Numerical study of a modified BBGY integral equation for the primitive model of an electrolyte solution

A modified Poisson-Boltzmann equation derived from the BBGY hierarchy of equations is solved numerically for the restricted primitive model of an electrolyte solution. Computations are carried out for 1 : 1 and 2 : 2 electrolytes, and results compared with 1 : 1 Monte Carlo data and results from the Poisson-Boltzmann extensions arising from the Kirkwood hierarchy. Satisfactory agreement with the Monte Carlo results are found in the 1 : 1 case, and the correct qualitative behaviour of thermodynamic properties obtained in the 2 : 2 case. At low concentrations in the 2 : 2 case the modified Poisson-Boltzmann equations from the Kirkwood and BBGY hierarchies predict the anomalous behaviour found in the thermodynamic properties of real electrolytes. The behaviour of the mean electrostatic potential changes from a damped exponential to a damped oscillatory form for 1·2 < κa < 1·3 (where κ is the Debye-Hückel constant and a is an ionic diameter) in the 1 : 1 case and at κa ～ 0·8 in the 2 : 2 case for the parameters considered.     

Thermodynamics of pseudo-hard body mixtures

Thermodynamic properties of binary mixtures of hard spheres of various size and pseudo-hard bodies, mimicking the short-range non-additive repulsive interactions in realistic models of water, have been determined over the entire concentration range using standard NVT Monte Carlo simulations. Virial coefficients of the mixture have also been computed. Having no other theoretical tool currently available, a perturbed virial expansion is examined with respect to its potential to estimate/predict the properties of the mixture without resorting to any fitting of simulation data. The perturbed virial expansion is found to perform quite accurately for the mixtures containing larger spheres, whereas for small spheres dissolved in water the result is only qualitatively correct.     

The effects of collisional quenching on degenerate four-wave mixing

We report a theoretical and experimental investigation of the effects of collisional quenching on resonant degenerate four-wave mixing (DFWM). Using single-mode laser radiation, peak signal intensity measurements were performed on an isolated line in the A – X transition of NO. By using appropriate mixtures of N₂ and CO₂ as buffer gases, we varied the collisional quenching rate over several orders of magnitude while maintaining a fixed total collisional dephasing rate. The mixtures had approximately 100 Torr total pressure and were at room temperature. For I/I _sat approximately equal to 0.02, DFWM intensities were found to be less affected by variations in quench rate than were laser-induced fluorescence (LIF) intensities (I and I _sat are the pump laser and one-photon saturation intensities, respectively). Moreover, for I/I _sat roughly equal to 0.5, DFWM intensities were observed to be nearly independent of quench rate. The results are compared to two theoretical predictions, with good agreement observed. Both theories indicate that the minimum sensitivity of DFWM to quenching occurs near I/I _sat1.     

Single-cluster Monte Carlo dynamics for the Ising model

We present an extensive study of a new Monte Carlo acceleration algorithm introduced by Wolff for the Ising model. It differs from the Swendsen-Wang algorithm by growing and flipping single clusters at a random seed. In general, it is more efficient than Swendsen-Wang dynamics ford>2, giving zero critical slowing down in the upper critical dimension. Monte Carlo simulations give dynamical critical exponentsz _w=0.33±0.05 and 0.44+0.10 ind=2 and 3, respectively, and numbers consistent withz _w=0 ind=4 and mean-field theory. We present scaling arguments which indicate that the Wolff mechanism for decorrelation differs substantially from Swendsen-Wang despite the apparent similarities of the two methods.     

The virial coefficients of hard hypersphere binary mixtures

The third, fourth and fifth virial coefficients of hard hypersphere binary mixtures with dimensionality d = 4, 5 have been calculated for size ratios R ≥ 0.1, R = ≡ σ₂₂/σ₁₁, where σ _ii is the diameter of component i. The composition independent partial virial coefficients have been evaluated by Monte Carlo integration of the corresponding Mayer modified star diagrams. The results are compared with the predictions of Santos, S., Yuste, S. B., and Lopez de Haro, M., 1999, Molec. Phys., 96, 1 of the equation of state of a multicomponent mixture of hard hyperspheres, and the good agreement gives strong support to the validity of that recipe.     

Calculation of the thermodynamic properties of electrolyte solutions using a modified Poisson-Boltzmann equation

Further results are presented of the numerical study made by Burley, Huston and Outwaite on a modified Poisson-Boltzmann equation in electrolyte solution theory. Activity and osmotic coefficients are calculated for 1 : 1 and 2 : 2 electrolytes using the Güntelberg charging process and comparisons made with the Poisson-Boltzmann coefficients. The consistency of the activity coefficients is checked by considering the Debye charging process and the second-moment condition of Stillinger and Lovett. Comparisons are also made with the Monte Carlo studies of Card and Valleau. Our results indicate that the long-range effects predicted by the modified Poisson-Boltzmann equation are considerably more accurate than those of the Poisson-Boltzmann equation.     

Excess properties of Lennard-Jones binary mixtures from computer simulation and theory

Monte Carlo simulation and theory are used to calculate the excess thermodynamic properties of binary mixtures of spherical Lennard-Jones molecules. We study the excess functions of three binary mixtures characterized by the following size and dispersive energy ratios: (1) (σ₂₂/σ₁₁)³ = 2 and ?₂₂/?₁₁ = 2; (2) (σ₂₂/σ₁₁)³ = 1 and ?₂₂/?₁₁ = 1/2 and (3) (σ₂₂/σ₁₁)³ = 1/2 and ?₂₂/?₁₁ = 2. In all cases, the unlike size parameter, σ₁₂, is kept constant and equal to the value given by the Lorentz combining rule (σ₁₂ = (σ₁₁ + σ₂₂)/2). However, different unlike dispersive energy parameter values are considered through the following combining rules: (a) ?₁₂ = (?₁₁?₂₂)^1/2 (Berthelot rule); (b) ?₁₂ = ?₁₁ (association); and (c) ?₁₂ = ?₂₂ (solvation). The pressure and temperature dependence of the excess volume and excess enthalpy is studied using the NpT Monte Carlo simulation technique for all the systems considered. Additionally, the simplest conformal solution theory is used to check the adequacy of this approach in predicting the excess properties in a wide range of thermodynamic conditions and variety of binary mixtures. In particular, we have applied the van der Waals one-fluid theory to describe Lennard-Jones binary mixtures through the use of the Johnson et al. [1993, Molec. Phys., 78, 591] Helmholtz free energy. Agreement between simulation results and theoretical predictions is excellent in all cases and thermodynamic conditions considered. This work confirms the applicability of the van der Waals one-fluid theory in predicting excess thermodynamic properties of mixtures of spherical molecules. Furthermore, since binary mixtures of spherical Lennard-Jones molecules constitute the reference fluid to be used in perturbation theories for complex fluids, such as the statistical association fluid theory (SAFT), this work shows clearly the applicability of the conformal solution theory within the framework of SAFT for predicting excess functions.     

MC simulation results for a hard core model of carbon tetrachloride

Monte Carlo simulations on a hard tetrahedron fluid (hard core model of CCl₄) have been performed. The average site-site correlation functions, their generalized (1, 0, 0) spherical harmonic expansion coefficients, equation of state, and virial coefficients have been calculated and compared with theoretical methods currently available. The RISM equation is less accurate for the model studied than for simpler models considered so far. For the equation of state the best results are obtained from the Boublik-Nezbeda equation which agrees with the simulation results throughout the density range considered.     

Is it possible to infer the equation of state of a mixture of hard discs from that of the one-component system?

Based on exact asymptotic properties of the composition-independent virial coefficients of a binary mixture of hard discs in the limits α = σ₂/σ₁ → 0, α → 1 and α → ∞, R. J. Wheatley (1998, Molec. Phys., 93, 965) has recently proposed an approximate interpolation equation for these coefficients. In this note, the equation of state equivalent to this interpolation is obtained, expressing the compressibility factor of the mixture in terms of that of the pure system. An extension to an arbitrary number of components is also given. The equation of state derived here is compared with another one recently proposed by following a different route (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) and with Monte Carlo simulation results. It is shown that the latter equation is more accurate than the former one, at least for not too disparate mixtures (0.7 < α < 1).     

Modified equation of state for pure and hard sphere mixtures in mean ionic activity coefficient calculations by mean spherical approximation model

A hard sphere equation of state (EOS) based on tetrakaidecahedron cell geometry (instead of spherical shape) and highly optimized molecular dynamic simulation data is proposed. The EOS is extended to hard sphere mixture and its performance for compressibility factor calculation at different diameter size of hard sphere mixtures by using various mixing rule is compared with Monte Carlo simulation data. The results indicated that for all mixing rules, the proposed EOS has minimum error comparing with computer simulation data. Also the residual prosperities are derived by using the proposed EOS. The residual properties are used in mean spherical approximation model (MSA) to evaluate the mean ionic activity coefficient of aqueous electrolyte solutions. The results are compared with those obtained by similar hard sphere equations of state and it is shown that the proposed EOS has a better performance in predicting the mean ionic activity coefficient.     

Calculation of bridge function coefficients via modified Mayer-samplings

Two modified Mayer-sampling methods are described based on Transition Matrix Monte Carlo (TMMC) and overlap sampling for calculating the integrated diagrams appearing in the coefficients of the bridge function; b_n (r) constructed in terms of the total correlation function h(r) for a hard-sphere system. Calculations are performed using prescribed h(r) for reduced densities at 0.2, 0.5 and 0.8 up to the third-order expansion in density. The results from these methods compared with the generic Monte Carlo Mayer-sampling are analysed in detail. It was found that the TMMC Mayer-sampling approach shows better precision over core and tail regions of b ₂(r) and b ₃(r), and the overlap sampling method shows overall improvement in the precision of the bridge coefficients. Both methods can be straightforwardly applied to calculate higher order bridge function coefficients and to any model systems with relatively simple modifications.     

Deterministic methods for numerical simulation of high-energy runaway electron avalanches

The possibilities of two deterministic methods for describing the kinetics of high-energy runaway electrons (REs) are analyzed as alternatives to stochastic methods requiring unrealistically large computing resources in problems of numerical simulation of electric discharges in dense gases involving REs. One of the methods being developed in recent years is based on multigroup equations for the moments of the electron distribution function, while the second method, which is conventionally used to solve problems in gas discharges, is based on the diffusion-drift equation. The modern method of multigroup equations allows one to calculate the RE energy distribution and the spatial RE distribution along the electric force, which are close to these distributions obtained by the Monte Carlo method if the number N of energy groups is chosen properly. The diffusion-drift equation does not give the energy distribution, but its advantage is the possibility of obtaining spatial RE distributions using small computing resources not only along but also perpendicular to the electric force, which are close to those calculated by the Monte Carlo method. To simulate discharges by the method of multigroup equations, it is necessary to know a priori the number N of groups providing good accuracy, the characteristic RE multiplication time t _e , and the energy runaway threshold ɛ_th as functions of the electric-field overvoltage. The diffusion-drift equation requires the specification, along with t _e , of the directed RE velocity and longitudinal and transverse diffusion coefficients calculated by the Monte Carlo method.     

Simulation study of random sequential adsorption of mixtures on a triangular lattice

Random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding random walks on the lattice. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the deposition processes in two-component mixtures. Approach to the jamming limit in the case of mixtures is found to be exponential, of the form: θ(t) ∼ θ_jam - Δθ exp(- t/σ), and the values of the parameter σ are determined by the order of symmetry of the less symmetric object in the mixture. Depending on the local geometry of the objects making the mixture, jamming coverage of a mixture can be either greater than both single-component jamming coverages or it can be in between these values. Results of the simulations for various fractional concentrations of the objects in the mixture are also presented.     

The fluctuation kinetics of formation of nanoparticles: Particle-size distribution

A stochastic simulation of the growth of particles on a uniform cubic lattice was performed by the Monte Carlo method. Changes in the width of the distribution (M _w /M _n ) as the size of particles increased were extremal in character. Distribution narrowing occurred much more slowly than in classic polymerization. An empirical equation relating the number of free vacancies of a growing particle and its mean size was obtained. The introduction of a stabilizer deactivating free vacancies of a growing particle caused the appearance of a critical phenomenon. At stabilizer concentrations higher than critical, large-sized particles could not form. At stabilizer concentrations close to critical, the particle-size distribution was bimodal. This resulted in an anomalously larger distribution width.     

Successive defects asymmetric simple exclusion processes with particles of arbitrary size

This paper uses various mean-field approaches and the Monte Carlo simulation to calculate asymmetric simple exclusion processes with particles of arbitrary size in the successive defects system. In this system, the hopping probability p (p<1) and the size d of particles are not constant. Through theoretical calculation and computer simulation, it obtains the exact theoretical results and finds that the theoretical results are in agreement with the computer simulation. These results are helpful in analysing the effect of traffic with different hopping probabilities p and sizes d of particle.     

Novel neon-hydrate of cubic ice structure

The stability of ice I cubic (ice I_c) whose voids are occupied by neon particles is investigated using a hybrid type of isobaric grand-canonical Monte Carlo simulation. We show that the resulting neon hydrate is stable under high pressure and temperature where ice I_c alone is unstable, suggesting the existence of a novel neon hydrate of ice I_c. We also show through chemical potential calculation that the neon hydrate of the ice I_c structure is more favorable than the neon hydrate of the ice II structure, whose existence was proven from experiment under high pressure condition.     

Simulated migration and reaction of correlated point defects

Monte Carlo techniques are applied to defect migration in the vicinity of a fixed reaction volume. Intended to simulate “stage I_D ” annealing of correlated interstitials and vacancies in fcc metals, large reaction spheres of up to 720 atomic volume are employed, with mobile defects in the (100) split configuration symmetrically diffusing from maximum distances of more than twice the capture radius r ₀ and for a maximum of 100 jumps. This discrete approach and continuum theory are judged to be equally valid. Random walk recovery probabilities are not a smooth function of initial distance r, but fQr large reaction volumes agree with continuum theory nearly as well for r ～ r ₀ as for r > 2r ₀. Absolute agreement is improved as the “dumbbell” separation of the split defect is increased. Recovery due to an extended distribution of defects is obtained by weighting individual walks from distances < 3r ₀: the resulting composite annealing curves disallow observation of structure and compare favorably with resistivity data. The number of symmetric random walk jumps N required to reach the maximum rate of correlated recovery-the “I_D peak”-is found for a wide range of initial distributions, the best parameter estimates for Cu giving N = 50 ± 15. This value and resistivity data give N for the uncorrelated recovery peak in agreement with theory. One dimensional migration is excluded by these results.     

The Fe–Cr system: atomistic modelling of thermodynamics and kinetics of phase transformations

To understand the behaviour of irradiated defects and kinetic pathways of micro-structural evolution in Fe–Cr alloys, we use a combination of density functional theory with statistical approaches involving cluster expansions and Monte Carlo simulations. A lowest negative mixing enthalpy is found at 6.25% Cr that is consistent with our DFT prediction of an ordered Fe₁₅Cr structure. At 50% Cr, it is found that the predicted enthalpy of formation is 4 times smaller than that calculated by the CPA approach. Thermodynamic and precipitation properties are then discussed in term of segregation between the Fe₁₅Cr and α^′-Cr phases and of vacancy-mediated kMC simulation. To cite this article: D. Nguyen-Manh et al., C. R. Physique 9 (2008).     

Equation of state of mixtures of simple fluids at high pressures

Recently Lee and Levesque have extended the Weeks-Chandler-Andersen pure fluid perturbation theory to mixtures and have compared the results to the Leonard-Henderson-Barker mixture theory. The results seem to favour the Leonard-Henderson-Barker theory. This and other previous comparisons of mixture theories have been mostly confined to the study of ‘zero pressure’ thermodynamic properties. In this paper we compare the Lee-Levesque, Leonard-Henderson-Barker and the van der Waals one-fluid theories to high pressure equation of state data for helium-xenon mixtures. This system is modelled by a binary mixture of Lennard-Jones fluids and the hard sphere reference system is characterized by the Grundke-Henderson hard sphere mixture radial distribution function. The Lee-Levesque theory compares favourably with experimental equation of state data up to pressures of 2000 atmospheres. The Leonard-Henderson-Barker and van der Waals one theories are satisfactory. Although the van der Waals one theory yields the poorest results, it does offer the advantage of having the greatest ease of computation.

Previous theoretical and Monte Carlo calculations of mixture properties have assumed the geometric mean rule for the mixed interaction energy parameter, ε ₁₂, with consequent disagreement with experimental results. We point out that ε ₁₂ can be determined from mixed second virial coefficient data and use such improved determinations of ε ₁₂. We show that this method yields significantly improved theoretical predictions.     

Calculation of the ordering of the atoms in alloys with lattices of the Cu3Au type by the Monte Carlo method

The Monte Carlo method is used to calculate the temperature dependences of the short- and long-range ordering parameters in alloys with lattices of the Cu₃Au type. A new local ordering parameter is introduced and calculated. The equilibrium structure of the ordered alloy has a clearly-expressed domain character at high temperature. Above a certain temperature T_c the alloy contains small ordered complexes with lattices of the Cu₃Au and CuAu types. The relative temperatures kT_c/w₁ derived from various theories are compared. The quantity kT_c/w₁ obtained on the basis of the best analytical approximations agrees closely with that calculated by the Monte Carlo method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 76–85, February, 1973.In conclusion, the authors wish to thank L. E. Popov for constant interest in the work and O. K. Gordeev for Idndly placing the program for computing and plotting the isolines at their disposal.