Noncritical interface near a critical end point II

doi:10.1016/0378-4371(80)90015-1

Physica A: Statistical Mechanics and its Applications

Volume 104, Issue 3, December 1980, Pages 595-620

https://doi.org/10.1016/0378-4371(80)90015-1 Get rights and content

Abstract

The tension of the interface between a noncritical phase α and a near-critical phase βγ, or between α and the individual β and γ phases into which βγ separates, is discussed. A thermodynamic description of a critical end point is given, earlier theoretical results on the interfacial tensions are reviewed, and the earlier theory is generalized and extended and some of its restrictions are relaxed. The theory is then reformulated, and made to incorporate nonclassical critical-point exponents. The critical isotherm of interfacial tension σ_α,βγ as a function of composition has a horizontal tangent at the critical point, where it is found to be of degree $μ β ≈ 3.9$ , with μ and β the conventional critical surface-tension and coexistence-curve exponents. The tensions σ_αβ and σ_αγ when T<T_c, and the tension σ_α,βγ at fixed, critical composition when T > T_c, as functions of $θ ∼ T T_{c} − 1$ , all have |θ|^γ and |θ|^μ as their leading singularities, with γ the conventional critical susceptibility exponent. Numerical results are presented for the realistic case γ ≈ μ. It is remarked that the liquid-vapor interface in ⁴He at its λ-point is also a noncritical interface at a critical end point.

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The non-ideal behaviour of the interfacial tension of the n-heptane + perfluoro-n-hexane mixture: A computational study
2015, Chemical Physics Letters
Citation Excerpt :
It is well known that mixtures of alkanes–perfluoroalkanes present a different behaviour from the ideal; for instance, the large positive deviation from Raoult's law, the large positive excess functions and the liquid–liquid immiscibility [1,2]. In particular, the vapour–liquid interface tension presents a horizontal inflection close to the critical composition which has been observed in actual experiments [3,4] and predicted by the square gradient theory of van der Waals [5,6]. In fact, those studies have suggested that the anomalous behaviour could be due to the unusually weak attractive interaction between the alkane and perfluoroalkane molecules [7–11].
Computer simulations of the n-heptane + perfluoro-n-hexane system is used to understand the unusual behaviour of the horizontal inflection of the vapour–liquid interfacial tension experiments of alkanes–perfluoroalkanes mixtures. Simulations are conducted and good agreement with experiments is obtained. From the results a change in curvature to a nearly constant value in the tensions at high perfluoro-n-hexane compositions is identified. Dynamic, thermodynamic and structural properties are analysed in that region and peculiar tendencies of those properties are found which help us to understand the nature of the uncommon inflection in the isotherms.
Modelling and understanding of the vapour-liquid and liquid-liquid interfacial properties for the binary mixture of n-heptane and perfluoro-n-hexane
2013, Journal of Molecular Liquids
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For example, mixtures of these components display substantial deviations from ideality as evidenced in a variety of interesting phenomena, including large regions of liquid–liquid immiscibility, large positive deviations from Raoult's law, and large positive excess functions [1,2]. Another physical property of particular interest in perfluoroalkane + alkane mixtures is the interfacial tension: isotherms of the vapour–liquid interfacial tension of these mixtures present a horizontal inflection close to the critical composition [3,4], as predicted by the square gradient theory of van der Waals [5,6]: moreover, the interfacial tension isotherms close to the critical region are found to be characterised by a variety of shapes, some of which suggest aneotropic behaviour [3,4]. Correspondingly, a large number of investigations have been carried out to address the unusual properties observed in these systems [7–16]; it has been suggested that the anomalous behaviour could be the result of an unusually weak attractive interaction between the alkyl and perfluoroalkyl moieties.
A detailed molecular-dynamics (MD) simulation study of the interfacial tension of the n-heptane + perfluoro-n-hexane (C₆F₁₄ + C₇H₁₆) binary mixture is carried out at different temperatures and compositions for the vapour–liquid and liquid–liquid interfaces. The molecules are represented as chains of united-atom segments using previously reported force-field pure-component parameters with no further modification, while the unlike interaction energy between the alkyl and perfluoroalkyl segments is adjusted to describe the experimental data for the fluid phase equilibria. In the case of the vapour–liquid interface a horizontal inflection at a critical composition, sometimes referred to as aneotropy, is represented by our model as found in experimental work. Interfacial tensions for both the liquid-liquid and vapour-liquid interfaces are determined using the test-area method. A prerequisite of the determination of the interfacial tension by direct MD simulation involves studies of the fluid phase equilibria (vapour–liquid (VLE) and liquid–liquid equilibria (LLE)). A unique unlike energetic interaction parameter is adjusted to improve the prediction of the LLE and used in all the other simulations. The resulting description of the fluid phase behaviour is compared with that calculated using an equation of state (SAFT), and with experimental data where available; good agreement is observed.
Bulk and interfacial properties of a cationic micellar system near the critical point
2007, Chemical Physics
The behaviour of the cationic micellar system [dodecylammonium chloride + water + KCl] has been studied at two different salt concentrations. Detailed measurements of density, of vapour–liquid surface tension, and of liquid–liquid interfacial tension have been performed. The temperature dependence of the density, and the use of extended scaling equations have allowed us to determine the binodal and pseudospinodal curves. The diameter of the coexistence curve and the order parameter can be described with the 3D-Ising critical exponents. The temperature dependence of the effective critical exponent B_eff, obtained from the density measurements, follows the same master curve previously found using the composition of the coexisting phases for many binary mixtures. The behaviour of the vapour–liquid surface tension data for both the homogeneous and for the two-phase system, is compatible with Widom’s equations. In particular the three curves have a common tangent at the critical point. In spite that the concentrations of the mixtures studied are above the critical micelle concentration, the difference of the surface tension of the two coexisting phases is well above the experimental uncertainty, and increases with |T_c − T|. The liquid–liquid interfacial tension data has been described accurately with Ising critical exponents. However, the combination of the critical amplitudes obtained from the surface and interfacial tension, and from light scattering are not in agreement with the universal values calculated from theory. This leads to the conclusion that Antonow’s rule does not hold for this micellar system. The analysis of the vapour–liquid surface tension in the one-phase region at T ≈ T_c leads to the conclusion that the system’s behaviour corresponds to the weak field approach.
Usual and unusual surface tensions of perfluorocarbon-containing binary liquid mixtures near a critical endpoint
1998, Journal of Colloid and Interface Science
The surface tension of binary mixtures near critical endpoints is discussed with detailed focus on the composition dependence. The similarity with the corresponding composition dependence of the vapor pressure is explored. The differences between mixtures in which the more volatile component is of lower surface tension, i.e. is less tense—known as usual mixtures—and those in which the more volatile component is more tense—known as unusual mixtures—are analyzed. As a basis of discussion, a limited review of the related literature is presented along with some fresh results for two usual mixtures (octamethylcyclotetrasiloxane + perfluoropentane and heptane + perfluorohexane) and two unusual mixtures (pentane + perfluoroheptane and 2-methylpentane + perfluorotributylamine). These mixtures display a variety of horizontal inflections characteristic of the behavior of the noncritical interface near a critical endpoint. The form of the horizontal inflection is uniformly related to the sign of the difference between the surface tensions of the pure components, i.e. with the overall negative slope towards the less tense component. A number of the mixtures also exhibit aneotropy or surface azeotropy. All these results are shown to be well described by the Surface Regular Solution theory.
Temperature dependence of vapor/liquid interfacial tensions in a binary liquid mixture with a miscibility gap near its critical point: Determination of universal surface amplitude ratios
1997, Physica A: Statistical Mechanics and its Applications
The inverted pendant drop method (rising bubble method) is used to measure the temperature dependence of the surface tensions σ_α,βγ, σ_α,β and σ_α,γ of a 2,6-dimethyl pyridine (abbreviated 2,6-DMP)/water mixture of critical composition in a temperature range −8.373 K < (T − T_c) < +5.634K. The experiments are carried out near the lower liquid/liquid critical point of the system. α is the vapor phase and β,γ the homogeneous liquid phases. At temperatures above the liquid/liquid critical temperature a 2,6-DMP rich liquid phase β coexists with a water rich liquid phase γ. The temperature dependence of the liquid/liquid interfacial tension σ_{β, γ} of that system is known. The surface tension data are used to calculate the material specific values of the critical amplitudes B_α,βγ, B_α,β and B_α,γ of the corresponding surface tensions using four different fitting procedures. They differ in the number of free parameters of the fit. In three of the four fits the values of the critical amplitudes B_α,β, B_α,γ and B_β,γ are consistent with Antonow's rule. The critical amplitudes are used to calculate the universal surface amplitude ratios Q, P₊, P, P₋ which are theoretically expected to have material independent values. The signs of the calculated amplitude ratios agree with the theoretical predictions. In three of the four fits the sequence of the amplitude ratios ordered in accordance with their magnitude agrees with the corresponding sequence of the theoretical values. The experimental values are close to the theoretical values. Visual observations confirm the results of the measurements of the temperature dependence of the surface tensions: The α, γ interface is completely wet by phase β in the range of temperatures studied ((T − T_c) < 5.6 K). A wetting transition is not observed.
Universal surface-tension and critical-isotherm amplitude ratios in three dimensions
1996, Physica A: Statistical Mechanics and its Applications
The universal surface-tension and critical-isotherm amplitude ratios are studied numerically for three-dimensional Ising models. Modern estimates of the critical temperature and exponents allow reliable evaluation of the critical surface-tension amplitude, K, using recent Monte Carlo data for the simple cubic lattice. Likewise, the amplitudes C^c, for the susceptibility and, $ƒ_{1}^{c}$ , for the second-moment correlation length, on the critical isotherm have been re-estimated using existing series expansions. The method of inhomogeneous differential approximants also yields a direct estimate of the correction-to-scaling exponent, θ_c, on the critical isotherm which, via scaling, corresponds to the thermal correction exponent θ = 0.55 ± 5 this supports previous estimates and the stronger conclusion θ = 0.54 ± 3. For the universal ratios, we estimate $K(ƒ_{1}^{−})^{2} = 0.096_{5} ± 2, C^{c} δ/(B^{δ−1} C^{+})^{1δ} = 0.93 ± 2_{5}$ , and $(C^{+} /C^{c})(ƒ_{1}^{c} /ƒ_{1}^{+})^{2−η} = 1.17 ± 2$ , where $B, ƒ_{1}^{−}, and C^{+}$ are the amplitudes of the spontaneous magnetization, and (second moment) correlation length, and of the susceptibility above T_c.

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^☆: Work supported by the National Science Foundation and the Cornell University Materials Science Center. F.R.G. acknowledges with thanks support by the Programa de Superacíon del Personal Académico de la UNAM, and by the Banco de México.

^†: Present address: Departamento de Física, Facultad de Ciencias, UNAM, Ciudad Universitaria, México, 20, D. F., México.

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Physica A: Statistical Mechanics and its Applications

Noncritical interface near a critical end point II☆

Abstract

J. Colloid Interf. Sci.

Phys. Rev.

J. Chem. Phys.

J. Chem. Phys.

Z. Phys. Chem.

J. Stat. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chim. Phys.

Kolloid-Z.

J. Chim. Phys.

The Physics and Chemistry of Surfaces

The Physics and Chemistry of Surfaces

J. Chem. Phys.

Soviet Phys. JETP

The non-ideal behaviour of the interfacial tension of the n-heptane + perfluoro-n-hexane mixture: A computational study

Modelling and understanding of the vapour-liquid and liquid-liquid interfacial properties for the binary mixture of n-heptane and perfluoro-n-hexane

Bulk and interfacial properties of a cationic micellar system near the critical point

Usual and unusual surface tensions of perfluorocarbon-containing binary liquid mixtures near a critical endpoint

Temperature dependence of vapor/liquid interfacial tensions in a binary liquid mixture with a miscibility gap near its critical point: Determination of universal surface amplitude ratios

Universal surface-tension and critical-isotherm amplitude ratios in three dimensions