role in the stabilization of ions at large separations. The structure of ion-induced clusters is analyzed in terms of binary correlation functions. The non-pair interactions influence essentially the structure of ion solvation shells. The results of simulation show that the separation of the charges in water clusters containing simple ions can take place under atmospheric conditions.     

A new method to calculate the thermodynamical properties of liquids from accurate speed-of-sound measurements

In this work, a new method for the determination of the thermodynamical properties of fluids, based on experimental speed-of-sound measurements, is described. This method consists in the solution of recursive equations (REM, Recursive Equations Method) for the determination of the density ρ(p,T) and specific heat capacity at constant pressure c_p(p,T), using the initial values of density ρ(p₀,T) and isobaric specific heat capacity c_p(p₀,T) known at a reference pressure p₀, as a function of the temperature, if the speed-of-sound function, u(p,T), is known, at least over a certain temperature and pressure range. A complete uncertainty analysis has also been developed. As an example of the good performances of this analysis method, firstly density and isobaric specific heat capacity have been calculated for water and the results have been compared with those predicted by the International Association for the Properties of Water and Steam 95 Formulation (IAPWS-95), as delivered by Wagner and Pruss. One more application has been made starting from experimental speed-of-sound values in pure acetone. These results have been compared with those calculated by the most advanced numerical integration methods and with the prevision of the dedicated NIST equation of state (EoS) by Lemmon and Span.     

A variational method to calculate static electronic properties

Using a coupled cluster form of the wave function, a variational method is formulated for calculation of static properties of any order. Corresponding to an appropriate perturbed hamiltonian H() including the relevant static property, a size consistent functional is set up. In a hierarchical fashion, properties of different orders may be found out using a variational method.     

A new method to calculate Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect

Calculations of Franck-Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. We have developed a new method for calculating Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect. Closed-form formulas of two-, three-, and four-dimensional Franck-Condon factors were derived straightforwardly by using the properties of Hermite polynomials and Gaussian integrals. This new method was applied to study the photoelectron spectra of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)), whose equilibrium geometries and harmonic vibrational frequencies were calculated by using the coupled cluster singles and doubles with perturbative triples [CCSD(T)] method together with the basis sets of 6-311++G(3df,2pd) and aug-cc-pVTZ. The adiabatic ionization energies were computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X=D,T,Q,5). It was found that the simulated photoelectron spectra were mainly composed of nu(2) progressions and the combination bands of nu(1) and nu(2), whereas pure nu(1) transitions should be too weak to be observable, contrary to the literature reports. It was also found that the first discernible peak in the experimental photoelectron spectra did not correspond to the adiabatic transition. The adiabatic ionization energies of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)) are proposed to be 16.78 and 16.83 eV, about 0.40 and 0.58 eV lower than the best experimental values, respectively. Conversely, the calculated ionization energies are in agreement with the proposed values within 0.02 eV.