Modeling Interfacial Tension of Heptane + Alcohol Mixtures Using Cubic Plus Association Equation of State Plus Simplified Gradient Theory

This work has been dedicated to modeling the interfacial tension of the heptane + alcohol mixtures in the temperature range of 288.15 K to 333.15 K. The cubic plus association equation of state is applied to the liquid–vapor phase equilibrium calculations. The binary interaction parameters are obtained according to the experimental isothermal and isobaric phase equilibrium data. For the binary interaction parameters, correlations have been obtained as a function of temperature for isothermal phase equilibrium, and a constant value for isobaric phase equilibrium. The linear gradient theory is used as a predictive and adjustment approach to describe the interfacial tension of the heptane + alcohol mixtures. The influence parameters of the pure components were constant and the symmetric parameters of the binary mixtures were correlated with the temperature. The results of this work show that the cubic plus association equation of state is capable of simultaneously representing the phase equilibrium and interfacial tension of the mixtures studied. The results obtained in the interfacial tension are in agreement with those published in the literature.

Covolume parameter in the CPA-EOS C Number of carbons in the alcohol c Influence parameter c 1 Adjustable parameter of the CPA EOS f 0 Helmholtz energy density g The radial distribution function k ij Interaction parameter in the CPA-EOS n Number of points n c Number of components P Absolute pressure R Universal gas constant T Absolute temperature x Mole fraction of the liquid phase The mole fraction of the molecule i not bonded at site A z Position in the interface

Introduction
The phase equilibrium of the alkane + alcohol mixtures is of great importance in engineering calculations [1]. On the other hand, interfacial tension calculations of the alkane + alcohol mixtures is a very important property for many physical, chemical, and biological processes [2]. In my previous work [3], phase equilibrium and interfacial tension of the hexane + alcohol mixtures were studied at different temperatures and results are in agreement with the experimental information on phase equilibrium and interfacial tension. Due to the good results obtained with the hexane + alcohol mixtures, the idea of this work is use the cubic plus association equation of state (CPA-EOS) to study the behavior of phase equilibrium and use the linear gradient theory (LGT) to study the interfacial tension for the heptane + alcohol mixtures at different temperatures. The gradient theory (GT) was initially developed by van der Waals [4] and later reformulated by Cahn and Hilliard [5]. This thermodynamic approach consists of expressing the Helmholtz energy density as the sum of two contributions. The first part considers the Helmholtz energy density in a hypothetically homogeneous medium, while the second part represents the inhomogeneous contributions of Helmholtz energy, namely through the product of concentration gradients and characteristic parameters [6]. Numerous authors thereafter have utilized GT with different equations of state (EOSs) to describe the interfacial behavior of systems with different types of phase equilibrium. GT has been widely used for calculating the interfacial tension because the same thermodynamic model, i.e., an equation of state can be directly used for the bulk phases and for the interface [7]. The predictive use of GT, i.e., symmetric parameter equal to zero requires the solution of a set of algebraic equations, and for the alkane + alcohol mixtures, the presence of multiple stationary points in density make the calculations more complex. In a previous work, GT was used as a predictive approach to calculate the interfacial tension of the alcohol + glycerol mixtures at different temperatures [8]. On the other hand, GT used as an adjustment approach, i.e., nonzero symmetric parameter requires the solution of a set of differential equations that makes the calculation of interfacial tension more complex. In order to simplify the gradient theory, Zuo and Stenby [9] developed the linear gradient theory, which eliminates the need to solve the set of time consuming density profile equations. The speeds up the calculation procedure without significantly losing accuracy in the interfacial tension results [10].
According to the literature, the interfacial tension of the alkane + alcohol mixtures studied with GT or LGT have been heptane + ethanol, and heptane + 1-propanol mixtures at 298.15 K. Mejía et al. [11] used the Peng-Robinson equation of state (PR-EOS) and the Huron-Vidal mixing rule (MHV) with gradient theory as an adjustment approach to represent the interfacial tension of the heptane + ethanol mixture at 298.15 K. The authors [11] obtained good results in adjusting the interfacial tension of the mixture. On the other hand, Zuo and Stenby [9] use Soave Redlich Kwong equation of state (SRK-EOS) with LGT to adjust the interfacial tension of the heptane + ethanol, and heptane + 1-propanol mixtures at 298.15 K. They obtained results that are in agreement with the experimental information of interfacial tension. However, the authors used a quadratic mixing rule unable to correctly model the phase equilibrium of the mixtures.
The articles published by Liang et al. [7] and Liang and Michelsen [12], suggest that study the hexane + ethanol mixture with the gradient theory is challenging from a numerical point of view, due to the presence of multiple stationary points in the density profiles (when the parameter symmetric is null), while more advanced numerical techniques are required to optimize the adjustable parameter of the mixture. Therefore, due to the authors' suggestions [7,12], it is expected to find similar characteristics for alkane + alkanol mixtures. Therefore, it has been decided to used LGT for two reasons: to avoid numerical problems with the use of the null symmetric parameter and to avoid a high computational cost when adjusting symmetric parameter. The novelty of the article is to show the application of linear gradient theory to complex mixtures (according to the literature when the gradient theory is used) and therefore, to study both the prediction and the correlation of the experimental surface tension. In addition, apply LGT with an equation of state capable of correctly modeling the homogeneous phases of the mixtures, because one of the requirements of GT is that the homogeneous phases are well modeled. Then, the results of this work would be consistent, unlike those obtained in the literature by Zuo and Stenby [9].
Finally, in this work, CPA-EOS with LGT is used to study the interfacial tension of the heptane + ethanol, heptane + 1-propanol, heptane + 1-butanol, heptane + 1-hexanol, and heptane + 1-octanol mixtures in the temperature range of 288.15 K to 333.15 K.

The CPA-EOS
The CPA-EOS is a powerful tool for thermodynamic modeling of associating components as well as non-associating components, and it has been successfully applied into many systems [7,13]. In this work, the cubic plus association equation of state [1,14] was selected for modeling the phase equilibrium for the heptane + ethanol, heptane + 1-propanol, heptane + 1-butanol, heptane + 1-hexanol, and heptane + 1-octanol mixtures. According to the literature [1,7,[15][16][17], good results in the representation of the phase equilibrium have been obtained for the alkane + alcohol mixtures. Yakoumis et al. [1] are the only authors who have modeled the phase equilibrium of the heptane + alcohol mixtures; they have obtained for the heptane + ethanol mixture a deviation of 1.7 % in vapor pressure, and 1.5 % in vapor phase mole fraction, and for the heptane + 1-butanol mixture, the have obtained a deviation of 1 % in vapor pressure, and 0.02 % in vapor phase mole fraction. The results are accurate because this equation of state models the association part of the components in the mixtures, unlike the cubic equations of state. CPA equation of state is given by Eq. 1: where P is pressure, R is the universal gas constant, T is temperature, is molar density, and x i is the mole fraction of component i.
The physical term is that of the SRK-EOS [18] and the association term is taken from SAFT [19].
For a pure component, a and b are the van der Waals attractive energy and covolume parameters. The attractive energy parameter is given by a Soave-type temperature dependency defined by Eq. 2: where a 0 and c 1 are adjustment parameters and T c is the critical temperature.
The key element of the association term is X A i , which represents the mole fraction of the molecule i not bonded at site A and can be obtained from Eq. 3: The term Δ A i B i (for self-associating molecules) is given by Eq. 4: where A i B i and A i B i are the association energy and the association volume, respectively, and g( ) is the radial distribution function [20] defined as Eq. 5 : This model has five adjustment parameters for each pure component ( A i B j ) and can be obtained by fitting vapor pressure and liquid density data. For nonpolar components (e.g. heptane), only the three parameters ( a 0 , b, c 1 ) of the SRK term are required and for associating components (e.g. alcohol) in addition to the mentioned parameters, it is required A i B j and A i B j . In this work, the parameters of the pure components are obtained from the literature, and for alcohol, a 2B scheme is used.
The classical van der Waals mixing rule [21] is used to calculate the energy and covolume parameters of the mixture, which are given by Eq. 6: where a i and b i are the energy and covolume parameters of the pure components, respectively, w i is the molar fraction in the liquid phase ( w i = x i ) or vapor phase ( w i = y i ), and k ij is the parameter of interaction between different molecules. In this work, the parameter of interaction is obtained by adjusting the experimental information of the equilibrium of phases.
On the other hand, for the 0-site/2-site (hydrocarbon/alcohol) mixtures no combining rules are needed for the association parameters A i B j and A i B j [1].

The Linear Gradient Theory
The linear gradient theory was first presented by Zuo and Stenby [9,22,23] and proposes that density profile of each component in the mixture are distributed linearly between the bulk phases in equilibrium, i.e., the density of component i at position z, i (z) , in the interface can be represented by Eq. 7: where V i and L i are the vapor and liquid bulk density at equilibrium condition of component i, respectively, n c is the number of components in the mixture, and n is the number of grids in which the interface is divided.
According to LGT, the interfacial tension, , can be obtained from Eq. 8: where ΔΩ( ) is the grand thermodynamic potential which is calculated at each grid point by Eq. 9: where f 0 ( ) is the Helmholtz energy density of a hypothetically homogeneous fluid with a uniform concentration, 0 i is the chemical potential of component i in the bulk equilibria phases, and P 0 is the pressure in the bulk equilibria phases.
In Eq. 8, c is the influence parameter of the mixture and s represent the independent variable. The subscript s is the component which has the maximum value of the density difference between the coexisting homogeneous phases defined as Eq. 10: The influence parameter of the mixture depends on the pressure range [22]. At low pressure, the influence parameter is correlated from Eq. 11: where c ij is the cross-influence parameter, which can be obtanined as Eq. 12: where c ii and c jj are the influence parameters of the pure components i and j, respectively, and ij is a symmetric parameter which can be obtained by adjusting the experimental information of interfacial tension of the binary mixture. According to the value adopted by the symmetric parameter, LGT can be used as a predictive ( ij = 0 ) or adjustment ( ij ≠ 0 ) approach.
In this work, a constant value for the influence parameter has been used for each fluid.

Pure Fluids
All pure-component parameters of CPA-EOS have been listed in Table 1. Table 2 indicates the influence parameters for all fluids. The absolute average deviation for interfacial tension ( AAD %) of pure fluid was obtained using Eq. 13: where n represent number of points, exp and theo are experimental and theoretical value, respectively.
From Table 2, it can be seen that the theoretical model correctly adjusts the interfacial tension for the pure fluids. The smallest deviation (0 %) is reached for ethanol, because the influence parameter was obtained at a single temperature (298.15 K), and the greatest deviation (0.973 %) is reached for 1-propanol.

Phase Equilibrium Calculations of the Heptane + Alcohol Mixtures
In this work, CPA-EOS was used to describe the phase equilibrium of the following mixtures: heptane(1) + ethanol(2), heptane(1) + 1-propanol(2), heptane(1) + 1-butanol(2), heptane(1) + 1-hexanol(2), and heptane(1) + 1-octanol (2).  Because linear gradient theory calculations require adjustment of the phase equilibrium binary parameter, this parameter was adjusted using the phase equilibrium information available in the literature. For the mixtures studied in this work, the binary interaction parameters of CPA-EOS have been listed in Table 3. For the heptane + ethanol mixture at 303.27 K, the binary interaction parameter was calculated using experimental phase equilibrium from Berro et al. [28]. For the heptane + 1-propanol mixture, a linear correlation of the binary parameter with temperature was obtained in the temperature range of 298.15 K to 333.15 K; this correlation was obtained with experimental phase equilibrium from Sayegh et al. [29]  The binary interaction parameters of the isothermal phase equilibrium were obtained by minimizing of objective function given by Eq. 14: The binary interaction parameter of the isobaric phase equilibrium was obtained using the objective function given by Eq. 15  The absolute average deviation in vapor pressure ( AADP% ) was calculated using Eq. 16: The absolute average deviation in vapor mole fraction ( AADy 1 % ) was calculated using Eq. 17: The absolute average deviation in boiling temperature ( AADT% ) was calculated using Eq. 18: From Table 3 it can be seen that CPA-EOS correctly adjusts the phase equilibrium for the studied mixtures. The smallest deviation (0.07 %) is reached for vapor mole fraction of the heptane + 1-octanol mixture at 353.15 K, and the greatest deviation (2.69 %) is reached for vapor pressure of the heptane + 1-octanol mixture at 313.15 K. Furthermore, the results are in agreement with those published by Yakoumis et al. [1]. Figures 1, 2, 3, 4 and 5 shows the phase envelope of the mixtures studied. From these Figures it is oberverd that the results obtained are in agreement with the experimental information. On the other hand, the heptane + ethanol, heptane + 1-propanol, heptane + 1-butanol, and heptane + 1-hexanol mixtures are azeotropic.

Interfacial Tension of the Heptane + Alcohol Mixtures
According to the results of the phase equilibrum obtained, the following binary interaction parameters were used in the interfacial tension calculations. For the heptane + ethanol mixture, k ij = 0.00624 was used for the interfacial tension at 298.15 K. For the heptane + 1-propanol mixture, k ij = −0.09442 + 0.00035T was used for the interfacial tension in the temperature range of 293.15 K to 333.15 K. For the heptane + 1-butanol mixture, k ij = −0.07152 + 0.00027T was used for the interfacial tension in the temperature range of 288.15 K to 308.15 K. For the heptane + 1-hexanol mixture, k ij = 0.04015 was used for the interfacial tension in the temperature range of 288.15 K to 308.15 K. Finally, for the heptane + 1-octanol mixture, k ij = −0.08854 + 0.00032T was used for the interfacial tension in the temperature range of 288.15 K to 308.15 K.
The deviation in the interfacial tension was obtained from of Eq. 13. The experimental information of the interfacial tension was obtained from Papaioannou and Panayiotou [25] for the heptane + ethanol mixture at 298.15 K, Papaioannou and Panayiotou [25] for the heptane + 1-propanol mixture at 298.15 K, McLure et al. [26] for the heptane + 1-propanol mixture in the temperature range of 293.15 K to 333.15 K, and Vijande et al. [27] for the heptane + 1-butanol, heptane + 1-hexanol, and heptane + 1-octanol mixtures in the temperature range of 288.15 K to 308.15 K. Table 4 indicates the statistical deviation for interfacial tension using LGT as a predictive approach ( ij = 0 ) and as an adjustmente approach ( ij = optimal). Table 5 show the optimal symmetric parameters, which were obtained by minimizing of objective function given by Eq. 19: The objective function of Eq. 19 was proposed by Weiland et al. [37,38] and was used in this work because weights all of the number of data points, n, (high and low interfacial tensions) equally.
From Table 4 it can be seen that LGT + CPA-EOS no correctly predicts the interfacial tension of the mixtures studied. The greatest deviation (5.47 %) is reached for the heptane + 1-octanol mixture at 303.15 K and 308.15 K. According to the literature [3,7,39], better results are obtained when the symmetric parameter is adjusted with the experimental information interfacial tension of the binary mixtures. Fig. 1 Phase envelope of the heptane + ethanol mixture at 303.27 K with CPA-EOS (Lines: theoretical results and Symbols: experimental data). ( • ) 303.27 K from Berro et al. [28], (---) using k ij = 0.00624 (Color figure online) According to Table 4, the smallest deviation (0.33 %) is reached for the heptane + 1-octanol mixture at 298.15 K, and the greatest deviation (1.10 %) is reached for the heptane + 1-propanol mixture at 333.15 K. Furthermore, the deviations obtained for the heptane + ethanol, and heptane + 1-propanol mixtures at 298.15 K are smaller than those published by Zuo and Stenby [9], who obtained 1.66 % (0.71 % in this work) for the heptane + ethanol mixture, and 0.75 % (0.50 % in this work) for the heptane + 1-propanol mixture. Figures 6,7,8,9 and 10 shows the interfacial tension for the heptane + alcohol mixtures. For the mixtures studied in the entire temperature range, it is observed that the linear gradient theory as a predictive approach is not capable of correctly representing the interfacial tension, this is because the linear theory cannot correctly predict the interfacial tension of mixtures with components that present accumulation in the interface [40]. On the other hand, the correlative use of linear gradient theory is capable of correctly modeling the interfacial tension of the mixtures studied. Finally, the good results using LGT as adjustment approach are in agreement with those obtained for the hexane + alcohol mixtures [3]. Figure 11 shows the variation of the interfacial tension of the mixtures at 298.15 K with the liquid phase mole fraction of heptane. From Fig. 11 it is observed that as the molecular chain of the alcohol increases, the interfacial tension of the mixtures also increases. A possible justification can be attributed to the increase in the cohesive energy of the mixture as the alcohol chain increases. The same characteristic was obtained in the previous work for the hexane + alcohol mixtures [3].