The singular points and phase diagram of the supercritical region of a substance

This paper suggests using the supercritical point and the maximum fluctuation point on the supercritical isotherm for the analysis of the behavior of various substances in the vicinity of the critical point. These three points lie at the vertices of a triangle that is formed by the supercritical isotherm, the line of the local minima of stability, and the line of maxima of fluctuations. In this triangle, which is called supercritical, the fluctuations and instability behave such that this part of the phase surface is most interesting from the viewpoint of performing various chemical reactions. Here, large fluctuations and the stability of the system rapidly decrease with increasing volume. This region is studied in the approximation of the van der Waals and Van Laar equations.     

The critical state and its thermodynamic stability

The Gibbs approach to study of the critical state is expanded. It is noted that the well known critical state condition D=0 was obtained by Gibbs as a necessary condition for a non-zero solution of a system of homogeneous linear equations and defines a line of spinoidal solutions. To determine the critical point on this line the explicit form of solutions of a system of homogeneous equations is considered and their relationship to the slope of the phase equilibrium line is established. For the condition D=0 four possible cases of critical system thermodynamic behavior are considered. The relationship between each of these cases and the slope of the phase equilibrium line at the critical point is established. General thermodynamic stability conditions for the critical point are considered as a function of the type of critical behavior at the point. It is shown that one of the consequences of critical point stability is a merger of the binodal line with the spinodal at the critical point.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 93–98, June, 1988.     

The concentration fluctuations and the mutual diffusion coefficient in the supercritical solution

Molecular dynamics simulations on Lennard-Jones mixture in the supercritical region are performed. The number of the molecules in the unit cell is 23328 and the mole fraction is 0.5. The concentration fluctuations near the critical point are large compared to the ideal mixing. The mutual diffusion coefficient normalized by the self-diffusion coefficient has a small value near the critical density because of the large concentration fluctuation. The heat capacity, partial molar volume and correlation lengths of density and concentration in the supercritical region are compared with those at the normal density.     

Three-dimensional phase diagram of a ferroelectric

On the basis of the molecular-dynamic theory, by phenomenologically generalizing the expression for the energy of a ferroelectric, an equation of state is derived here for a ferroelectric in electric field intensity as well as temperature and mechanical stress. The σ(mechanical stress) — T(temperature) phase diagram is analyzed, whereupon ths spinodal, the critical point, and the supercritical line extending into the low-temperature range are plotted. A superposition of this diagram with the E (electric field intensity) — T(temperature) diagram yields a three-dimensional phase diagram in all three state variables. Its elements are the spinodal surface, the critical line, and the supercritical surface bordering on the critical line.     

SINGULAR PERTURBATION EXPANSION FOR MARKOVIAN MASTER EQUATIONS

Asymptotic properties of the master equations for chemical reactive systems whose macroacopic rate equations have more than one stationary state are discussed Itsing generating function method. The systematic singular perturbation expansion method for equation of generating function is generalized to include the case of multi-stationary system beyond the bifurcation point and the following conclusions are proved: For such systems, there is a Gaussian fluctuation before next genuine bifurcation point] a critical fluctuation at a genuine bifurcation point; and a macroscopic fluctuation when the system is on the coexistence line which is determined by the master equation. Furthermore, the relation between the cumulant of generat-ing function and the stochastic potentia 1 is also established. Our discussion, however, is limited to homogeneous system of one variable only.     

Hydrothermal Synthesis of Metal Oxide Nanoparticles at Supercritical Conditions

Hydrothermal synthesis of CeO₂ and AlO(OH) were conducted using a flow type apparatus over the range of temperature from 523 to 673 K at 30 MPa. Nanosize crystals were formed at supercritical conditions. The mechanism of nanoparticle formation at supercritical conditions was discussed based on the metal oxide solubility and kinetics of the hydrothermal synthesis reaction. The reaction rate of Ce(NO₃)₃ and Al(NO₃)₃ was evaluated using a flow type reactor. The Arrhenius plot of the first order rate constant fell on a straight line in the subcritical region, while it deviated from the straight line to the higher values above the critical point. The solubility of Ce(OH)₃ and AlO(OH) was estimated by using a modified HKF model in a wide range of pH and temperature. In acidic conditions, where hydrothermal synthesis reaction is concerned, solubility gradually decreased with increasing temperature and then drastically dropped above the critical point. The trend of the solubility and the kinetics around the critical point could be explained by taking account of the dielectric constant effect on the reactions. There are two reasons why nanoparticle are formed at supercritical conditions. Larger particles are produced at subcritical conditions due to Ostwald ripening; that could not be observed in supercritical water because of the extremely low solubility. Second reason is the faster nucleation rate in supercritical water because of the lower solubility and the extremely fast reaction rate.     

Critical behavior and 1/f noise in lumped systems undergoing two phase transitions

It is shown that the interaction of order parameters when subcritical and supercritical phase transitions take place simultaneously may result in a self-organized critical state and cause a 1/f ^α fluctuation spectrum, where 1≤α≤2. Such behavior is inherent in potential and nonpotential systems of nonlinear Langevin equations. A numerical analysis of the solutions to the proposed systems of stochastic differential equations showed that the solutions correlate with fractional integration and differentiation of white noise. The general behavior of such a system has features in common with self-organized criticality.     

The calculation of singular points in the supercritical region for a system with a Lennard—Jones interaction potential

In this work the positions of the critical point, the supercritical point, and the maximum fluctuation point in a supercritical isotherm were found for a system with the Lennard—Jones interaction potential. Virial coefficients and methods based on accelerated convergence of the perturbation-theory series, which are well known for such systems, were used. The results were compared with computer-simulation data. As has been established, if one uses the positively defined Weeks—Chandler—Andersen potential as a reference system, the calculated parameters tend monotonically to exact values as a function of the number of virial coefficients. This decomposition is favorably different from the virial one, where the aspiration is not monotonic. These results indicate that this method makes it possible to determine the positions of the three vertices of the supercritical triangle with an accuracy that is comparable to that of a simulated experiment.     

Distinctive features of random local motions in critical and supercritical fluids

Equations for large-scale local fluctuations in fluids, from an ideal gas to an incompressible fluid, including the critical and supercritical state are derived for the first time based on the first principles. The modern phenomenological representation of the critical state of fluids is confirmed and essentially refined; in particular, it is demonstrated that that local density fluctuations in a compressible fluid are accompanied by nonthermodynamic fluctuations in the collective velocity and temperature of the fluid. Distinctive features of the development of these fluctuations near the critical point determine the specific behavior of fluids in the critical and supercritical states.     

Structure of liquid solutions near the critical point

Specific features of the structure of the critical state of binary liquid solutions leading to an anomalous behavior of the Rayleigh line due to a dramatic increase in concentration and density fluctuations are considered. It is shown that an experimental treatment must deal with two fluctuation regions near the critical point of solvent vaporization. In the first region, one can achieve a sufficient degree of accuracy by using theories like self-consistent field theory. In the second region, which is closer to the critical point than the first region, scaling theory of second-order phase transitions may be applied. It is found that the anomalous behavior of the Rayleigh line associated with kinetic coefficients is determined by the equilibrium thermodynamic properties and by the radius of fluctuation correlation (r_c). A general theory is developed for calculating thermodynamic potentials, especially the chemical potential and its concentration derivative in the fluctuation region. The results of these calculations are compared with the experimental data briefly described in the paper. Translated fromZhurnal Struktumoi Khimii, Vol. 39, No. 4, pp. 655–668, July–August, 1998.     

Amplitude expansions for instabilities in populations of globally-coupled oscillators

We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes.     

Liquids: Dynamics of liquid structure

The study of the liquid state is predominately an experimental subject so any useful definition of “dynamical structure” in liquids should be closely related to the types of experiments employed in these studies. Most studies of the liquid state have been carried out for fluids lying on or near the liquid-vapor coexistence curve, the line connecting the triple point (T.P.) and the critical point (C.P.). The shaded portion of Figure 1 indicates this region. The term dynamical structure suggests measurements on systems which are not in thermodynamic equilibrium. Even so, equilibrium properties are determined through the application of the fluctuation dissipation theorem. Before we present a heuristic definition of dynamical structure, let us first consider an example which illustrates this definition.     

A study on the correlations of supercritical temperature of fluid alkali metals based on the generalised van der Waals equation of state

The supercritical temperature, the inversion temperature and the critical compressibility factor of caesium, rubidium and potassium are correlated through the generalised van der Waals equation of state. This three-parameter equation differs from the known van der Waals equation of state by the introduction of a third parameter in the expression for molecular pressure. For caesium, rubidium and potassium, the ratios of the supercritical temperature to the inversion temperature are estimated. The results of estimation are in agreement with the results from other equations which are based on experimental data. It has been established that caesium, rubidium and potassium obey the single-parameter law of corresponding states with the ratio of the supercritical temperature to the inversion temperature as the thermodynamic similarity parameter.     

Effects of pair correlation on mean-field theory of BTW sand pile model

We extend the mean-field calculation of BTW sand pile model to one that includes the correlation between pairs of nearest neighbors. Specifically, we derive dynamical equations of both one-site and two-site densities, and solve the equations order by order starting with the mean-field solution. The investigation provides analytical results for both stationary and dynamic states of the sand pile near the critical point, which are valid in the regime where h?ε²?1 (h= incoming rate of sand grains, ε=bulk dissipation rate of sand grains). In the stationary case, we evaluate the pair correlation and the correction to the mean-field single-site densities due to the correlation. The correction is found to be of the same order as the mean-field solution. In the dynamic case, the initial state deviates from the stationary state by a small fluctuation, which subsequently decays exponentially, with the time constant being reduced from the corresponding mean-field value. Again, the correction to the time constant in this case is found comparable to the mean-field value itself.     

Shear-thinning-induced chaos in taylor-couette flow

The effect of weak shear thinning on the stability of the Taylor-Couette flow is explored for a Carreau-Bird fluid in the narrow-gap limit. The Galerkin projection method is used to derive a low-order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional nonlinear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the base (Couette) flow, becomes lower as the shear-thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow. Similar to Newtonian fluids, there is an exchange of stability between the Couette and Taylor vortex flows, which coincides with the onset of a supercritical bifurcation. However, unlike the Newtonian model, the Taylor vortex cellular structure loses its stability in turn as the Taylor number reaches a critical value. At this point, a Hopf bifurcation emerges, which exists only for shear-thinning fluids.     

Mixed spin Ising model with four-spin interaction and random crystal field

The effects of fluctuations of the crystal field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions are investigated within the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the two-dimensional square lattice. It has been found that the system exhibits a variety of interesting features resulting from the fluctuation of the crystal field interactions. In particular, for low mean value D of the crystal field, the critical temperature is not very sensitive to fluctuations and all transitions are of second order for any value of the four-spin interactions. But for relatively high D, the transition temperature depends on the fluctuation of the crystal field, and the system undergoes tricritical behaviour for any strength of the four-spin interactions. We have also found that the model may exhibit reentrance for appropriate values of the system parameters.     

Magnetic fluids in an external field

Within mean field approximation we investigate the phase diagrams of magnetic fluids in presence of a magnetic field. In a finite field the magnetic phase transition is absent, but instead a line of first order liquid-liquid transitions ending in a critical point occurs for a magnetic interaction, which is sufficiently strong. Varying the magnetic field these critical points extend from the tricritical point at H=0 to a critical endpoint. For a fluid with Ising spins we calculate the critical lines and several tricritical exponents analytically. For Heisenberg fluids we obtain the phase diagrams from a numerical solution of the mean field equations of state. Received 20 March 1998     

声学时间尺度下活塞效应的数值模拟研究

流体在气液临界点附近热平衡重新建立的时间大大缩短,这主要依赖于“活塞效应”对能量输运的加速作用,它本质上是一种热声波动作用。本文从基本控制方程组出发,通过数值求解流体动力学控制方程组,对声学时间尺度上超临界CO2中的动量和能量输运过程展开研究,结果表明流体温度空间分布不遵循经典傅里叶导热模型,其随时间的演化过程呈现明显...