Binodal Surface Research Articles

Temperature dependence of liquid–liquid equilibria in nonaqueous two–phase systems based on formamide

The cloud–point method was used to determine temperatures located on the binodal surface for several nonaqueous two-phase systems containing formamide, and: {1–pentanol + NaBr} or {1–pentanol + NaNO3}, or {1–butanol + NaNO3}, or {1–butanol + D–sorbitol}, or {acetonitrile + D–fructose}. For comparison purposes, the measurements were made in the water + acetonitrile + D-fructose system. The measured temperatures ranged from about 0 to 90 °C. The experimental data were described by the NRTL model as well as through several empirical equations with the parameters depending on temperature. Finally, the paper discusses the advantages and disadvantages of individual solutions.

Towards Tropically Counting Binodal Surfaces

We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor plans to recover the count of multinodal curves. We then prove that for $$\delta =2$$ or 3 nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second-order term of the polynomial giving the degree of the family of complex projective surfaces in $${\mathbb {P}}^3$$ of degree d with $$\delta $$ nodes. We classify when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and prove that this only happens for projective degree d surfaces satisfying point conditions in Mikhalkin position when $$d>4$$ .

A tropical count of binodal cubic surfaces

There are 280 binodal cubic surfaces passing through 17 general points. For the typically used tropical point conditions, we show that 214 of these give tropicalizations such that the nodes are separated on the tropical cubic surface.

Solubility in the system acetic acid – n-amyl alcohol – n-amyl acetate – Water at 293.15 K, 303.15 K, 313.15 K and 323.15 K and atmospheric pressure

Solubility of the quaternary system acetic acid – n-amyl alcohol – n-amyl acetate – water was investigated at 293.15 K, 303.15 K, 313.15 K, 323.15 K and atmospheric pressure. Binodal curves and binodal surfaces were determined using isothermal titration (cloud-point technique) and presented in composition simplices including 3D phase diagrams in composition tetrahedron. Binary mixtures were carried out by the gas chromatography method. Experimental data were predicted using UNIFAC model, and it was found that the calculating results of liquid–liquid equilibria (LLE) are in inconsistent agreement with the experimental data on solubility at all temperatures.

Full field dependence of primary aberrations in perturbed double-plane symmetric systems with a circular pupil.

This paper deals with the theory of primary aberrations for perturbed double-plane symmetric optical systems consisting of a combination of tilted and decentered surfaces and a circular pupil. First, the analytical expressions describing the full field behavior of Zernike polynomials are derived from the fourth-order wavefront aberration function for this class of optical systems. Then, such expressions are combined to retrieve the full field dependence of primary coma, primary astigmatism, and field curvature. They are described by an elliptical conic-shaped surface with a variable apex location over the field of view, by a binodal surface with two nodes over the field of view, and by a general elliptical surface with one node. The proposed analytical expressions provide a better understanding of the primary aberration behavior for these systems and can be of great use in their optical design and aberration correction. An optical system constituted by a pair of tilted and decentered biconic lenses is studied to validate the proposed expressions.

Liquid-liquid equilibria, solubility and critical states in the system propionic acid – 1-propanol – n-propyl propionate – water at 303.15 K

New experimental data on liquid-liquid equilibria (LLE), solubility and critical states in quaternary system propionic acid – 1-propanol – n-propyl propionate – water at 303.15 K and atmospheric pressure are presented. LLE was studied by Gas Chromatography (GC). Solubility and critical (plait) points were investigated by “cloud-point technique” method. Binodal curves and surfaces are presented in concentration tetrahedron. The critical curve is presented in 3d concentration space. A comparative analysis of the results obtained in the work with literature data is carried out. The experimental data are correlated by NRTL model. Standart deviation σ for quaternary mixture does not exceed 0.344%.

Liquid-liquid equilibrium of acetic acid – ethanol – ethyl acetate – water quaternary system: Data review and new results at 323.15 K and 333.15 K

A brief review of data on solubility and liquid-liquid equilibrium (LLE) in the acetic acid – ethanol – ethyl acetate – water system is presented. A new set of experimental data on solubility, LLE, and critical phases in this quaternary system, ternary subsystems (acetic acid – ethyl acetate – water, ethanol – ethyl acetate – water) and binary subsystem ethyl acetate – water are presented for 323.15 K and 333.15 K at atmospheric pressure. The gas chromatography and the cloud point technique were used to obtain experimental data. Binodal curves, surfaces, and critical manifolds are presented including 3D phase diagrams in composition tetrahedron. Experimental LLE data were correlated using the NRTL model.

Quaternary liquid-liquid equilibrium, solubility and critical states: Acetic acid – n-butanol – n-butyl acetate – water at 318.15 K and atmospheric pressure

Experimental data on solubility, critical states and liquid–liquid equilibrium (LLE) for the quaternary system acetic acid – n-butanol – n-butyl acetate – water and ternary subsystems acetic acid – n-butanol – water, n-butanol – n-butyl acetate – water and acetic acid – n-butyl acetate – water were obtained at 318.15 K and at atmospheric pressure. The solubility and critical states were studied by isothermal titration method (“cloud-point technique”). The LLE was determined using gas chromatographic analysis. Binodal surface in quaternary system and binodal curves in ternary systems were constructed. Compositions of critical points (plait points) and critical curve were determined. The NRTL model was applied to calculate the LLE data for the quaternary and ternary systems. The results of NRTL modeling are in good agreement with the experimental data.

Solubility, liquid-liquid equilibrium and critical states for the quaternary system formic acid – ethanol – ethyl formate – water at 298.15 K and 308.15 K

Solubility, critical states and liquid-liquid equilibrium (LLE) in the quaternary reacting system formic acid – ethanol – ethyl formate – water and in the ternary subsystems formic acid – ethyl formate – water and ethanol – ethyl formate – water were studied experimentally at 298.15 K and 308.15 K and atmospheric pressure. Binodal curves, binodal surfaces, tie-lines and critical manifolds were determined. In order to construct binodal surfaces and critical curves of the quaternary system, five cutting planes with several different ratios of concentration of formic acid to ethanol were studied. Experimental LLE data were compared with the data calculated by NRTL model, and it was found that experimental and calculated data are in a good agreement.

The QCD Phase-Diagram Obtained from the NJL and the Extended-NJL Models for Quark and Hadron Phases

We analyse the hadron/quark phase transition described by the Nambu-Jona-Lasinio (NJL) model [quark phase] and the extended Nambu-Jona-Lasinio model (eNJL) [hadron phase]. While the original formulation of the NJL model is not capable of describing hadronic properties due to its lack of confinement, it can be extended with a scalar-vector interaction so it exhibits this property, the so-called eNJL model. As part of this analysis, we obtain the equations of state within the SU(2) versions of both models for the hadron and the quark phases and determine the binodal surface.

LLE of Methanol-Methylcyclohexane-Isooctane-Benzene Quaternary System

A quaternary system {methanol + methylcyclohexane + isooctane + benzene} was studied at T = 303.15 K. Only after all ternary equilibria data were determined experimentally, the quaternary binodal surface could be characterized fully. For this purpose, tie line data of {methanol + methylcyclohexane + benzene} ternary systems were obtained at the same temperature, while data for {methanol + isooctane+ benzene}, and {methanol + + isooctane + methylcyclohexane} were taken from literature. Experimental results show that the binodal surface with four quaternary sectional planes P1,P2,P3,P4 with several methylcyclohexane/isooctane in the solid diagram is small and that the highest benzene mass fraction values beyond which only one phase is present for the methanol-rich phase and hydrocarbon-rich one, respectively, are: 0.046 and 0.053 for P1, 0.053 and 0.057 for P2, 0.055 and 0.049 for P3, 0.034 and 0.041 for P4. So, if this quaternary system contains the correct methanol and hydrocarbons concentrations, this blend can be used as a reformulated gasoline because no phase separation should be observed. The distribution of benzene between both phases was also analysed. Ternary experimental results were correlated with the UNIQUAC and NRTL equation.The UNIQUAC equation is more accurate than the NRTL equation for the ternary systems studied here. The equilibria data of the three ternary systems were used to determine interactions parameters for the UNIQUAC equation. The UNIQUAC equation fitted to the experimental data is more accurate than the UNIFAC method for this quaternary system. The main reason for this is that F is fit in the size of the quaternary binodal surface.

Liquid-liquid equilibrium for the quaternary system propionic acid + n-propanol + n-propyl propionate + water at 293.15, 313.15 and 333.15 K

Liquid-liquid equilibrium (LLE) for the quaternary system propionic acid + n-propanol + n-propyl propionate + water and for the ternary sub-systems propionic acid + n-propyl propionate + water and n-propanol + n-propyl propionate + water were studied at 293.15, 313.15 and 333.15 K and atmospheric pressure. Binodal surface in quaternary system and binodal curves in ternary systems were determined. Experimental LLE data were compared with the values calculated by NRTL models, these data are in good agreement.

Characteristics of three phase separation in the ternary system of a polymer in a mixed solvent

The three phase separation was obtained for the ternary system of ethylene glycol diacetate (EGDA) + methylcyclohexane (MCH) + polystyrene (PS) with the molecular weight (Mw = 4.8 × 104 g/mol). A phase equilibrium experiment was carried out to determine binodals and three phase triangles. With decreasing temperature the three phase separation appears at 35.0 °C in the two phase region spreading from the EGDA-MCH side protruding a new binodal with a critical point on a composition triangle. The binodal spreading from the EGDA-PS side merges with the binodal from EGDA-MCH side at 34.0 °C via a critical point, while the binodal from MCH-PS side merges with the new binodal at 32.9 °C. Thus, the binodal surface in a prism diagram of composition triangle and temperature has two saddle points at which a critical line has a minimum.

Binodal varieties of the systems with four-component azeotrope

The regularities of the formation of binodal surfaces and atlas of their orgraphs for four-component systems with two pairs of stratified components are presented. The evolution of orgraphs of binodal surfaces is studied upon changing external conditions.

Stable and metastable silicate liquid immiscibility in ferrobasalts

The onset of immiscibility in ferrobasaltic systems has been the subject of much research recently. The compositional space of the two-liquid field and the maximum temperature of the binodal surface have been investigated experimentally, but results from static and centrifuge experiments are controversial. In the article by Hou and Veksler (2015, May–June issue) entitled “Experimental confirmation of high-temperature silicate liquid immiscibility in multicomponent ferrobasaltic systems,” the authors present experimental evidence for immiscibility between silica- and iron-rich melts at 1150–1200 °C, which are significantly higher to previous studies (ca. 1000–1025 °C). These results have important implications for potential large-scale differentiation of magmas by liquid unmixing and for the formation of both Fe-Ti-P-rich melts and rhyolites.

Formation and lateral migration of nanodroplets via solvent shifting in a microfluidic device

Solvent shifting is a process in which a non-solvent is added to a solvent/solute mixture and extracts the solvent. The solvent and the non-solvent are miscible. Because of solution supersaturation, a portion of the solute transforms to droplets. In this paper, based on this process, we present an investigation on droplet formation and their radial motion in a microfluidic device in which a jet is injected in a co-flowing liquid stream. Thanks to the laminar flow, the microfluidic setup enables studying diffusion mass transfer in radial direction and obtaining well-defined concentration distributions. Such profiles together with the ternary phase diagram give detailed information about the conditions for droplet formation as well as their radial migration in the channel. The ternary system is composed of ethanol (solvent), de-ionized water (non-solvent), and divinylbenzene (solute). We employ analytical/numerical solutions of the diffusion equation to obtain concentration profiles of the components. We show that in the system under study droplets are formed in a region of the phase diagram between the binodal and the spinodal, i.e., via a thermally activated process. The droplets are driven to the channel centerline by the solutal Marangoni effect but are not able to significantly penetrate into the single-phase region, where they get rapidly dissolved. Therefore, the radial motion of the binodal surface carries the droplets to the centerline where they get collected.

Chemical equilibrium for the reactive system acetic acid + n-butanol + n-butyl acetate + water at 308.15 K

Chemical equilibrium (CE) for the quaternary reactive system of acetic acid–n-butanol–n-butyl acetate–water was studied at 308.15K and atmospheric pressure. The experimental data on CE were considered in comparison with the data on liquid–liquid equilibrium: the CE is reached both in homogeneous and heterogeneous area of reactive mixture compositions. The crossing of CE and binodal surfaces is presented in a composition tetrahedron and a square of transformed composition variables. The thermodynamic constant of CE at 308.15K was determined with the use of NRTL and UNIQUAC models.

Liquid–Liquid Equilibria of the Methanol + Hexane + Methylcyclohexane + Toluene Quaternary System at T = 303.15 K

Tie line data of the {methanol + methylcyclohexane + hexane} and {methanol + toluene + hexane} ternary systems were obtained at T = 303.15 K, while data for {methanol + toluene + methylcyclohexane} were taken from the literature. A quaternary system {methanol + methylcyclohexane + hexane + toluene} was also studied at the same temperature. In order to obtain the binodal surface of the quaternary system, four quaternary sectional planes with several hexane/methylcyclohexane ratios were studied. Experimental results show that the binodal surface in the solid diagram is small and that the highest toluene mass fraction values, beyond which only one phase is present for the methanol-rich phase and hydrocarbon-rich one, respectively, are: 0.052 and 0.064 for P1, 0.068 and 0.052 for P2, 0.047 and 0.107 for P3, 0.029 and 0.030 for P4. So, if this quaternary system contains the correct methanol and hydrocarbon concentrations, this blend can be used as a reformulated gasoline, because no phase separation should be observed. Ternary experimental results were correlated with the UNIQUAC and NRTL equations. The equilibrium data for the three ternary systems were used to determine interaction parameters for the UNIQUAC equation. The NRTL equation is more accurate than the UNIQUAC for the ternary systems studied here. The UNIQUAC equation fitted to the experimental data is more accurate than the UNIFAC method for this quaternary system.

Liquid–liquid equilibrium and critical states for the system acetic acid + n-butanol + n-butyl acetate + water at 308.15 K

Liquid–liquid equilibrium (LLE) for the quaternary system acetic acid–n-butanol–n-butyl acetate–water and for the ternary sub-systems acetic acid–n-butyl acetate–water, n-butanol–n-butyl acetate–water and acetic acid–n-butanol–water were studied at 308.15K and atmospheric pressure. Binodal surface and curves, tie-lines and compositions of critical points of LLE were determined. Experimental LLE data were correlated by NRTL model. Experimental LLE data were compared with the values calculated by NRTL models, these data are in good agreement.

Analysis of binodal manifolds for four-component systems

The results of a thermodynamic and topological analysis of closed binodal manifolds for fourcomponent systems are presented. The basic structures of the digraphs of binodal surfaces are reported, and their classification is suggested.