Critical Fluctuations in Models with van der Waals Interactions

In the classical ideal gas there are no interparticular forces. The Dutch scientist Johannes Diderik van der Waals derived a correction leading to the van der Waals equation. The interparticular forces in the van der Waals equation can not be explained by covalent bonds (characterized by the sharing of electron pairs) of the particles (atoms and molecules) in the gas. Those forces can only be explained via weak interactions and can be attractive or repulsive. The weak interactions between atoms and molecules have later been named after Johannes Diderik van der Waals. Weak interactions are not restricted to intermolecular interactions but can also occur intramolecular. In both cases the weak interactions can be responsible for the shape of molecules, supersystems and clusters. A famous example for this is the protein folding in RiboNucleic Acid (RNA) and DeoxyriboNucleic Acid (DNA). The effect of medical compounds often is founded on weak interactions with something in the human body. Unfortunately, the calculation of weak interactions is a challenging task. Very accurate ab initio quantum mechanical methods and a good description of the molecular orbitals are needed. Both requirements lead to high computational demands (calculation time and storage usage for main memory and hard disc). In ab initio quantum chemistry one normally only calculates the (total) energy E of a system. This is done with an accuracy of approximately 99% (when compared to the exact result) if the Hartree-Fock method (HF) is used. For many cases this accuracy is not sufficient to achieve the chemical accuracy of 1 kcal/mol. In contrast to thermochemistry where (reaction) enthalpies are typically calculated from standard enthalpies of formation (which are of the same order of magnitude as the reaction enthalpies and a certain accuracy of the formation enthalpies leads to a result of similar quality of the relative energies) in ab initio quantum chemistry calculations, the calculated total energies are larger by orders of magnitude than the energy of the desired property. For weak interactions the ratio of calculated values and the desired property is even larger and dynamic electron correlation, which is not covered at all by HF, plays a larger role or is sometimes responsible for the complete interaction energy. In particular the dispersion energy is a pure electron correlation effect. Methods that give a correction to HF to include dynamic electron correlation in the calculated energy need larger basis sets for the description of the molecular orbitals than HF itself. In the example of the interaction energy of the benzene dimer, the total electronic energy of the dimer is approximately 300000 kcal/mol, whereas the interaction energy is approximately 3 kcal/mol. This ratio of 105 illustrates why an accuracy of more than 99% and hence methods that cover dynamic electron correlation are necessary. The Coupled Cluster method with single and double excitations and triple excitations at perturbation theory level (CCSD(T)) (see Section 3.10) is considered to be highly accurate for most weak interactions. The convergence of the total energy with the basis set size can be greatly improved by the introduction of explicit correlation. Unfortunately CCSD(T) is rather expensive and systems with more than 30 atoms are very hard to calculate. Many systems of interest with weak interactions are larger, and therefore CCSD(T) is not feasible for their calculation. The MP2 method is much cheaper[c] than the CCSD(T) method, but unfortunately not as accurate. Earlier attempts to improve MP2 like SCS-MP2 have been successful to some extent but were never totally satisfying. Other attempts such as MP2.5 are more expensive and therefore also restricted to smaller systems. The goal of this work was to test the effects of explicit correlation in the CCSD(T) method and to develop a new method for the calculation of weak interactions that is able to improve on MP2 and SCS-MP2 towards CCSD(T). A new method - Dispersion-Weighted-MP2[d] (DW-MP2) has been developed in this work. It combines MP2 and SCS-MP2 and will be shown to achieve the accuracy needed for chemical accuracy.

The inherent structures (IS) are the local minima of the 3N-dimensional potential energy surface, or landscape, of an N-atom system. Stillinger has given an exact IS formulation of thermodynamics. Here the implications for the equation of state are investigated. It is shown that the van der Waals (vdW) equation, with density-dependent a and b coefficients, holds if the averaged IS energy is close to its high-temperature plateau value. The density-dependence alone significantly enriches the equation of state. Furthermore, an additional "landscape" contribution to the pressure is found at lower T. The resulting extended vdW equation is capable of yielding a waterlike density anomaly, flat isotherms in the coexistence region vs vdW loops, and several other desirable features. The plateau IS energy, the width of the distribution of IS, and T(TOL), the "top of the landscape" temperature at which the plateau is reached, are simulated over a broad reduced density range, 2.0>or=rho>or=0.20, in the Lennard-Jones fluid. Fits to the data yield an explicit equation of state, which is argued to be plausible at high density. Nevertheless, a(rho(c)) and b(rho(c)), where rho(c) is the critical density, are in excellent agreement with the standard values obtained by fitting the vdW equation at the critical point.

The van der Waals equation is used to study the properties of a liquid-gas system in the region just above the critical point. The $P\text{\ensuremath{-}}T$ curve of maximum isothermal compressibility, which persists to 1.068 times the critical temperature, is found to be a natural extension of the liquid-vapor equilibrium curve. Since the van der Waals equation is analytic, the present results do not support the view, recently advanced by Douglass, that there might be a fundamental difference between a liquid and a gas.

It is shown that in the grand canonical ensemble for the case of a lattice system the fluctuations in energy and particle number have an asymptotic (for large volume) normal distribution. A similar statement is valid for fluctuations in energy and particle number for a sufficiently large region of the lattice even in the case of a limiting Gibbs distribution. Both theorems are proved provided the activity z is small enough or the temperature T is sufficiently high.

Zeolitic encapsulation of small non-polar gas molecules under appropriate p–T conditions (Vr > 1) can be conveniently dealt with in terms of van der Waals' equation of state for real gases. Thus, the amount of gas trapped in the zeolitic cages can be predicted if the cage volumes are known; otherwise, void-volume values may be obtained from encapsulation experiments. These void-volume values, as obtained from hydrogen encapsulation in A-type zeolites, are ca. 20% larger than those reported in the literature, but in agreement with calculated void volumes derived from crystallographic data and from a refined structural model of zeolite A. The ‘jump’ in encapsulation capacity at ca. 2.5 univalent blocking cations per unit cell is explained in terms of the percolation theory as transition from β-cage encapsulation to encapsulation in both the α- and β-cages. Based on the van der Waals equation, calculated available void volumes and the percolation theory, a theoretical encapsulation curve is proposed which agrees fairly well with experimental results reported so far for H2 in Cs, Na-A zeolites and for Ar, Kr and CH4 in K, Na-A zeolites.

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions, which generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

The study of detonation processes in gaseous media is of interest for the coal mining, aviation, aerospace industries, and hydrogen energy. Under high pressure conditions during the passage of a detonation wave, it is necessary to take into account the equation of state of a real gas, the effects of endothermicity and exothermicity. The paper considers the flow of real gas through a plane detonation wave. A modified Rankine-Hugoniot equation has been obtained to describe the dynamics of changes in the parameters of a van der Waals gas flow when it passes through a detonation wave. The relations that determine the limiting pressure for the existence of detonation in a Van der Waals gas were obtained. The dependence of the limiting pressure on the parameters A and B of the Van der Waals equation of state is shown. An increase in parameter A slows down the growth of pressure in the detonation wave, and an increase in parameter B intensifies it. An equation for determining the velocity of combustion products in a real gas is obtained. Compared with the velocity of combustion products in an ideal gas. For an ideal gas, combustion products flow from the detonation front at a critical (sonic) velocity. For a Van der Waals gas, the velocity of combustion products can be greater than the critical one. Moreover, both parameters characterizing the additional pressure (A) and the additional volume (B) lead to the acceleration of combustion products. The additional pressure transfers its energy to the flow and thus causes acceleration. The additional volume leads to a decrease in the degree of compression in the same way as the limiting compression (28). Therefore, the flow velocity increases.

Thermal gravity convection in a horizontal layer of compressible perfect gas heated from below and a van der Waals gas near the critical state is investigated. The characteristics of the isentropic equilibrium of a compressible medium with a van der Waals equation of state are considered. The known conditions of convection onset in the perfect and van der Waals gases are checked on the basis of a solution of the complete and linearized equations. The restrictions imposed in deriving the known formulas for the adiabatic temperature gradients used in the conditions of absence and onset of convection are discussed. The characteristics of the convective heat transfer are examined, including the causes of the heat-transfer deterioration in the near-critical medium above the hydrostatic equilibrium threshold.

A comparative survey is made of the neutron and light scattering techniques as applied to structural investigations in the vicinity of critical points in fluids. For neutron scattering as applied to fluctuations in binary mixtures, some criteria for a good contrast in order to optimize the scattering parameters such as the isotopic composition or the concentration are considered. This leads us to show how to write the partial structure factors in binary mixtures in order to remove the coupling between concentration and density fluctuations artificially introduced bv Bhatia and Thornton when the components are of different size. Finally, the amplitudes of the concentration fluctuations near a critical point are discussed.

On the basis of the van der Waals equation we show that the maxima in the specific heat at constant pressure (infinite below the critical point and finite above) lie on one continuous line through the critical point. Experimental data fit the curve, above as well as below the critical point, although the van der Waals equation is known to be not entirely correct. The calculation is based on the rules of Ehrenfest for higher order transitions and does not make use of the caloric properties of the substance.

We explore the properties of a simple analytic equation of state, proposed by Heyes and Woodcock [D. M. Heyes and L. V. Woodcock, Mol. Phys. 59, 1369 (1986)] which is a development of the van der Waals equation of state, improved with an analytically simple yet accurate hard-core component for the compressibility factor. We call this the hard-sphere van der Waals (HSvdW) equation of state. We show, for the first time, that the HSvdW equation of state gives analytic values for the critical parameters and that it gives significantly more realistic values for the critical density than the van der Waals and other cubic equations of state. For single-component square wells of arbitrary well width, the HSvdW equation of state also produces a critical temperature and pressure in agreement with those from a recent Gibbs Monte Carlo simulation study. The chief advantage over a similar equation of state with an alternative form for the hard-core compressibility factor, such as Carnahan–Starling, is that it leads to more tractable analytic expressions for thermodynamic properties without loss of accuracy. Extensive Monte Carlo computer simulations of square-well (SW) and square-shoulder (SS) fluids with variable well width have been performed. We have calculated the compressibility factor, internal energy, pair distribution functions, and coordination number (average number of particles within the soft-shell interaction range). We have also determined the gas–liquid coexistence curve using the particle insertion chemical potential method for the square-well case, λ=1.5σ, where λ is the diameter of the soft interaction shell. This agrees well with the Gibbs ensemble simulation coexistence curve. For the purpose of improving the attractive component of the HSvdW equation of state, we have made use of the SW and SS simulation data and that of other workers to develop another new simple equation of state based on the quasichemical local coordination number approximation (QCA) which is a lattice model of a fluid. The QCA is improved to include the local coordination of the hard-sphere fluid. Present adaptations extend the range of earlier QCA models to arbitrary well width and sign of the well (either attractive well or repulsive shoulder). The simulation coordination numbers are reproduced well by the present model, although there is little improvement in the overall equation of state over the simpler HSvdW.

The van der Waals equation is well known for the description of two-dimensional monolayers. The formation of a monolayer is the result of a compromise between the process of self-organization on the surface and the probabilities of spatial configurations of adsorbate molecules near the surface. The main reasons for the geometric heterogeneity of the monolayer are the geometric disorder and the energy inhomogeneity of the surface profile. A monolayer is a statistically related system and its symmetry causes correlations of processes at different spatial scales. The classical van der Waals equation is written for the two-dimensional, completely symmetric Euclidean space. In the general case, the geometry of the monolayer must be defined for the Euclidean space of fractional dimension (fractal space) with symmetry breaking. In this case, the application of the classical van der Waals equation is limited. Considering the fractal nature of the monolayer–solid interface, a quasi-two-dimensional van der Waals equation is developed. The application of the equation to experimental data of an activated carbon is shown.

On equations of state in a lattice Boltzmann method

This study evaluated the monthly average of the enthalpy, geopotential, latent heat content, moist static energy, and dry static energy at several levels of the atmosphere as a Van Der Waals gas from radiosonde data at 00:00 UTC and 12:00 UTC over the metropolitan region of Natal-RN-Brazil. The years 2015 and 2017 were considered because they presented more than 70% of complete data, with days coinciding at both times. To construct the vertical profiles of the variables, the density, specific humidity, and geopotential were calculated at each level through a mathematical approximation based on the Calculus. The Van der Waals-like atmosphere results showed that the latent heat content compensated for any change in enthalpy and potential energy from the surface to the mid-troposphere. Energy was exchanged among enthalpy, geopotential, and latent heat content, showing a roughly constant moist static energy. The moist static energy was a conserved quantity under adiabatic and hydrostatic transformations, although the dry static energy was not conserved. The results of the atmosphere as a Van Der Waals gas and the atmosphere as an ideal gas were compared, proving to be similar.

Many models and real complex systems possess critical thresholds at which the systems shift dramatically from one sate to another. The discovery of early-warnings in the vicinity of critical points are of great importance to estimate how far the systems are away from the critical states. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the multifractal and geometrical properties of the magnetization time series of the two-dimensional Ising model. Multifractality of the time series near the critical point has been uncovered from the generalized Hurst exponents and singularity spectrum. Both long-term correlation and broad probability density function are identified to be the sources of multifractality. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties. Evolution of the topological quantities of the visibility graph, along with the variation of multifractality, serve as new early-warnings of phase transition. Those methods and results may provide new insights about the analysis of phase transition problems and can be used as early-warnings for a variety of complex systems.