Handling a very large data set for determination of surface tension of chemical compounds using Quantitative Structure–Property Relationship strategy

Abstract

In this work, the Quantitative Structure–Property Relationship (QSPR) strategy is applied to represent/predict the surface tension of pure chemical compounds at (66.36–977.40) K temperature range. To propose a comprehensive, reliable, and predictive model, 18298 data belonging to experimental surface tension values of 1604 chemical compounds at different temperatures are studied. The Sequential Search mathematical method has been observed to be the only variable search method capable of selection of appropriate model parameters (molecular descriptors) regarding this large data set. To develop the final model, a three-layer Artificial Neural Network has been optimized using the Levenberg–Marquardt (LM) optimization strategy. Using this dedicated strategy, we obtain satisfactory results quantified by the following statistical parameters: absolute average deviations of the represented/predicted properties from existing experimental values: 3.8%, and squared correlation coefficient: 0.985.

Introduction

The surface effects on naturally occurring phenomena and industrial applications, e.g. reactions over the surface of a catalyst, boiling heat transfer, condensation, and microscale channel flow processes such as lubrication, corrosion, adherency, detergency, and reactions in electrochemical cells have been of great interest in the past decade (Ramírez-Verduzcoa et al., 2006, Wemhoff and Carey, 2006). Liquid–vapor interfaces are vital for the performance of detergents, in chemical engineering separations such as absorption and distillation, and in the performance of biological membranes (Ramírez-Verduzcoa et al., 2006, Wemhoff and Carey, 2006).

Effects of different physical forces on fluid phase equilibria can be never denied in practical applications (Balasubrahmanyam, 2008). There are unequal asymmetric forces acting upon a molecule, which are zero at equilibrium (Danesh, 1998, Poling et al., 2001, Escobedo and Mansoori, 1996). At low gas densities, the molecules at the surface experience a sidewise attraction toward the bulk liquid meanwhile they are attracted a little in the direction of the bulk gas (Danesh, 1998, Poling et al., 2001, Escobedo and Mansoori, 1996). These attractive forces tend to pull the surface towards the bulk liquid phase. Therefore, the surface layer is in tension and, at equilibrium, it tends to minimize its area compatible with the mass of material, container restraints, and external forces (Poling et al., 2001, Escobedo and Mansoori, 1996). This molecular tension at the surface is quantitatively expressed as surface (interfacial) tension, which refers to the force exerted at the interface per unit length (Danesh, 1998). As a matter of fact, it is a macroscopic thermophysical property that affects behavior of fluids in a variety of processes, as already mentioned (Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e).

One of the applications of this inhomogeneous property is to determine the capillary pressure, which is used to investigate the effects of surface forces on fluid distribution within a reservoir. Furthermore, the relative permeability of the fluids that is a significant factor for describing the fluids flow and phase behavior (in a dynamic way) in hydrocarbon reservoirs is related to the interfacial tension (Danesh, 1998, Balabin and Syunyaev, 2008, Balabin et al., 2011c, Syunyaev et al., 2009; Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e). Of particular interest are the effects of the values of interfacial tensions of gas condensates on condensate recovery in the case of retrograde condensation in gas condensate reservoirs (Danesh, 1998, Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e). Furthermore, to improve production and increase oil yields, in the extraction of crudes by adding surfactants to modify the interfacial properties between crude oil and the geological reservoir, the significance of surface tension phenomena is literally demonstrated (Danesh, 1998, Balabin and Syunyaev, 2008, Balabin et al., 2011c, Syunyaev et al., 2009, Queimada et al., 2001; Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e).

Regarding the aforementioned significance of surface tension values of pure compounds, several calculation/estimation methods have been to date proposed for this purpose. In 1923, Macleod presented an empirical equation to correlate the experimental values of surface tension based on the difference between the density of a liquid and vapor of a chemical compound at equilibrium at a given temperature and a constant characteristic of the liquid phase. Later, Sugden, 1924, Sugden, 1930 reported the constant characteristic of the Macleod's (1923) correlation (Macleod, 1923) to be a function of molecular weight and another parameter called the “parachor” as follows (Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e):

$${\sigma }^{1/4}=(P)({\rho }_L-{\rho }_V)/M$$

where σ is the surface tension in (dyn/cm), P denotes the parachor in (kg^1/4 m³ s^−1/2 kmol⁻¹), ρ is the density in (g/cm³), M is the molecular weight, and subscripts L and V refer to the liquid and vapor phases, respectively. He stated that parachor is a number that represents the molar volume of a compound when the temperature is such that its surface tension is unity (Balasubrahmanyam, 2008). In other words, parachor parameter addresses the molar volume of a substance ignoring the effects of temperature. Therefore, there are unique values of this property for each chemical compound (Wemhoff and Carey, 2006). Bayliss (1937) has calculated the parachor values by fitting the experimental data of n-Paraffins surface tensions using the least squares method. A correlation based on molecular weight has been presented by Baker and Swerdloff (1955) and Schechter and Guo (1995) for evaluation of the parachors of n-Paraffins.

However, parachor property can be related to the critical properties of compounds including critical temperature and molar volume as follows (Fanchi, 1990; Gharagheizi et al., 2011c, Gharagheizi et al., 2011d, Gharagheizi et al., 2011e):

$$P=0.324T_c^{1/4}{v}_c^{7/8}$$

where T is the temperature in K, v is the molar volume in m³/kmol and subscript c denotes the critical value. Another approach has been presented by Quayle (1953), who reported group contributions for calculation of parachor properties. However, the reported group contributions are incomplete and many functional groups are not represented, i.e. the model is not applied to many chemical compounds (Poling et al., 2001).

Although the method suggested by Macleod (1923), which relates the surface tension to the densities has some drawbacks, several authors have used this approach to evaluate the surface tensions of various compounds (Weinaug and Kalz, 1943, Lee and Chien, 1984, Hugill and van Welsenes, 1986, Gasem et al., 1989, Fanchi, 1985, Ali, 1994) mainly due to its simplicity. The main disadvantages of applying this relation are as follows:

• The values of the parachors are not always available experimentally and the estimation techniques have not been developed for enough chemical families.

• The mean absolute deviations (MAD) of the calculated/estimated surface tensions of different substances from experimental data are too high for complex chemical structures (Poling et al., 2001, Escobedo and Mansoori, 1996). The obtained results for different kinds of chemical compounds (Balasubrahmanyam, 2008) show 8.6% deviation from experimental values.

Corresponding state principles have been also developed for correlating the surface tension data of pure compounds (Escobedo and Mansoori, 1996, Brock and Bird, 1955, Curl and Pitzer, 1958, Pitzer, 1995, Zuo and Stenby, 1997, Rice and Teja, 1982, Sastri and Rao, 1995, Riedel, 1954). For instance, Brock and Bird (1955) applied the reduced temperature, the Riedel (1954) parameter at the critical point, critical temperature, and critical pressure to correlate the surface tension experimental data for nonpolar liquids. This correlation also results in high MAD (14%) from the same experimental surface tension values, upon which the Macleod's correlation (Escobedo and Mansoori, 1996) was tested. Another attempt has been made by Curl and Pitzer (1958) and Pitzer (1995) that proposed critical temperature, pressure, and acentric factor (ω) as the parameters of a correlation for calculation of the surface tensions. The obtained results using this correlation show MAD of 17% (Poling et al., 2001).

Two-reference corresponding states methods have been firstly applied by Rice and Teja (1982) who used critical temperature and volume to derive a correlation, and later Zuo and Stenby (1997) that reported a correlation based on the critical pressure and temperature to calculate surface tensions. These equations do not lead to satisfactory results for chemical compounds with strong hydrogen-Bonding forces (Poling et al., 2001). To overcome these shortcomings, Sastri and Rao (1995) evaluated the surface tensions by a correlation based on critical pressure and temperature, normal boiling point, reduced temperature, and reduced boiling temperature. Their proposed correlation leads to MAD (4%) in comparison with the surface tensions experimental values of 30 pure compounds (Poling et al., 2001). The most recent effort on these kinds of models were performed by Queimada et al. (2001) who applied two different corresponding states approaches to predict the surface tension of the series of the n-alkanes. The first method has been a second-order perturbation model based on a Taylor series expansion of the surface tension using the Pitzer acentric factor (Curl and Pitzer, 1958), and the second one uses shape factors to consider the non-Conformalities (Queimada et al., 2001). Comparisons of the results of two models with experimental data show the MAD% range of 5.9–13% (Queimada et al., 2001).

A preliminary literature review shows that there are two general theoretical strategies for evaluating the surface tensions of pure chemical compounds based on deriving special equations using computer simulations (Wemhoff and Carey, 2006). The first method in this category uses molecular dynamics (MD) simulations to determine the pressure tensor variation through the interfacial region of fluids (Wemhoff and Carey, 2006), and the difference between the normal and tangential components of the pressure tensor across the interfacial region is integrated to attain the surface tension as follows (Wemhoff and Carey, 2006):

$$\gamma ={\int }_{z=-\infty }^{z=+\infty }[p_n(z)-p_t(z)]dz$$

where γ denotes the interfacial tension (surface tension), z is the coordinate normal to the interface, p is the pressure, d is the derivative operator, and subscripts n and t refer to the normal and the tangential components of the simulation. Due to the fact that the interfacial tension is the only source of anisotropy in the pressure tensor, this equation has gained theoretical validity among the researchers (Wemhoff and Carey, 2006). Several authors (Wemhoff and Carey, 2006, Weng et al., 2000, Enders et al., 2004, Harris, 1992, Alejandre et al., 1995, Nijmeijer et al., 1988, Dunikov et al., 2001, Barker, 1993, Sinha et al., 2003, Holcomb et al., 1993, Mecke et al., 1997, Binder and Muller, 2000) have applied/modified this strategy (known as “virial”) (Wemhoff and Carey, 2006) for evaluation of the surface tension of various kinds of fluids.

The second theoretical strategy based on computer simulation is the finite-size scaling approach (Wemhoff and Carey, 2006). In this method, the difference between the probability of observing bulk liquid and vapor states and the minimally probable intermediate state is used as a measure of the excess free energy due to the presence of the interfacial region for a known domain characteristic dimension, instead of simulating the interfacial region in virial approach (Wemhoff and Carey, 2006). This method has the advantage over the virial approach since it has been found that the size of the domain affects the interfacial region profile (Wemhoff and Carey, 2006, Binder and Muller, 2000). Similarly, Wemhoff and Carey (2006) have developed the free energy integration (FEI) method as a simplified variation of density functional theory to evaluate the system net excess free energy by performing a numerical integration of the local free energy density. The obtained values of the interfacial region thickness through MD (molecular dynamics) simulations were considered as the inputs of their proposed model (Wemhoff and Carey, 2006). They stated that the FEI method provides a simple means of estimating surface tension from a knowledge of the system temperature and density profile characteristics (Wemhoff and Carey, 2006). Applying this approach for prediction of surface tension of argon, nitrogen, and water contributed to satisfactory results in wide range of temperatures (form very low temperatures up to the near critical region) (Wemhoff and Carey, 2006).

In a recent attempt done by Benet et al. (2010) the surface tension of ethane was calculated in the transferable potentials for phase equilibria (TraPPE) and optimized potentials for liquid simulations (OPLS) force fields by means of computer simulations. They indicated that their presented technique may be combined easily with the wandering interface method (WIM) (Benet et al., 2010), which is a well-known model for evaluation of surface tension of polymers, in order to calculate surface free energies and consequently surface tension values (Benet et al., 2010). Their obtained results show that the TraPPE model produces a much better description of ethane's surface tension than the OPLS model (Benet et al., 2010).

Equations of state have also been applied for calculation/estimation of surface tension of various fluids. In a 1949’s study, Tolman calculated surface tension by applying the van der Waals (2008) equation of state to the interfacial region by assuming empirical relations regarding the structure of the interface. Based on the van der Waals (2008) fluid model, a simple equation of state, unique for non-uniform fluids, has been developed by Sher et al. (2005). They applied their model to the liquid–vapor interface to derive the density profile in the interfacial region (Sher et al., 2005). They stated that their proposed EoS leads to acceptable predictions of surface tensions of ammonia, carbon tetrachloride, ethanol, nitrogen, and water.

In 1998, Abbas et al. tried to evaluate the surface tension values of pure fluids using the following approaches: (1) the generalized van der Waals (GvdW) theory in the effective potential approximation; (2) molecular dynamics (MD) simulation with potential truncation; and (3) hybrid MD/GvdW theory. Having described the capabilities of the applied methods in prediction of surface tensions of the investigated compounds, they concluded that the orientational ordering of the hydrogen halides (HCl and HBr) is of minor importance for the surface tension values.

The novel molecular layer structure theory (MLST) (Do et al., 2003) was applied by Do et al. (2003) to develop a new model for prediction of surface tensions of chemical compounds based on thermodynamic and molecular interaction between molecules to describe the vapor–liquid phase equilibria and surface tension of pure compounds. They concluded high model prediction capabilities for determination of surface tensions of several compounds including argon, krypton, ethane, n-Butane, iso-Butane, ethylene, and sulfur hexafluoride due to acceptable agreements between the predicted results and the experimental surface tension values (Do et al., 2003)

Davis and Scriven (1976) have also tried a different method based on developing a correlation between surface tension and liquid compressibility, based on a molecular model. However, the most accurate equation for representation/prediction of the surface tensions of pure liquids has been proposed by Escobedo and Mansoori (1996), who have derived an expression based on statistical-mechanic and corresponding state principle. Their results show MAD slightly higher than 1% and around 2.6% for representation and prediction of surface tensions of 94 various organic compounds, respectively (Escobedo and Mansoori, 1996). It should be pointed out that for presenting a precise comparison among the described methods, the same data sets should be applied to check their accuracy, reliability and comprehensiveness.

Recently, Gharagheizi et al. (2011c) have proposed an accurate Artificial Neural Network-Group Contribution (ANN-GC) method for calculation/estimation of surface tensions of 752 chemical compounds at different temperatures. They have reported absolute average deviations of the calculated/estimated properties from existing experimental values: 1.7%, and squared correlation coefficient: 0.997. The applied experimental data values (Project 801, 2006) for developing+validating the model have been about 4700 temperature-dependent data points.

However, there is still need for presentation of more comprehensive methods for determination of surface tensions of very large group of pure chemical compounds, which allow evaluation of surface tension values at different temperatures. In this study, we propose a novel approach based on the Quantitative Structure–Property Relationship (QSPR) strategy to represent/predict the surface tensions of pure chemical compounds at wide ranges of temperatures using a very large available dataset of experimental values.