Rectilinear Diameter Research Articles

Estimation of the critical behavior in an active colloidal system with Vicsek-like interactions.

We study numerically the critical behavior of a modified, active Asakura-Oosawa model for colloid-polymer mixtures. The colloids are modeled as self-propelled particles with Vicsek-like interactions. This system undergoes phase separation between a colloid-rich and a polymer-rich phase, whereby the phase diagram depends on the strength of the Vicsek-like interactions. Employing a subsystem-block-density distribution analysis, we determine the critical point and make an attempt to estimate the critical exponents. In contrast to the passive model, we find that the critical point is not located on the rectilinear diameter. A first estimate of the critical exponents β and ν is consistent with the underlying 3d-Ising universality class observed for the passive model.

Vapour-liquid coexistence of an active Lennard-Jones fluid.

We study a three-dimensional system of self-propelled Lennard-Jones particles using Brownian dynamics simulations. Using recent theoretical results for active matter, we calculate the pressure and report equations of state for the system. Additionally, we chart the vapour-liquid coexistence and show that the coexistence densities can be well described using simple power laws. Lastly, we demonstrate that our out-of-equilibrium system shows deviations from both the law of rectilinear diameters and the law of corresponding states.

Thermodynamics of Gas–Liquid Criticality: Rigidity Symmetry on Gibbs Density Surface

The thermodynamic state function rigidity, defined simply as $$(\mathrm{d}p/\mathrm{d}\rho )_{T}$$ , where p is the pressure, $$\rho $$ is the density and T is the temperature, is the work required to reversibly increase the density of a fluid. Along any isotherm, rigidity $$(\omega )$$ decreases with density for a gas phase and increases with density for a liquid. Thermodynamics, therefore, can define a distinction between gas and liquid. For any one-phase system, rigidity is everywhere positive, in any two-phase region $$\omega =0$$ . For temperatures above critical coexistence, the rigidity has a constant value in the mesophase that separates the percolation loci, which bound the limits of existence of liquid and gas phases in the supercritical region. The law of rectilinear diameters extends in the supercritical region as a defining line of the colloid-like inversion between gas-in-liquid and liquid-in-gas. Every equilibrium state of gas phase has a corresponding isothermal state on the liquid phase with the same rigidity. We illustrate this symmetry between gas and liquid empirically using literature $$\rho (p,T)$$ equations-of-state for some real fluids, notably carbon dioxide, water and steam, and argon. At the molecular level, the symmetry can be explained by a correspondence between statistical properties of available holes in a liquid and sites of molecular clusters in the gas with equivalent number density fluctuations for complementary states of gas and liquid.

Wide-Pore Cryogels Prepared Using the Combination of Liquid-Liquid Phase Separation and Cryotropic Gel-Formation Processes

Novel, previously unknown, wide pore poly(vinyl alcohol) cryogels (PVACGs) have been prepared through the cryotropic gelation approach, when water–PVA–gum Arabic (GuAr) ternary liquid systems were used as feeds. The following set of conditions necessary for obtaining wide-porous, permeable for a water flow, and, simultaneously, mechanically strong enough PVACGs was established: the total concentration of gelling component – PVA (MW of 86 kDa) and non-gelling polymer – GuAr (MW of ~650 kDa) should exceed ~14 wt. %, GuAr/PVA ratio should be near 1 : 1 (w/w), and the feed’s pH should be within the range of ~5-11. The phase diagrams for the water–PVA–GuAr ternary liquid systems at pH 5.3 and 10.0 have demonstrated that such “optimum” compositions are in the vicinity of the rectilinear diameters of the respective diagrams, and that the limit of GuAr solubility in the PVA-rich phase is small. In such cases this GuAr fraction strongly binds with PVA, especially under alkaline conditions, within the forming gel phase of heterophase PVACG. The light microscopy studies revealed the presence of at least three distinct kinds of pores in these cryogels. Firstly, there are interconnected channel-like gross pores with cross-section of 100-200 μm which are the replicas of continuous GuAr-rich phase in the two-phase polymeric systems appearing due to the liquid-liquid phase separation in the ternary water–PVA–GuAr system. The second type is represented by isolated roundish large pores ca. 10-70 μm in diameter being the replicas of GuAr-rich phase dispersed as liquid droplets in the PVA-rich phase. And the third type is represented by the smaller rounded pores ~1-5 μm in diameter being the replicas of ice polycrystals. Such sophisticated “poly-porous” morphology of the studied PVA cryogels is their unique feature distinguishing them from other known PVA cryogels.

ВПЛИВ НАНОЧАСТИНОК НА ПАРАМЕТРИ ФАЗОВОЇ РІВНОВАГИ РІДИНА-РІДИНА. ЧАСТИНА 2.

Issues of nanoparticles impact on the parameters of the phase liquid-liquid equilibria in the general vicinity of the critical point are considered in the paper. Binary system exfoliating nitrobenzene/n-heptane and nitrobenzene / nanogeptan are been objects of study. Simulation results of phase equilibrium liquid-liquid using the extended scaling equations are given. It is shown that for an adequate description only three coefficients and critical system parameters are necessary. The rule of rectilinear diameter for exfoliating systems is applied. Critical phenomena in the curves for the three-component separation nanofluid s ( nitrobenzene / heptane / nanoparticles TiO 2 ) are isomorphic phenomena on the curves of the bundle of binary solutions (nitrobenzene / heptane) .

Calculation of critical point coordinates of ternary miscibility gap from experimental tie-lines

Demixing phenomena in liquid ternary systems are widely used in chemical engineering processes. When a critical point is observed, the composition of this invariant point of major interest is experimentally difficult to determine. A simple calculation method for the critical point is proposed using the experimental tie-lines. The computing treatment is based on the application of the barycentric weighting of the binodal points in order to extend the rectilinear diameter method and the exploitation of the modulus of the experimental tie-lines. A systematic study was carried out on a large set of ternary systems taken from DECHEMA and NIST. The results are compared with those available in literature or calculated by standard methods using binary interaction parameters. Very good agreements are obtained when the experimental tie-lines are located close to the critical point.The method can be extended to wide miscibility gaps by using a polynomial deviation instead of the rectilinear diameter line. A typical case is detailed (2-butanone+2-butanol+water at various temperatures). The results lead to (i) the point by point calculation of the entire binodal curve and (ii) the thermal evolution of the critical point and the possible disappearance of the critical point when temperature increases.

Liquid–Liquid Phase Behavior of Solutions of 1,3-Dimethylimidazolium- and 1-Methyl-3-propylimidazolium Bis((trifluoromethyl)sulphonyl)amide (C1,3mimNTf2) in n-Alkyl Alcohols

The phase behavior of binary liquid mixtures of the ionic liquids (ILs) 1,3-dimethylimidazolium bis((trifluoromethyl) sulfonyl) amide (C1mimNTf2) and 1-methyl-3- propyl imidazolium bis ((trifluoromethyl) sulfonyl) amide (C3mimNTf2) with n-alkyl alcohols (CnOH; 3 ≤ n ≤ 8) was investigated. The phase diagrams are obtained by applying the cloud-point method on a set of synthetically prepared samples. Partial miscibility with an upper critical solution temperature (UCST) was observed. The study focuses on the determination of the liquid–liquid phase behavior in a temperature range of (267–423) K at atmospheric pressure. UCSTs were observed in between 280 K and 415 K. The results are complementing prior measurements on similar IL-CnOH systems concerning ILs with longer side chain. A numerical analysis of the phase diagrams on the basis of Ising criticality was applied presuming the validity of the rectilinear diameter or a nonlinear diameter as requested by the theory of complete scaling. The results (UCST, cr...

The saturation pressure for different objects in reduced variables and the justification of some empirical relations set from the van der Waals equation

We analyse the saturation pressure (SP) behaviour along the coexistence curve liquid–gas in reduced variables, containing only the Zeno line parameters. Using experimental and simulation data for substances (Ar, N2, Hg, Cs) and two models (van der Waals and Lennard–Jones) we create a global 3D picture of SP dependence on temperature and density. We also show that the well-known empirical Timmermans relation and partly Riedel equation are the consequences of the van der Waals equation and the rectilinear diameter law.

Similarities and contrasts between critical point behavior of heavy fluid alkalis and d-dimensional Ising model

Abstract A brief summary is first given of recent progress in establishing the near-critical point behavior of the fluid alkalis Rb and Cs. Departure from the law of Rectilinear Diameters is emphasized, along with its consequences for theories emphasizing homogeneity and scaling. The behavior as the critical points of Rb and Cs are approached is compared and contrasted with the d -dimensional Ising model.

Phase diagram and universality of the Lennard-Jones gas-liquid system

The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.

The General Phase Behavior of Mixtures of 1‐Alkyl‐3‐Methylimidazolium Bis[(trifluoromethyl)sulfonyl]amide Ionic Liquids with n‐Alkyl Alcohols

The liquid-liquid equilibrium (LLE) phase behavior of mixtures of 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]amide (C(2)mimNTf(2)) with the n-alkyl alcohols (C(n)OH; 3≤n≤8) is described. By applying the cloud-point method, the LLEs were determined over a temperature range of 275-423 K at ambient pressure. Partial miscibility with upper critical solution temperatures (UCST) was observed. The UCST increase with increasing chain length of the alcohols. The phase diagrams were analyzed numerically by presuming Ising criticality. Concepts for the description of the asymmetry of the phase diagram by presuming the validity of the rectilinear diameter rule or a nonlinear diameter requested by the theory of complete scaling were applied. The results (UCST, critical composition, width and diameter of the phase diagrams) are compared with similar systems and discussed in terms of the corresponding state behavior; they map on a single curve. From at least 45 individual phase diagrams, general aspects of the behavior of this ionic liquid-alcohol mixture type were extracted. A simple empirical relationship was formulated to allow the description of the UCST with an accuracy of about 10 K when taking the ratio of the molar volumes of the alcohol and the 1-alkyl-3-methylimidazolium cation of the ionic liquid into account.

Cryostructuring of polymer systems. Wide pore poly(vinyl alcohol) cryogels prepared using a combination of liquid–liquid phase separation and cryotropic gel-formation processes

Novel, previously unknown, wide pore poly(vinyl alcohol) cryogels (PVACGs) have been prepared through a cryotropic gelation approach when water–PVA–gum arabic (GuAr) ternary liquid systems were used as feeds. The following set of conditions necessary for obtaining wide-porous, permeable for a water flow, and, simultaneously, mechanically strong enough PVACGs was established: the total concentration of gelling component—PVA (MW of 86 kDa) and non-gelling polymer—GuAr (MW of ∼650 kDa) should exceed ∼14 wt%, the GuAr : PVA ratio should be near 1 : 1 (w/w), and the feed pH should be within the range of ∼5 to 11. The phase diagrams for the water–PVA–GuAr ternary liquid systems at pH 5.3 and 10.0 have demonstrated that such “optimum” compositions are in the vicinity of the rectilinear diameters of the respective diagrams, and that the limit of GuAr solubility in the PVA-rich phase is small. In such cases this GuAr fraction strongly binds with PVA, especially under alkaline conditions, within the gel forming phase of heterophase PVACG. Light microscopy studies revealed the presence of at least three distinct kinds of pores in these cryogels. First, there are interconnected channel-like gross pores with a cross-section of 100–200 μm and larger which are the replicas of continuous GuAr-rich phase in the two-phase polymeric systems appearing due to the liquid–liquid phase separation in the ternary water–PVA–GuAr system. The second type are isolated roundish large pores ca. 10–70 μm in diameter being the replicas of the GuAr-rich phase dispersed as liquid droplets in the PVA-rich phase. In the third, there are smaller rounded pores ∼1 to 5 μm in diameter being the replicas of ice polycrystals. Such a sophisticated “poly-porous” morphology of the studied PVA cryogels is their unique feature distinguishing them from other known PVA cryogels.

Unified Picture for the Classical Laws of Batschinski and the Rectilinear Diameter for Molecular Fluids

The explicit relations between the thermodynamic functions of the lattice gas model and the fluid within the framework of an approach proposed earlier [ Kulinskii , V. L. J. Phys. Chem. B 2010 , 114 , 2852 ] are derived. It is shown that the Widom line serves as the natural border between the gas-like and the liquid-like states of the fluid. The explanation of the global cubic form of the binodal for the molecular liquids is proposed, and the estimate for the amplitude of the binodal opening is obtained.

Émile Ovide Joseph Mathias. Thermodynamics of saturated fluids

Émile Ovide Joseph Mathias (1861-1942) was a multifaceted scientist who started his activities in the area of thermodynamics and then switched to magnetic phenomena and electrical phenomena in the atmosphere. He collaborated with Cailletet in the development of methods for measuring and correlating the density of liquefied gases and their saturated vapors. His name is particularly known for the so-called law of rectilinear diameters that allows an easy determination of the critical volume.

A method for the calculation of the adsorbed phase volume and pseudo-saturation pressure from adsorption isotherm data on activated carbon

We propose a new method for evaluating the adsorbed phase volume during physisorption of several gases on activated carbon specimens. We treat the adsorbed phase as another equilibrium phase which satisfies the Gibbs equation and hence assume that the law of rectilinear diameters is applicable. Since invariably the bulk gas phase densities are known along measured isotherms, the constants of the adsorbed phase volume can be regressed from the experimental data. We take the Dubinin-Astakhov isotherm as the model for verifying our hypothesis since it is one of the few equations that accounts for adsorbed phase volume changes. In addition, the pseudo-saturation pressure in the supercritical region is calculated by letting the index of the temperature term in Dubinin's equation to be temperature dependent. Based on over 50 combinations of activated carbons and adsorbates (nitrogen, oxygen, argon, carbon dioxide, hydrocarbons and halocarbon refrigerants) it is observed that the proposed changes fit experimental data quite well.

Settling the bound on the rectilinear link radius of a simple rectilinear polygon

We improve the best known bound on the rectilinear link radius of a simple rectilinear polygon with respect to its rectilinear link diameter. The new bound is tight and is compatible with the known bound on the (regular) link radius of a (regular) simple polygon with respect to its (regular) link diameter. The previous bound on the rectilinear link radius of a simple rectilinear polygon was proven by Nilsson and Schuierer in 1991.

Global isomorphism between the Lennard-Jones fluids and the Ising model

The interpretation of the linear character of the observable classic rectilinear diameter law and the linear character of the Zeno-line (unit compressibility line Z=1) on the basis of global isomorphism between Ising model (lattice gas) and simple fluid is proposed. The correct definition of the limiting nontrivial Zeno state is given and its relation to the locus of the critical point is derived within this approach. We show that the liquid-vapor part of the phase diagram of the molecular fluids can be described as the isomorphic image of the phase diagram of the lattice gas. It is shown how the position of the critical points of the fluids of the Lennard-Jones type can be determined based on the scaling symmetry. As a sequence, the explanation of the well-known fact about "global" cubic character of the coexistence curve of the molecular fluids is proposed.

Evolution of the liquid-vapor coexistence of the hard-core Yukawa fluid as a function of the interaction range

The present work is devoted to the study of the liquid-vapor coexistence curve of hard-core Yukawa fluids for range parameter lambda, going from 0.5 to 7 by means of an integral equation approach. Both binodal and spinodal lines are computed and compared to available simulation data, and the integral equation used appears to be accurate. We also compare two methods for determining the coordinates of the critical point. The first one, using the rectilinear diameter law, appears to be less accurate than the second one based on the heat capacity at constant volume. It is found that the critical temperature decreases as the range of the interactions increases and that the liquid-vapor coexistence disappears for lambda greater than 6.

Simple Geometrical Interpretation of the Linear Character for the Zeno-Line and the Rectilinear Diameter

The unified geometrical interpretation of the linear character of the Zeno-line (unit compressibility line Z = 1) and the rectilinear diameter is proposed. We show that recent findings about the properties of the Zeno-line and striking correlation with the rectilinear diameter line as well as other empirical relations can be naturally considered as the consequences of the projective isomorphism between the real molecular fluids and the lattice gas (Ising) model.

The history of the rectilinear diameter law

The formulation of the so-called law of rectilinear diameter for the determination of the critical volume of substances in the concluding decades of the nineteenth century became in a very useful and acceptably exact alternative tool for researchers in the field of critical phenomena. Its corresponding original expression, and even those of its early few modifications, were so mathematically simple that their use did not limit to exclusively contribute to remove the by then experimental obstacle for the estimating of this critical parameter, but also extended along several decades in the increasing applications of the principle of corresponding states.