The phase transition of the first order in the critical region of the gas-liquid system

This is a summarising investigation of the events of the phase transition of the first order that occur in the critical region below the liquid-gas critical point. The grand partition function has been completely integrated in the phase-space of the collective variables. The basic density measure is the quartic one. It has the form of the exponent function with the first, second, third and fourth degree of the collective variables. The problem has been reduced to the Ising model in an external field, the role of which is played by the generalised chemical potential $\mu^*$. The line $\mu^*(\eta) =0$, where $\eta$ is the density, is the line of the phase transition. We consider the isothermal compression of the gas till the point where the phase transition on the line $\mu^*(\eta) =0$ is reached. When the path of the pressing reaches the line $\mu^* =0$ in the gas medium, a droplet of liquid springs up. The work for its formation is obtained, the surface-tension energy is calculated. On the line $\mu^* =0$ we have a two-phase system: the gas and the liquid (the droplet) one. The equality of the gas and of the liquid chemical potentials is proved. The process of pressing is going on. But the pressure inside the system has stopped, two fixed densities have arisen: one for the gas-phase $\eta_{g} = \eta_{c} ( 1 - {d}/{2})$ and the other for the liquid-phase $\eta_{l} = \eta_{c} (1 + {d}/{2} )$ (symmetrically to the rectlinear diameter), where $\eta_{c} = 0.13044$ is the critical density. Starting from that moment the external pressure works as a latent work of pressure. Its value is obtained. As a result, the gas-phase disappears and the whole system turns into liquid. The jump of the density is equal to $\eta_{c} d$, where $d = \sqrt{{D}/{2G}} \sim \tau^{\nu/2}$. $D$ and $G$ are coefficients of the Hamiltonian in the last cell connected with the renormalisation-group symmetry.Comment: 27 pages, 9 figure

Mesoscopic theory for inhomogeneous mixtures

Mesoscopic density functional theory for inhomogeneous mixtures of sperical particles is developed in terms of mesoscopic volume fractions by a systematic coarse-graining procedure starting form microscopic theory. Approximate expressions for the correlation functions and for the grand potential are obtained for weak ordering on mesoscopic length scales. Stability analysis of the disordered phase is performed in mean-field approximation (MF) and beyond. MF shows existence of either a spinodal or a $\lambda$-surface on the volume-fractions - temperature phase diagram. Separation into homogeneous phases or formation of inhomogeneous distribution of particles occurs on the low-temperature side of the former or the latter surface respectively, depending on both the interaction potentials and the size ratios between particles of different species. Beyond MF the spinodal surface is shifted, and the instability at the $\lambda$-surface is suppressed by fluctuations. We interpret the $\lambda$-surface as a borderline between homogeneous and inhomogeneous (containing clusters or other aggregates) structure of the disordered phase. For two-component systems explicit expressions for the MF spinodal and $\lambda$-surfaces are derived. Examples of interaction potentials of simple form are analyzed in some detail, in order to identify conditions leading to inhomogeneous structures.Comment: 6 figure

Statistical field theory for simple fluids: the collective variables representation

An alternative representation of an exact statistical field theory for simple fluids, based on the method of collective variables, is presented. The results obtained are examined from the point of another version of theory that was developed recently by performing a Hubbard-Stratonovich transformation of the configurational Boltzmann factor [J.-M. Caillol, Mol. Phys. 101 (2003) 1617]. The analytical expressions for the pressure and the free energy are derived in two-loop approximation for both versions of theory and it is shown that they are indeed equivalent.The results yield a new type approximation within an untested approximation scheme

A mesoscopic field theory of ionic systems versus a collective variable approach

We establish a link between the two functional approaches: a mesoscopic field theory developed recently by A.Ciach and G.Stell [A. Ciach and G. Stell, J. Mol. Liq. 87 (2000) 253] for the study of ionic models and an exact statistical field theory based on the method of collective variables.Comment: 7 page

Liquid-gas phase transition at and below the critical point

This article is a continuation of our previous works (see Yukhnovskii I.R. et al., J. Stat. Phys, 1995, 80, 405 and references therein), where we have described the behavior of a simple system of interacting particles in the region of temperatures at and about the critical point, T \geqslant T_{c}. Now we present a description of the behavior of the system at the critical point (T_{c}, \eta_{c}) and in the region below the critical point. The calculation is carried out from the first principles. The expression for the grand canonical partition function is brought to the functional integrals defined on the set of collective variables. The Ising-like form is singled out. Below T_{c}, when a gas-liquid system undergoes a phase transition of the first order, i.e., boiling, a "jump" occurs from the "extreme" high probability gas state to the "extreme" high probability liquid state, releasing or absorbing the latent heat of the transition. The phase equilibria conditions are also derived.Comment: 23 pages, 9 figure

Ab initio study of the vapour-liquid critical point of a symmetrical binary fluid mixture

A microscopic approach to the investigation of the behaviour of a symmetrical binary fluid mixture in the vicinity of the vapour-liquid critical point is proposed. It is shown that the problem can be reduced to the calculation of the partition function of a 3D Ising model in an external field. For a square-well symmetrical binary mixture we calculate the parameters of the critical point as functions of the microscopic parameter r measuring the relative strength of interactions between the particles of dissimilar and similar species. The calculations are performed at intermediate ($\lambda=1.5$) and moderately long ($\lambda=2.0$) intermolecular potential ranges. The obtained results agree well with the ones of computer simulations.Comment: 14 pages, Latex2e, 5 eps-figures included, submitted to J.Phys:Cond.Ma

Gas-liquid critical point in ionic fluids

Based on the method of collective variables we develop the statistical field theory for the study of a simple charge-asymmetric $1:z$ primitive model (SPM). It is shown that the well-known approximations for the free energy, in particular DHLL and ORPA, can be obtained within the framework of this theory. In order to study the gas-liquid critical point of SPM we propose the method for the calculation of chemical potential conjugate to the total number density which allows us to take into account the higher order fluctuation effects. As a result, the gas-liquid phase diagrams are calculated for $z=2-4$. The results demonstrate the qualitative agreement with MC simulation data: critical temperature decreases when $z$ increases and critical density increases rapidly with $z$.Comment: 18 pages, 1 figur

Field theory for size- and charge asymmetric primitive model of electrolytes. Mean-field stability analysis and pretransitional effects

The primitive model of ionic systems is investigated within a field-theoretic description for the whole range of size-, \lambda, and charge, Z, ratios of the two ionic species. Two order parameters (OP) are identified, and their relations to physically relevant quantities are described for various values of \lambda and Z. Instabilities of the disordered phase associated with the two OP's are determined in the mean-field approximation. A gas-liquid separation occurs for any Z and \lambda different from 1. In addition, an instability with respect to various types of periodic ordering of the two kinds of ions is found

Spatial inhomogeneities in ionic liquids, charged proteins and charge stabilized colloids from collective variables theory

Effects of size and charge asymmetry between oppositely charged ions or particles on spatial inhomogeneities are studied for a large range of charge and size ratios. We perform a stability analysis of the primitive model (PM) of ionic systems with respect to periodic ordering using the collective variables based theory. We extend previous studies [A. Ciach et al., Phys. Rev.E \textbf{75}, 051505 (2007)] in several ways. First, we employ a non-local approximation for the reference hard-sphere fluid which leads to the Percus-Yevick pair direct correlation functions for the uniform case. Second, we use the Weeks-Chandler-Anderson regularization scheme for the Coulomb potential inside the hard core. We determine the relevant order parameter connected with the periodic ordering and analyze the character of the dominant fluctuations along the $\lambda$-lines. We show that the above-mentioned modifications produce large quantitative and partly qualitative changes in the phase diagrams obtained previously. We discuss possible scenarios of the periodic ordering for the whole range of size- and charge ratios of the two ionic species, covering electrolytes, ionic liquids, charged globular proteins or nanoparticles in aqueous solutions and charge-stabilized colloids