Berthelot Rule Research Articles

Kinetic simulations of laser-induced plume expansion from a copper target into a vacuum or argon background gas based on ab initio calculation of Cu–Cu, Ar–Ar, and Ar–Cu interactions

The kinetic simulations of plume expansion induced by pulsed laser heating of a copper target in a vacuum or low-pressure argon background gas are performed based on the direct simulation Monte Carlo (DSMC) method and ab initio quantum mechanical calculation of interactions between copper and argon atoms. The potential energy curves (PECs) for Cu–Cu, Ar–Ar, and Ar–Cu interactions are obtained in density functional theory (DFT) calculations with the van der Waals (vdW) correction. The computed Cu–Cu PEC is strikingly different from the Lennard-Jones (LJ) potentials with semi-empirical parameters, which were previously suggested for kinetic simulations of the copper vapor flows. It is found that the Lorentz–Berthelot rule cannot reliably predict the parameters of the LJ potential for cross-species Ar–Cu interaction. The DFT-vdW PECs are fitted by the Morse long-range (MLR) potentials. The MLR potentials are used to compute the outcomes of binary collisions in the DSMC method based on the solution of the classical scattering problem and to parameterize the variable hard sphere (VHS) collision model. The results of the DSMC simulations based on DFT-vdW PECs are compared with the results obtained based on various parameterizations of the VHS model. It is shown that the previously developed parameterizations of the VHS model can either over- or underestimate the plume temperature and density compared to the results obtained based on the DFT-vdW PECs. The simulations also reveal the strong effect of the cross-species collision model parameters on the flow structure in the mixing layer, which is dominated by molecular diffusion.

Determining the three-phase coexistence line in methane hydrates using computer simulations

Molecular dynamics simulations have been performed to estimate the three-phase (solid hydrate-liquid water-gaseous methane) coexistence line for the water-methane binary mixture. The temperature at which the three phases are in equilibrium was determined for three different pressures, namely, 40, 100, and 400 bar by using direct coexistence simulations. In the simulations water was described by using either TIP4P, TIP4P/2005, or TIP4P/Ice models and methane was described as simple Lennard-Jones interaction site. Lorentz-Berthelot combining rules were used to obtain the parameters of the cross interactions. For the TIP4P/2005 model positive deviations from the energetic Lorentz-Berthelot rule were also considered to indirectly account for the polarization of methane when introduced in liquid water. To locate the three-phase coexistence point, two different global compositions were used, which yielded (to within statistical uncertainty) the same predictions for the three-phase coexistence temperatures, although with a somewhat different time evolution. The three-phase coexistence temperatures obtained at different pressures when using the TIP4P/Ice model of water were in agreement with the experimental results. The main reason for this is that the TIP4P/Ice model reproduces the melting point of ice I(h).

Nanoscale Poiseuille flow and effects of modified Lennard–Jones potential function

Numerical simulation of Poiseuille flow of liquid Argon in a nanochannel using the non-equilibrium molecular dynamics simulation (NEMD) is performed. The nanochannel is a three-dimensional rectangular prism geometry where the concerned numbers of Argon atoms are 2,700, 2,550 and 2,400 at 102, 108 and 120 K. Poiseuille flow is simulated by embedding the fluid particles in a uniform force field. An external driving force, ranging from 1 to 11 PN (Pico Newton), is applied along the flow direction to inlet fluid particles during the simulation. To obtain a more uniform temperature distribution across the channel, local thermostating near the wall are used. Also, the effect of other mixing rules (Lorenthz–Berthelot and Waldman–Kugler rules) on the interface structure are examined by comparing the density profiles near the liquid/solid interfaces for wall temperatures 108 and 133 K for an external force of 7 PN. Using Kong and Waldman–Kugler rules, the molecules near the solid walls were more randomly distributed compared to Lorenthz–Berthelot rule. These mean that the attraction between solid–fluid atoms was weakened by using Kong rule and Waldman–Kugler rule rather than the Lorenthz–Berthelot rule. Also, results show that the mean axial velocity has symmetrical distribution near the channel centerline and an increase in external driving force can increase maximum and average velocity values of fluid. Furthermore, the slip length and slip velocity are functions of the driving forces and they show an arising trend with an increase in inlet driving force and no slip boundary condition is satisfied at very low external force (<1 PN).

Molecular models for 267 binary mixtures validated by vapor–liquid equilibria: A systematic approach

By assessing a large number of binary systems, it is shown that molecular modeling is a reliable and robust route to vapor–liquid equilibria (VLE) of mixtures. A set of simple molecular models for 78 pure substances from prior work is taken to systematically describe all 267 binary mixtures of these components for which relevant experimental VLE data is available. The mixture models are based on the modified Lorentz–Berthelot combining rule. Per binary system, one state independent binary interaction parameter in the energy term is adjusted to a single experimental vapor pressure. The unlike energy parameter is altered usually by less than 5% from the Berthelot rule. The mixture models are validated regarding the vapor pressure at other state points and also regarding the dew point composition, which is a fully predictive property in this work. In almost all cases, the molecular models give excellent predictions of the mixture properties.

Non-Lorentz–Berthelot Lennard-Jones mixtures: A systematic study

Binary mixtures of two identical Lennard-Jones fluids with non-Lorentz–Berthelot combining rules have been simulated over the entire concentration range at three state points in order to examine the effect of cross interactions on the mixing properties, excess volumes and enthalpies, and partial molar volumes, and on the structure. Various combinations of deviations, in both the energy and size cross parameters, from the Lorentz–Berthelot rule, have been considered and the results analyzed using a recently proposed method based on the Tikhonov regularization. It is found that the most important is the size cross parameter and that by varying the combining rules a variety of qualitatively different behavior can be produced including, e.g., a minimum in the partial molar volumes observed otherwise only in complex mixtures of associating fluids; occurrence of this minimum seems to be associated with changes in the second coordination shell as witnessed by the pair correlation function.

Calculation of Solid−Liquid Work of Adhesion Patterns from Combining Rules for Intermolecular Potentials

The thermodynamic work of solid−liquid adhesion is a well-defined property. Their phenomenological patterns relating to surface tensions, however, have not been well-characterized. We employed a hard sphere model to determine the solid−liquid work of adhesion patterns from a mean-field theory through calculations of the liquid−vapor, solid−vapor, and solid−liquid interfacial tensions. By plotting the work of adhesion with the liquid−vapor interfacial tension, we constructed curves that appear to behave in a very regular manner for a variety of combining rules; the curves shift regularly when we increase the strength of solid−solid interaction and hence the solid−vapor surface tension. Contact angle patterns were also constructed via Young's equation. We found that, except Berthelot's rule, the (9:3), Steele, and (12:6) combining rules yield essentially similar adhesion patterns. The regularity of the patterns is remarkable and in reasonable agreement with recent experimental findings. We have shown that m...

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 ε22/ε11 = 2; (2) (σ22/σ11)3 = 1 and ε22/ε11 = 1/2 and (3) (σ22/σ11)3 = 1/2 and ε22/ε11 = 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) ε12 = (ε11ε22)1/2 (Berthelot rule); (b) ε12 = ε11 (association); and (c) ε12 = ε22 (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.

New BET-like models for heterogeneous adsorption in microporous adsorbents

Abstract The paper presents a package of isotherm equations for heterogeneous adsorption aimed at the analysis of pore structure of sub- and microporous materials. One considers adsorption of small nearly spherical molecules in irregular pores of molecular size. The generalized BET theory is exploited respecting restrictions for multilayer adsorption (LBET approach). The model is based on thermodynamic relationships expressing changes of internal energy and configurational entropy due to the process. The adsorption energy is evaluated by using the Berthelot rule, and corrected with a factor Z a representing a fraction of effective contacts enabling full adsorbent–adsorbate interaction. Side adsorbate–adsorbate interactions are neglected and constrained multilayer adsorption is considered. One assumes the values for Z a to be uniformly distributed over the first layer adsorption sites within a range depending on the pore size. New models make it possible to obtain information on structure of pores and adsorption mechanisms on the basis of adsorption isotherms of small molecule adsorbates. Exemplary results of new models application for adsorption of CO 2 and CH 4 in an activated carbon are discussed.

Generalized van der Waals Theory of Interfaces in Simple Fluid Mixtures

An efficient implementation of the generalized van der Waals theory of fluids is presented for the calculation of surface tension in simple fluid mixtures. While detailed correlation analysis is avoided the dominant binding energy contribution and the negative contribution due to the nonlocal entropy are accounted for in the free energy density functional by simple physical approximations of the type originally introduced by van der Waals. Efficient computation is achieved by the use of a single-parameter optimization of a tanh-shaped profile representing the total density as well as the composition variation across the interface. This simple profile nevertheless incorporates the expected adsorption to the interface of the volatile component. Application is made to argon/krypton mixtures represented by Lennard-Jones potentials and Lorentz–Berthelot combining rules. Surface tension predictions compare well with both experimental observations and computer simulation results which also indicated close agreement in particle density profiles, especially if the Berthelot rule is amended with a binary interaction parameter slightly (3%) less than unity.

Contact angle measurement and contact angle interpretation

Recent progress in the correlation of contact angles with solid surface tensions are summarized. The measurements of meaningful contact angles in terms of surface energetics are also discussed. It is shown that the controversy with respect to measurement and interpretation of contact angles are due to the fact that some (or all) of the assumptions made in all energetic approaches are violated when contact angles are measured and processed. For a large number of polar and non-polar liquids on different solid surfaces, the liquid–vapor surface tension times cosine of the contact angle, γ lv cosθ, is shown to depend only on the liquid–vapor surface tension γ lv , and the solid–vapor surface tension γ sv when the appropriate experimental techniques and procedures are used. Equations which follow these experimental patterns and which allow the determination of solid surface tensions from contact angles are discussed. Universality of these experimental contact angle patterns is illustrated; other reasons which may cause data to deviate from the patterns slightly are discussed. It is found that surface tension component approaches do not reflect physical reality. Assuming the fact that solid surface tension is constant for one and the same solid surface, experimental contact angle patterns are employed to deduce a functional relationship to be used in conjunction with Young's equation for determining solid surface tensions. The explicit form of such a relation is obtained by modifying Berthelot's rule together with experimental data; essentially constant solid surface tension values are obtained, independent of liquid surface tension and molecular structure. A new combining rule is also derived based on an expression similar to one used in molecular theory; such a combining rule should allow a better understanding of the molecular interactions between unlike solid–liquid pairs from like pairs. Existing static contact angles for 34 different types of solid surfaces from Zisman et al. are evaluated in terms of their solid surface tensions using experimental contact angle patterns. A FORTRAN computer program has been implemented to automate these procedures. It is found that literature contact angles do not have to be discarded completely; they can be used to determine solid surface tensions, with caution. The surface tensions for the 34 solid surfaces from Zisman et al. are also reported.

Accurate high-pressure and high-temperature effective pair potentials for the systems N2–N and O2–O

Statistical mechanical chemical equilibrium calculations of N2 and O2 show that these molecules dissociate behind strong shock waves. Our determination of accurate intermolecular potentials has required the consideration of the dissociation products N and O. Our previous theoretical efforts to predict the thermodynamic properties of these molecules relied in part on corresponding states theory and shock wave data of argon, without consideration of the dissociation products. Recent high-pressure Hugoniot measurements, however, allowed a more accurate determination of the potentials and the explicit inclusion of the dissociation products. The best fit to the data is obtained with the exponential-6 coefficients, for O2–O2: ε/k=125 K, r*=3.86 Å, α=13.2; for O–O: ε/k=700 K, r*=2.40 Å, α=11.0; for N2–N2: ε/k=293 K, r*=3.91 Å, α=11.5; and for N–N: ε/k=600 K, r*=2.47 Å, α=10.0. The unlike pair interactions are obtained from these like interactions with a modified Lorentz–Berthelot rule. The coefficients in the modified Lorentz–Berthelot equations are k/l/m=1/1/0.93 for O2–O– and k/l/m=1/1/0.90 for N2–N interactions.

Phase equilibria in ternary Lennard-Jones systems

Gibbs ensemble Monte Carlo simulations are used to test the ability of the van der Waals one-fluid theory to predict phase equilibria for ternary systems of particles interacting with Lennard-Jones potentials. Simulation results for asymmetric systems are compared to theoretical results for mixtures that obey the Lorentz-Berthelot combining rules. Good agreement is found between theory and simulation far form the plait point. The theory is found to overestimate the plait point, as expected from previous studies of binary systems. Liquid-liquid equilibria in a symmetric ternary system with energy parameters deviating from the geometric mean (Berthelot) rule are also investigated at three different pressures. A three-phase region is found at high pressures. The agreement between theory and simulation is excellent for the symmetric systems studied here.

Phase equilibria of binary Lennard-Jones mixtures: simulation and van der Waals l-fluid theory

Abstract The Gibbs ensemble simulation technique is used to investigate the ability of van der Waals 1-fluid theory to predict phase equilibria for binary Lennard-Jones mixtures. Simulation data for highly asymmetric mixtures with size and energy parameter ratios equal to 1.00, 0.5, 0.4, 0.35 and to 0.5, 0.33, 0.25 respectively are compared to theoretical results for cases in which unlike-pair interactions follow the Lorentz-Berthelot combining rules. Additional comparisons are made for vapour-liquid and liquid-liquid equilibria of mixtures with energy parameter deviating from the geometric mean (Berthelot) rule, and for vapour-liquid and gas-gas equilibria of mixtures with size parameter deviating from the arithmetic mean (Lorentz) rule. Good agreement was found between theory and simulation when the energy parameter deviates from the Berthelot combining rule. The agreement is less satisfactory when the size parameter deviates from the Lorentz rule, especially for the case of gas-gas equilibria.

Simplified RPA Bhatia—Young Calculations of the Phase Diagram for a Model Binary Mixture. I. Equal Size Molecules

Abstract We present the results of calculations of the phase diagram for a model binary mixture of equal size hard-core molecules interacting via square-well potentials, but assuming that the system does not satisfy Berthelot rule. In particular, we analyse the effects that the parameter which measures departures from this rule has on the phase diagram of the system. The calculations are based on the Bhatia—Young model for the bulk properties of liquid mixtures.

Density profile and surface tension of a model binary alloy

We present the results of calculations of the density profiles and surface tension at the liquid-vapour interfaces of a model liquid binary alloy. The components of the model binary alloy interact via square-well potentials, but it is assumed that the system does not satisfy Berthelot's rule. The calculations are based on the density functional theory of inhomogeneous systems and the Bhatia-Young model for the bulk properties of liquid mixtures.

Estimation of the parameters in the Kirkwood-Buff theory of solution using Percus-Yevick fluid mixtures

Parameters G ij in the Kirkwood-Buff theory of solution were calculated for binary fluid mixtures of Lennard-Jones (LJ) 6–12 molecules by using the Percus-Yevick theory. Calculations were carried out for various parameters in the LJ potential. Under the Lorentz-Berthelot rules, G 11 and/or G 22 -composition curves do not show a maximum for any parameters in the LJ potential. When the intermolecular interaction between different species becomes much weaker than that expected from the Berthelot rule, G 11 and G 22 show a maximum and G 12 a minimum. The pressure effect on G ij was examined and calculations at constant pressure were also carried out. G ij is almost independent of the pressure when the ratio of the molecular volume of two components is in the range 1.0 to 2.5. Comparison was made between experimental and calculated G ij for cyclohexane-2,3-dimethylbutane and acetonitrile-toluene systems. For the latter system, the quantitative agreement between the calculated and experimental could not be obtained but showed that the characteristics of G ij -composition curves can be explained qualitatively by using the PY theory.

Negative Hamaker coefficients I. Particle engulfment or rejection at solidification fronts

Calculated Hamaker coefficients were obtained from free energies of adhesion, derived from contact angle data, for several polymers, as a function of the separation. Comparing these values with published Hamaker coefficients, an equilibrium separationd0=1.82 A for various systems was established. It was demonstrated that the Hamaker coefficients A12 calculated from the free energy of adhesion agreed very well with those calculated from the free energy of cohesion, using Berthelot's rule. Visser's procedure of modifying the combining rule to calculateA132 fromAij values was tested and confirmed. Hamaker coefficientsA132 for the interaction of various polymer particles, embedded in naphthalene melt, with the naphthalene solidification front turned out positive in some cases and negative in others, thus predicting attraction of some particles by the solidification front and repulsion of others. These predictions were compared with microscopic observations of the behaviour of particles at the solidification front by which they become either engulfed (attraction) or rejected, i. e. pushed along (repulsion). There was good agreement between prediction and experimental observation.

Monte Carlo study of a phase-separating liquid mixture by umbrella sampling

Recently developed nonphysical sampling methods—umbrella sampling—have been used to obtain the free energy and other properties of a binary liquid mixture exhibiting phase separation with an upper critical solution point. The system is a mixture of two identical Lennard-Jones liquids in which the interactions between the components are characterized by a range parameter σ12 obeying the Lorentz rule, but an energy parameter ε12= (1−α) (ε11ε22)1/2 violating the Berthelot rule. The sampling methods enable one to cover wide ranges of temperature and composition with few Monte Carlo experiments, successfully sampling the metastable regions and obtaining the coexistence curve. The case α=0.25 is studied in detail, and compared with some theoretical predictions. An estimate is made of the minimum value of α required for phase separation at low pressures.

Perturbation theory for mixtures of simple liquids

The perturbation approach developed by Weeks, Chandler, and Andersen (WCA) and by Verlet and Weis (VW) for pure systems is here generalized to the case of mixtures. We study binary mixtures of molecules interacting with the 12–6 Lennard-Jones potential, for which Monte Carlo simulations are available for comparison. The work is divided into two parts: The first part presents results of Monte Carlo calculations on mixtures of hard spheres of 864 and 1000 particles. The radial distribution functions generated are used to test the VW representation for the correlation functions of hard-sphere mixtures. This representation is found to work satisfactorily within the expected error limits. The second part deals with the two-step perturbation procedure for calculating the thermodynamic quantities of the Lennard-Jones system. The Lennard-Jones potential is divided into a reference potential, which is strictly repulsive, and an attractive part. The system of the reference potential is represented by a system of hard-sphere mixture with equivalent diameters determined by the WCA rule. Analytical expressions are given for evaluating these equivalent diameters. The Lennard-Jones system is then recovered to the first order by a λ expansion over the reference system. Comparison with Monte Carlo results for a mixture of Lennard-Jones molecules, obeying the Berthelot rule, shows that the total thermodynamic properties are reproduced by the perturbation theory to 1 per cent, while the agreement in excess properties is only moderately successful, similar to some other analytical theories compared here. To reproduce these excess properties, which are extremely small, a precision of 0·1 per cent in the theory is required. The present theory is estimated to be accurate to 1 per cent in view of the successive approximations made.

Thermodynamics of liquid mixtures of krypton and methane

The vapour pressure of liquid mixtures of krypton and methane has been measured as a function of composition at 103.94, 115.77 and 144.79 K. The excess Gibbs energy of mixing GE has been evaluated from the vapour pressure measurements, and the heat of mixing HE and the excess entropy of mixing SE have been estimated from the temperature dependence of GE. The volume change on mixing, VE, has been measured at 115.77 K, and is very small and negative. Values of the excess functions have been calculated from several recent theories. Even though the system only departs slightly from ideal behaviour, none of the theories applied predicts excess functions in good agreement with experiment if the geometric mean (Berthelot) rule is adopted for the intermolecular energy of a pair of unlike molecules. The differences between the theoretical and experimental values of GE can be annulled by departures from the geometric mean rule of only a few tenths of 1 %. Boron trifluoride, which was used as a cryostat material to give the constant temperature of 144.79 K, appears to crystallize in two forms with triple-points only 0.09 K apart.