# Probing the Interfacial Behavior of Type IIIa Binary Mixtures Along the Three-Phase Line Employing Molecular Thermodynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}or VL

_{2}), liquid–liquid (L

_{1}L

_{2}), and vapor–liquid–liquid (VL

_{1}L

_{2}) equilibria. For these mixtures, the interplay between bulk phases and their interfaces constitutes a challenging task for applied thermodynamics due to the sharp asymmetry between the intermolecular forces at bulk and interfacial regions. One of the prototypical examples of this Type III equilibria are the water + n-alkane [n ≥ 2] mixtures, where the relation between bulk phases and their interfaces is also a function of the n-alkane molecular chain length (n). Specifically, for n = 2 to 26, the critical line connecting the critical point of the n-alkane ends in an UCEP characterized by a vapor–liquid critical point and an aqueous-rich liquid phase; this behavior is classified as Type IIIa. For n = 28 to 32, the critical line connecting the critical point of pure water ends in an UCEP characterized by a vapor–liquid critical point and an n-alkane-rich liquid phase; this behavior is classified as Type IIIb. Finally, 26 < n < 28, water + n-alkane mixtures display a mass density (or barotropic) inversion, where the relative position of immiscible phases changes. Figure 1 displays a schematic representation of the coexistent density of Types IIIa and IIIb

_{.}(see Ref. [2] for further details on critical lines and subcritical phase equilibria for water + n-alkane [n ≥ 2] mixtures).

_{1},VL

_{2}, L

_{1}L

_{2}), three interfacial tensions (IFTs) can be defined: γ

^{VL1}, γ

^{VL2}, and γ

^{L1L2}, where L

_{1}refers to the water rich and L

_{2}to the n-alkane rich liquid phases, and V is the vapor phase at equilibrium. Assuming for instance that γ

^{VL1}> γ

^{L1L2}> γ

^{VL2}(case a, Figure 1), these IFTs are interrelated by the expression [5]: γ

^{VL1}≤ γ

^{L1L2}+ γ

^{VL2}. The equality denotes the situation of total (or perfect) wetting of the L

_{2}bulk phase in VL

_{1}interface (Antonow’s rule), whereas the inequality describes the situation of partial wetting of the L

_{2}bulk phase in VL

_{1}interface (Neumann inequality). The transition from total to partial wetting (or vice versa) is recognized as a wetting transition [6] and occurs at a certain point along the three-phase line. (see Refs. [7,8] and references therein). Additionally, when temperature increases and approaches the UCEP, the densities of two bulk phases becomes equal, ρ

^{L2}≈ ρ

^{V}for Type IIIa (case a, Figure 1), or ρ

^{L1}≈ ρ

^{L2}for Type IIIb (case b, Figure 1) causing that the interfacial tensions at the UCEP to approach γ

^{VL1}= γ

^{L1L2}and γ

^{VL2}= 0 or γ

^{VL1}= γ

^{VL2}and γ

^{L1L2}= 0, respectively.

_{1}or L

_{2}) is suspended on the tip of a needle. This drop is immersed in the other phase (be it L

_{2}or L

_{1}or V) in a closed chamber where temperature and/or pressure can be controlled. Interchanging the liquid of the drop and its surroundings, it is possible to independently measure the three interfacial tensions (γ

^{VL1}; γ

^{VL2}; γ

^{L1L2}). This tensiometric technique has been used by Mori et al., [11] for measuring some Type IIIa of water + n-alkane mixtures, where γ

^{VL1}> γ

^{L1L2}> γ

^{VL2}.

_{zz}AT ensembles. In MD, the interfacial tension is calculated by the Hulshof integral [21], where the pressure along the interfacial region is described by the Irving–Kirkwood (IK) tensor [22]. This approach has been successfully used by us in previous works to describe the three-phase interfacial tension for Lennard–Jones [18,19,23], and coarse-grained Mie [24] fluids. In the case of MC, the interfacial tension can be calculated using similar approach or alternatively using the grand canonical ensemble (μ, T, V). In the grand canonical ensemble, the interfacial tension is calculated by using the average height of the peaks of the probability distribution as a function of the number of particles together with the definition of interfacial free energy. For further details concerning to this methodology, the reader is redirected to Ref. [3] and references therein.

_{NB}) which are significantly lower than the UCEP temperatures (T

_{UCEP}). Therefore, it is necessary to use alternative procedures (theory and/or simulation) to fill the gap from T

_{NB}to T

_{UCEP}. On the other hand, the SGT coupled to any equation of state is usually unable to predict interfacial tensions in liquid-liquid equilibrium without input from experimental data. The reader is referred to the extensive digressions by Carey [13] and the implementations by Cornelisse et al. [20] in selected systems (water + n-hexane, water + benzene and n-hexane + water + ethanol mixtures). Consequently, SGT needs to use the interfacial tension data of the mixture to fine-tune the description of the interfacial behavior in liquid-liquid and liquid-liquid-vapor equilibrium. In contrast, MD simulations are capable of predicting interfacial properties via explicit simulation of the vapor-liquid-liquid (VLL) interfaces, where the required force field parameters can be parametrized by the use of pure component thermophysical and mixture phase equilibria data rather than having to recourse to interfacial tension data.

_{NB}= 341.88 K [25]) which is relative low in comparison to T

_{UCEP}= 495.82 K [26]. Additionally, Bertrand et al., [27] reported an experimental wetting transition at T

_{w}= 345.4 K, which was verified by Cornelisse et al. [20] by employing the Peng-Robinson EoS with SGT, but not seen with the APACT EoS.

## 2. Theory

#### 2.1. Square Gradient Theory for Mixtures

_{1}; VL

_{2}or L

_{1}L

_{2}), and is given by the following integral expression [13]:

_{c}is the number of species (n

_{c}= 2 in this work), ρ

_{i}is the interfacial molar concentration of species i and z is the spatial coordinate perpendicular to the planar interface. The superscripts α and β correspond to the two different bulk phases. c

_{ii}is the influence parameter of the pure fluid i. κ

_{ij}is a symmetric cross interfacial tension parameter for the mixture (κ

_{ij}= κ

_{ji}; κ

_{ii}= κ

_{jj}= 1), which is obtained by fitting Equation (1) with the experimental IFT values of the mixture.

_{ii}is calculated from the pure fluid coarse-grained (CG)-Mie parameters using the following expression [31]:

_{av}is the Avogadro’s constant, ε

_{ii}is the energy scale corresponding to the Mie potential well depth, σ

_{ii}length scale, corresponding with an effective segment diameter of the Mie potential, m

_{si}is the molecular chain length for the pure fluid i described by the CG approach, and α

_{ii}is the van der Waals constant given by:

_{r,ii}and λ

_{a,ii}are the repulsion and attraction parameters of the intermolecular (Mie) potential for pure fluid i, respectively. C

_{ii}is a constant, which is defined as:

_{ii}values through Equation 2. These values will be simultaneously used in theoretical calculation and molecular simulations to obtain phase equilibrium and interfacial properties.

_{i}(z) is calculated by solving the following system of differential equations [13]:

_{0}is the Helmholtz energy density of the homogenous system, which is given by the SAFT-VR Mie EoS for non-associating chain fluids [30], μ

_{i}

^{0}is the chemical potential of species i evaluated at the phase equilibrium conditions, calculated from its definition in the canonical ensemble μ

_{i}

^{0}= (∂a

_{0}/∂ρ

_{i})

_{T,V,}

_{ρj}.

_{i}(z) are obtained solving Equation (5), from which the interfacial tension can be calculated using Equation (1). Furthermore, with the information of the ρ

_{i}(z) profiles, it is possible to characterize the surface activity of the species along the interfacial region. Figure 2 illustrates four possible patterns of ρ

_{i}(z). Figure 2a shows the common biphasic interfacial profile for pure fluids or fluid mixtures without surface activity. In Figure 2b,c, the interfacial profile displays a stationary point, which can be a maximum (point A) or a minimum (point D). The stationary points reflect adsorption (A) or desorption (D) of species along the interface region (i.e., the surface activity). Finally, Figure 2d displays a possible density profile ρ

_{i}(z) for VL

_{1}L

_{2}equilibrium. In the latter, the interfacial concentration along the z coordinate shows three interfaces without surface activity, namely VL

_{1}, L

_{1}L

_{2}, VL

_{2}. The ρ

_{i}(z) profile also provides a route to explore the interfacial concentration of the mixture and its thermal evolution for three-phase systems. As illustrated in Figure 1 and reported in previous works (see Refs. [16,17,18,19]) ρ

^{L1}(z) ≠ ρ

^{L2}(z) + ρ

^{V}(z) at T < T

_{w}, whereas ρ

^{L1}(z) = ρ

^{L2}(z) + ρ

^{V}(z) at T > T

_{w}. For the case that T → T

_{UCEP}, ρ

^{L2}(z) ≈ ρ

^{V}(z) or ρ

^{L2}(z) ≈ ρ

^{L1}(z).

#### 2.2. The Statistical Associating Fluid Theory Model

^{Mie}, represented by:

_{ij}is the center-to-center distance of the interacting segments. The other terms have the same meaning described before but extended to mixtures applying combination rules [30]. Specifically, the unlike size parameter, σ

_{ij}is obtained using an arithmetic mean:

_{ij}is obtained using a Berthelot-like geometric average:

_{ij}is a binary interaction parameter, which is obtained from experimental data of phase equilibria. The cross attractive (λ

_{a,ij}) and repulsive (λ,

_{rij}) parameters involved in the Mie potential are calculated as:

_{ij}is given in Equation (4), where the attractive and repulsive exponents are replaced by λ

_{a,ij}and λ,

_{rij}, respectively. Finally, the homogeneous Helmholtz energy density for the SAFT-VR Mie EoS for non-associating chain fluid is given by [30]:

_{0}= A/(N k

_{B}T), A being the total Helmholtz energy, N the total number of molecules, N

_{av}the Avogadro constant, T the temperature, k

_{B}the Boltzmann constant, β = 1/(k

_{B}T), and ρ the molar density of the mixture. a

^{mono}is the Helmholtz energy contribution of the unbounded monomers forming a chain of m

_{s}tangential segments, a

^{chain}accounts for the Helmholtz energy of chain formation, and a

^{ig}is the Helmholtz energy of the perfect gas reference.

_{si}, λ

_{r,ii}, λ

_{a,ii,}ε

_{ii}, σ

_{ii}), and the mixing parameter (k

_{ij}). In this work, the pure fluid parameters are taken from previous works, where n-hexane is modeled as two tangent spheres [28], whereas water is modeled as a single sphere without electrostatic interactions [29]. In the latter case, the level of coarse graining averages out many important directional and long-range interactions present in water as a consequence of its very asymmetric molecular charge distribution. Such effects cannot be reproduced by a simplistic spherical isotropic potential, hence uniquely, the SAFT water molecular parameters are defined as a function of temperature and the type of properties (bulk or interfacial) that is targeted. In this work, we selected the set of molecular parameters based on interfacial tensions (see Table 1 for numerical values). Finally, for the alkane/water mixture, we used k

_{ij}= 0.3205 [33].

#### 2.3. The Three-Phase Equilibrium from SAFT-VR Mie EoS

_{0}) and its derivate properties, such as chemical potential, μ

_{i}, and the stability function, ℑ. a

_{0}and μ

_{i}are needed to calculate the phase equilibrium condition as well as input to SGT, whereas ℑ is used to validate the phase equilibrium results. In this work, the three-phase equilibrium conditions in the canonical ensemble are described by the following expressions [34]:

_{1}, L

_{2}, and V denote the three different bulk phases, x

_{i}and μ

_{i}are the mole fraction, and the chemical potential of component i, respectively. A

_{nm}is a shorthand notation for the derivative of A with respect to n and m. As an example, A

_{V}= (∂A/∂V), which gives the equilibrium pressure P

^{0}= - A

_{V}.

^{L1}= P

^{L2}= P

^{V}= P

^{0}) while Equations (11b) and (11c) express the chemical potential constraint (μ

_{i}

^{L1}= μ

_{i}

^{L2}= μ

_{i}

^{V}). Equation (11d) is a differential stability condition for phase equilibrium, comparable to the Gibbs energy stability constraint of a single phase [34]. Solving Equations (11a)–(11c) restricted to Equation (11d), it is possible to find the stable bulk densities and their mole fractions (ρ

^{L1}, ρ

^{L2}, ρ

^{V}, x

_{1}

^{L1}, x

_{1}

^{L2}, and x

_{1}

^{V}).

_{12}) is obtained by fitting Equation (1) with the experimental IFT values of the mixture [11]. The numerical value used in this work is κ

_{12}= 0.336, which is similar to the values reported by Cornelisse et al. [20] for the case of Peng-Robinson and APACT EoSs.

## 3. Molecular Dynamics Simulations

_{1}+ N

_{2}), and volume, V, at the simulation temperature, T, are estimated using the SAFT-VR Mie EoS, where the pure fluids and the fluid mixture are described by using the CG-Mie approach, i.e., the same Mie parameters taken from SAFT-VR Mie EoS (see Table 1). For a complete discussion of CG SAFT-VR Mie methodology and its top-down parameterization, the reader is directed to Müller and Jackson’s work [38].

_{1}, N

_{2}) is fixed, and the systems are set up in such a way that the volume fractions of the resulting bulk phases are comparable. The simulations cell employs a L

_{x}× L

_{y}× L

_{z}parallelepiped with periodic boundary conditions in all three directions, with L

_{x}= L

_{y}= 55 Å (L

_{x}= L

_{y}> 10σ) and L

_{z}= 8 L

_{x}= 440 Å. This simulation box is built and filled with two liquid phases and a vapor phase assembled through the z axis (i.e., the L

_{1}, L

_{2}, and V interfaces are located at the xy plane). In this assembly, each phase has an initial volume of L

_{x}= L

_{y}= 55, Lz = 145 Å, and is filled with a minimum of 19,000 Mie beads (in the whole simulation) at the three-phase line conditions. These values are chosen in order to have a large enough cell to accommodate two liquids, and the vapor regions with enough molecules to ensure a sensible statistics when calculating the densities of the bulk (non-interfacial) phase. To reduce the truncation and system size effects involved in the phase equilibrium and interfacial tension calculations, a cut-off radius of 27 Å (r

_{cut}≈ 6σ) is used throughout this work.

_{bond}= K

_{b}(r – σ)

^{2}, where K

_{b}= 7.583 kcal mol

^{−1}Å

^{−2}[33]. The velocity-Verlet integrator is used with a time step of 5 fs, and the temperature is controlled by a Nosé–Hoover thermostat with a relaxation constant of 0.2 ps.

_{i}(z), are obtained by splitting the simulation box along the z direction in L

_{x}× L

_{y}× 1 Å

^{3}bins and time averaging the number of molecules in each bin. Additionally, the center of mass of the system is fixed to its initial position to avoid profile smearing due to dynamical fluctuations. MD simulations allow evaluating the surface activity of species from the ρ

_{i}(z) profiles and the equilibrium pressure as well as the IFTs from Irving-Kirkwood method. Calculation of the two latter quantities requires obtaining the diagonal elements of the pressure tensor profiles along the direction of the box, which can be obtained in each bin using the virial expression [22,36]:

_{kk}are the diagonal pressure tensor elements, where the subscript k represents the spatial coordinate, either x, y, or z. k

_{B}is Boltzmann’s constant, T is the absolute temperature, S is the interfacial area, N is the total number molecules, f

_{ij}is the force on molecule i due to molecule j, and r

_{ij}represents the distance between molecules i and j. f

_{ij}and r

_{ij}contributions have been equally distributed among the slabs corresponding to each molecule, and all the slabs between them. In Equation (12), the first term takes into account the kinetic contribution to the pressure, and it represents the perfect (ideal) gas term, while the second term corresponds to the configurational contribution, which is evaluated as ensemble averages, < >, and not at instant values.

_{zz}element, while the interfacial tension γ, between each pair of bulk phases αβ (i.e., VL

_{1}; VL

_{2}or L

_{1}L

_{2}) can be calculated by integrating the pressure elements of Equation (12) through the z dimension [21]:

^{VL1}; γ

^{VL2}; γ

^{L1L2}) can be calculated from Equation (13) with the appropriate integration limits and are related to each other through the Antonow’s rule or the Neumann inequality.

_{zz}(z) – (P

_{xx}(z) + P

_{yy}(z))/2]dz = 0 within all bulk phases is verified.

^{VL1}≈ 54.03 mN/m, γ

^{L1L2}≈ (99.44–54.03) mN/m ≈ 46.41 mN/m, and γ

^{VL2}≈ (108.56–99.44) mN/m ≈ 9.12 mN/m. Further details related to the technical implementation of the previous expressions and their evaluation at three-phase conditions have been discussed in our previous works. [18,19,23,24]

## 4. Results and Discussions

_{w}and T

_{UCEP}results are compared to the available experimental measurements [11,26,27].

#### 4.1. Interfacial Tension Along a Three-Phase Equilibrium

^{VL1}, γ

^{VL2}and γ

^{L1L2}. Figure 4 illustrates the thermal evolution of the interfacial tensions (T – γ) for the mixture at three-phase line. This figure includes the tensiometry data reported by Mori et al., [11], MD results (the numerical data are summarized in Table 2), and SGT + SAFT-VR Mie EoS calculations.

_{NB-C6H14}= 341.88 K [25]). From Figure 4, an excellent agreement between SGT + SAFT-VR Mie EoS calculations and experimental data is observed. Noticeably, these theoretical calculations are not predictions as this agreement is a consequence of fitting the cross-influence parameter (κ

_{ij}). For the case of MD results, one observes an overprediction of IFT values but with low absolute average deviation; AADγ

^{VL1}= 5%, AADγ

^{VL2}= 7%; AADγ

^{L1L2}= 7%. However, it is important to note that the tensiometry data reported by Mori et al., [11] employed the density of the pure fluids rather than that of the mixture, and this choice influences the result inducing an AADγ ≈ 2%, as was discussed by us in a previous work. (See Ref. [10] and references therein).

^{VL2}→0, and γ

^{VL1}→ γ

^{L1L2}when the mixture approaches its UCEP. As was described in Sec.2.1, this is the expected behavior of Type IIIa binary mixtures due to ρ

^{L2}≈ ρ

^{V}as the UCEP is approached, then γ

^{VL2}≃ 0, and γ

^{VL1}≈ γ

^{L1L2}. Following the IFT extrapolations, the temperature of the UCEP (T

_{UCEP}) is estimated as 539.8 K from SGT, and 486.3 K from MD. Both results compare well with the value of 517 K calculated from theoretical predictions of interfacial tensions [20] and the experimental value 495.82 K [26]. The MD results suggest that γ

^{VL1}= γ

^{L1L2}= 27.93 mN/m at T

_{UCEP}. One possible reason for the overprediction of seen by the SGT is attributed to the fact that in general, while the SAFT-VR Mie model excels at predicting pure component critical points, it overestimates the critical condition of the mixtures [42]. The theoretical extrapolations are also in good agreement to ones reported by Cornelisse et al., [20] who calculated the IFTs from SGT combined to the Peng-Robinson and the APACT EoS.

_{w}), as the transition from equality to inequality in the relationship γ

^{VL1}≤ γ

^{L1L2}+ γ

^{VL2}. This mixture exhibits a wetting transition (T

_{w}) at 353.8 K (as estimated from SGT) and 347.2 K (as estimated from MD), which are both in close agreement with the value of 345.4 K reported by Bertrand et al. [27]. As it appears that γ

^{VL1}≤ γ

^{L1L2}+ γ

^{VL2}and using the criteria stated by Costas et al. [8], it is possible to conclude that the wetting transition is of first order.

#### 4.2. Bulk Densities and Interfacial Concentration Profiles Along a Three-Phase Equilibrium

_{i}) along the interfacial region (z) for three-phase system is characterized by three bulk zones and their interfaces: VL

_{1}, VL

_{2}, and L

_{1}L

_{2}, as was schematically illustrated in Figure 2d. The z – ρ

_{i}diagrams are used to describe the interfacial concentration, the surface activity of species, and the wetting behavior. In this section, the z – ρ

_{i}diagrams are described at three selected isothermal conditions: (i) T < T

_{w}, (ii) T > T

_{w}, (iii) T ≈ T

_{UCEP}.

_{UCEP}= 479.10 K, and ρ

_{UCEP}= 0.2667 g cm

^{−3}, which show good agreement with the experimental values (495.82 K and 0.2599 g cm

^{−3}[26]). Additionally, the extrapolated T

_{UCEP}(from bulk phase calculations) agrees with those obtained from IFTs. Table 3 includes the corresponding extrapolated conditions for the UCEP.

_{i}profiles at the isothermal condition of partial wetting of 290 K (T < T

_{w}). The concentration profiles in this figure reveal that both MD simulations and SGT results are in very good agreement with each other. Focusing on the VL

_{1}interfacial behavior (i.e., close to z = 0 and left hand in the insert snapshot) the aqueous bulk liquid region (L

_{1}) and the bulk vapor region (V) are partially separated by an organic bulk liquid region (L

_{2}). This type of structure is a clear evidence that L

_{2}partially wets the VL

_{1}interface at this thermodynamic condition. In other words, the interfacial concentration profile that connects the bulk phases VL

_{1}is different than the sum of the L

_{1}L

_{2}and the VL

_{2}interfacial concentration profiles (i.e., ρ

^{VL1}(z) ≠ ρ

^{L1L2}(z) + ρ

^{VL2}(z)). It can also be observed that the density of n-hexane shows a high peak that reflects a strong positive surface activity (accumulation or adsorption) at the VL

_{1}interface, whereas the density of water displays a weak positive surface activity at the VL

_{2}.

_{i}profiles at the isothermal condition total wetting of 380 K (T > T

_{w}). Similar to Figure 5a, Figure 5b displays a very good agreement between MD simulations and SGT results. Positive surface activity of n-hexane and water is observed at the VL

_{1}and VL

_{2}interfaces, respectively. It is important to point out that the surface activity of n-hexane notoriously increases from 290 K to 380 K. From Figure 5b, it is possible to observe that as L

_{2}completely wets the VL

_{1}interface at these thermodynamic conditions, then ρ

^{VL1}(z) = ρ

^{L1L2}(z) + ρ

^{VL2}(z). Finally, Figure 5c shows the z – ρ

_{i}profiles and a snapshot of the system at an isothermal condition of 485 K, which is near the T

_{UCEP}. At this temperature, ρ

^{L2}≈ ρ

^{V}, and ρ

^{L1}≠ 0 which means that only VL

_{1}interfaces are observed. Additionally, the surface activity of n-hexane is still noticeable, whereas water displays no surface activity. This figure includes the MD results only because the SGT predicts three phases at this temperature.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Sample Availability: Not available. |

**Figure 1.**Schematic representation of Temperature, T—Density, ρ and Interfacial Tension, γ—Temperature, T diagrams for three-phase equilibrium in Type III binary mixture. Case a: Type IIIa; Case b. Type IIIb.

**Figure 2.**Schematic representation of the multiphasic interfacial concentration profiles, ρ

_{i}, as a function of the z coordinate. (

**a**) pure fluids or fluid in mixtures without surface activity in a biphasic system; (

**b**) fluids with a positive surface activity (adsorption) in a biphasic system; (

**c**) fluids with a negative surface activity (desorption) in a biphasic system; (

**d**) fluids without surface activity in a triphasic system.

**Figure 3.**Cumulative interfacial tension, γ, as a function of the z coordinate for water (1) + n-hexane (2) mixture at the isothermal three-phase condition of 380 K. γ

^{VL1}≈ 54.03 mN/m, γ

^{L1L2}≈ (99.44–54.03) mN/m ≈ 46.41 mN/m, and γ

^{VL2}≈ (108.56–99.44) mN/m ≈ 9.12 mN/m.

**Figure 4.**Interfacial tension, γ—temperature, T diagram for water (1) + n-hexane (2) mixture. Experimental data [11]: (▲) γ

^{VL1}; (■) γ

^{L1L2}; (●) γ

^{VL2}; MD results: (△) γ

^{VL1}; (☐) γ

^{L1L2}; (○) γ

^{VL2}; SGT + SAFT-VR Mie EoS calculations: (—) γ

^{VL1}; (– • –) γ

^{L1L2}; (– •• –) γ

^{VL2}. (•••) Estimated wetting temperature (MD-T

_{w}= 347.20 K; SGT-T

_{w}= 353.81 K); (- - -) Estimated Upper Critical End Point (UCEP) temperature (MD-T

_{UCEP}= 486.30 K; SGT-T

_{UCEP}= 539.78 K). (◆) γ

^{UCEP}.

**Figure 5.**Interfacial concentration distribution along the interfacial region at three different isothermal conditions. (

**a**). 290 K; (

**b**). 380 K; (

**c**). 485 K. Top: Snapshot: (●) water, (●●) n-hexane. Bottom: Interfacial concentration profiles, ρ

_{i}, along the interfacial region, z. SGT + SAFT-VR Mie EoS calculations: (– • –) water, (– •• –) n-hexane. MD results: (•••) water, (•••) n-hexane.

Fluid | m_{si} | λ_{r,ii} | ε_{ii}/k_{B}/K | σ_{ii}/Å |
---|---|---|---|---|

n-hexane (n-C_{6}H_{14}) | 2 | 19.57 | 376.35 | 4.508 |

water (H_{2}O) | 1 | 8.00 | −4.806 × 10^{−4} T^{2} + 0.6107 T + 165.9 | −6.455 × 10^{−9} T^{3} + 9.1 x 10^{−6} T^{2} − 4.291 x 10^{−3} T + 3.543 |

^{a}The pure fluid Mie parameters are taken from Mejía et al. [28] for n-hexane, and Lobanova et al. [29] for water, T in K.

^{b}The attractive exponent was fixed at 6 (λ

_{a,ii}= 6) for both pure fluids.

^{c}The influence parameters (c

_{ii}) of pure fluids for SGT are calculated from Equation (2).

**Table 2.**Interfacial tensions result from Molecular Dynamics for water (1) + n-hexane (2) along three-phase equilibrium

^{a,b}.

T/K | γ^{VL1}/mN m^{−1} | γ^{L1L2}/mN m^{−1} | γ^{VL2}/mN m^{−1} |
---|---|---|---|

290 | 68.00_{1} | 53.00_{1} | 20.24_{1} |

320 | 64.20_{1} | 50.90_{1} | 16.64_{3} |

350 | 59.06_{2} | 46.47_{3} | 12.36_{2} |

380 | 54.68_{3} | 43.20_{2} | 9.20_{2} |

410 | 45.74_{3} | 40.61_{2} | 5.13_{4} |

440 | 40.75_{1} | 35.96_{5} | 2.83_{3} |

470 | 32.05_{2} | 30.30_{3} | 1.00_{2} |

486.3^{ c} | 27.93 | 27.93 | 0.00 |

^{a}The subscripted number is the uncertainty in the last digits. (i.e., 20.24

_{1}means 20.24 ± 0.01).

^{b}L

_{1}: aqueous (water) rich phase; L

_{2}: oil (n-hexane) rich phase; V: vapor phase.

^{c}Extrapolated upper critical conditions.

**Table 3.**Mass bulk densities and mole fractions results from Molecular Dynamics for water (1) + n-hexane (2) mixture along three-phase equilibrium

^{a}.

Organic (n-hexane rich) phase | ||||

T/K | x_{1} | ρ_{1}/g cm^{−3} | ρ_{2}/g cm^{−3} | ρ/g cm^{−3} |

290 | 0.047_{1} | 0.0071_{1} | 0.661_{2} | 0.668_{3} |

320 | 0.081_{1} | 0.0121_{1} | 0.626_{3} | 0.638_{3} |

350 | 0.116_{2} | 0.0163_{3} | 0.587_{1} | 0.603_{2} |

380 | 0.169_{1} | 0.0233_{2} | 0.543_{4} | 0.566_{2} |

410 | 0.232_{3} | 0.0312_{2} | 0.489_{1} | 0.520_{3} |

440 | 0.338_{1} | 0.0442_{1} | 0.412_{2} | 0.456_{3} |

470 | 0.455_{4} | 0.0581_{2} | 0.331_{1} | 0.389_{1} |

479.10 ^{b} | 0.501 | 0.134 | 0.134 | 0.268 |

Aqueous (water rich) phase | ||||

T/K | x_{1} | ρ_{1}/g cm^{−3} | ρ_{2}/g cm^{−3} | ρ/g cm^{−3} |

290 | 1.000 | 1.003_{1} | 0.000 | 1.003_{1} |

320 | 1.000 | 0.994_{2} | 0.000 | 0.994_{2} |

350 | 1.000 | 0.979_{2} | 0.000 | 0.979_{2} |

380 | 1.000 | 0.958_{1} | 0.000 | 0.958_{1} |

410 | 1.000 | 0.934_{3} | 0.000 | 0.934_{3} |

440 | 1.000 | 0.905_{1} | 0.000 | 0.905_{1} |

470 | 1.000 | 0.875_{2} | 0.000 | 0.875_{2} |

479.1^{b} | 1.000 | 0.864 | 0.000 | 0.864 |

vapor phase | ||||

T/K | x_{1} | ρ_{1}/g cm^{−3} | ρ_{2}/g cm^{−3} | ρ/g cm^{−3} |

290 | 0.964_{1} | 0.004_{1} | 0.001_{1} | 0.005_{1} |

320 | 0.925_{1} | 0.006_{1} | 0.003_{1} | 0.009_{1} |

350 | 0.901_{2} | 0.010_{2} | 0.006_{3} | 0.016_{2} |

380 | 0.849_{1} | 0.015_{2} | 0.013_{2} | 0.028_{2} |

410 | 0.807_{3} | 0.024_{1} | 0.027_{2} | 0.051_{1} |

440 | 0.755_{1} | 0.034_{3} | 0.053_{2} | 0.119_{2} |

470 | 0.571_{3} | 0.127_{4} | 0.095_{6} | 0.223_{1} |

479.1 ^{b} | 0.501 | 0.134 | 0.134 | 0.268 |

^{a}The subscripted number is the uncertainty in the last digits. (i.e., 0.661

_{2}means 0.661 ± 0.002).

^{b}Extrapolated upper critical conditions.

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**MDPI and ACS Style**

Alonso, G.; Chaparro, G.; Cartes, M.; Müller, E.A.; Mejía, A.
Probing the Interfacial Behavior of Type IIIa Binary Mixtures Along the Three-Phase Line Employing Molecular Thermodynamics. *Molecules* **2020**, *25*, 1499.
https://doi.org/10.3390/molecules25071499

**AMA Style**

Alonso G, Chaparro G, Cartes M, Müller EA, Mejía A.
Probing the Interfacial Behavior of Type IIIa Binary Mixtures Along the Three-Phase Line Employing Molecular Thermodynamics. *Molecules*. 2020; 25(7):1499.
https://doi.org/10.3390/molecules25071499

**Chicago/Turabian Style**

Alonso, Gerard, Gustavo Chaparro, Marcela Cartes, Erich A. Müller, and Andrés Mejía.
2020. "Probing the Interfacial Behavior of Type IIIa Binary Mixtures Along the Three-Phase Line Employing Molecular Thermodynamics" *Molecules* 25, no. 7: 1499.
https://doi.org/10.3390/molecules25071499