Development of a method to model the mixing energy of solutions using COSMO molecular descriptors linked with a semi-empirical model using a combined ANN-QSPR methodology

Highlights

• Relationships between the model parameters and the molecular interactions is assumed.

• A link is established between the model-parameters/COSMO molecular descriptors pairs.

• A combined ANN-QSPR procedure is used to quantify the relation parameter/ descriptor.

• The work shows the feasibility of the methodology developed.

Abstract

A methodology linking the interaction parameters of a semi-empirical model to COSMO molecular descriptors to represent the h^E(x) is presented. A database of 4395 values of ester-alkane is used to check the procedure. Relationships are established for the model coefficients, a_i1,i2,…ip, although only for a₁₂ and for a second parameter k₂₁, which governs the asymmetry of the representation of the mixing process. A discretization of the σ-profile and its statistical moments constitute two descriptors, as a vector, ${\mathbf{S}}_{\sigma -profile}^{\text{j}}$ and ${\mathbf{S}}_{\sigma -moment}^{\text{p}}$, for a₁₂. A third molecular descriptor/vector, is described for k₂₁, based on the divergence of Kullback-Leibler and the molecules size, which is presented for the first time as:

$$\mathbf{S}{\mathbf{R}}_{\sigma -profile}=\left\{A_1,A_2,D_{\text{KL}}[p_1\left(\sigma \right)\left|\right|p_2\left(\sigma \right)],D_{\text{KL}}[p_2\left(\sigma \right)\left|\right|p_1\left(\sigma \right)]\right\}$$

The clustering of systems in a a₁₂/k₂₁-space is predictable with molecular descriptors in relation to their main interactions. An ANN-QSPR binomial estimates the parameters a₁₂ and k₂₁ from molecular descriptors. The methodology generalizes the procedure, acceptably representing the energetic effects of solutions (R² > 0.9).

Introduction

The estimation of physicochemical properties of pure components and solutions is one of the most interesting topics in the field of thermodynamics applied to chemical engineering. The existing contributions in the current literature differ in nature, some are: (a) theoretical models, developed from the thermodynamics formalism and quantum chemistry, using information on the molecular structure of the compounds, (b) semi-empirical models obtained either with a purely empirical mathematical profile, or those that combine both sciences, which can be called thermodynamic-mathematical. The parametrization of the latter is carried out using a limited amount of experimental data of some properties, such as phase equilibria, solution properties, etc, (c) models based on data-learning, where the information provided by theoretical thermodynamics is omitted; some of these models could be included in (b).

Within the first group, there are some important contributions (Renon and Prausnitz, 1968), although, for this work we are especially interested in those that arise from quantum-chemical-based models, such as COSMO-RS (Klamt, 1995) or COSMO-SAC (Lin and Sandler, 2002). These models present an interesting scientific base, minimizing the empirical proposals. Of the two COSMO-based models (Klamt and Schüürmann, 1993), the one proposed by Klamt (1995) stands out in the modelling of liquid phase properties of a wide range of solutions, including non-polar, polar and associating compounds, just using theoretical quantum-mechanic calculations of the moleculeś screening charges. However, so far, the methods based on the first principle (a) do not produce accurate results, especially when there are specific interactions in the solutions. In such cases, an alternative to overcome certain inconveniences it to use semi-empirical models, as indicated in (b). These models are usually defined for the excess Gibbs energy, g^E, as NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975), and others. These are functions supported by a certain degree of theory but, in practice, they are used almost exclusively as correlative models of experimental data. The third case commented above (c) is when a mathematical relationship is generated, often without physical sense, to correlate the experimental data.

One of the most important aspects to consider when dealing with this topic is the experimentation, due to any modelling requires of good quality experimental data. One of the quantities obtained directly with precision in the solutions field, is the mixing enthalpy, h^E. Therefore, many theoretical models of different nature use estimates of this property (Lai et al., 1978, Gmehling et al., 1993) to verify the model consistency, and even the interpretation of its behavior, since the h^Es are sensitive to molecular interactions. The h^Es are a derivative function in relation to temperature of the Gibbs function, and the information it provides is useful, however, the estimates are still poor, increasing the discrepancies when the second derivatives are calculated, such as $c_{\text{p}}^{\text{E}}$ and others. These properties are not only important from the perspective of interactive analysis, but also for their role in the calculation of energy balances in the process simulation.

In an attempt to provide an adaptable equation for the g^E-function, a new model (Ortega et al., 2010) was introduced years ago, and since then, many works have attempted to validate its usefulness. Some focused on the multiproperty correlation obtained from the phase equilibria and mixing effects (energetic and/or volumetric nature) (Ríos et al., 2014, Ríos et al., 2018). It was proved that the model offers a great versatility by improving the representations made by other semi-empirical functions proposed for g^E, or for the Helmholtz function, a^E, but at the expense of using a greater number of parameters, which must be fitted to experimental data.

Assuming that the molecular interactions between the compounds in solution give rise to excess properties, these macroscopic quantities can be associated with molecular descriptors which, in turn, will be related to the characteristic parameters of the modelling of said quantities. This procedure has been used in other cases, such as: estimation of the binary interaction parameter (k_ij) of the Peng-Robinson equation of state (EoS) (Abudour et al., 2014, Abudour et al., 2017), that of PC-SAFT (Stavrou et al., 2016), and estimation of the energetic parameters of NRTL (Gebreyohannes et al., 2014). For these works, the structural descriptors of DRAGON and CODESSA databases were used with good results.

The σ-profiles obtained from COSMO have proven useful as molecular descriptors to estimate properties of pure components and mixtures, as well as in the modelling with an empirical extension (Zissimos et al., 2002) of COSMO-RS for complex mixtures. Some studies have shown the dependence of the σ-profiles with certain properties, such as the densities of ionic liquids (IL), with environmental properties such as toxicity, and with other solvent properties (Palomar et al., 2008a, Palomar et al., 2008b, Torrecilla et al., 2010). Even, a recent application of σ-profiles involves the evaluation of surface tensions of pure components (Kondor et al., 2014) which have a key role in interfacial transport rates. Also, several researchers (Ortega et al., 2008, Vreekamp et al., 2011, Zaitseva et al., 2016) analyzed the general suitability of using the COSMO-RS methodology to estimate the h^Es of solutions containing several molecular solvents. Therefore, the proposal for this work is to establish relationships between the parameters of a semi-empirical mathematical-thermodynamic model, applied to h^E, and the COSMO σ-profiles used as a basis for molecular descriptors. It covers the importance analysis of each descriptor obtaining the explained variance of model parameters. The relationships are raised by generating a Quantitative Structure-to-Property Relationship (QSPR) (Balaban, 2001, Diudea, 2001) using a multilinear analysis and Artificial Neural Networks (ANN). The study is focused on the mixing energies of a set of homologous series of ester-alkane. Why have these solution sets been chosen? There are several reasons, one of them is due to the large amount of experimental information available in the literature (Fernández et al., 2010, Fernández et al., 2013, Fernández et al., 2014, Gonzalez et al., 1993, Gonzalez et al., 1994, Gonzalez and Ortega, 1993, Gonzalez and Ortega, 1994, Lorenzana et al., 1989, Lorenzana et al., 1990, Ortega, 1990, Ortega, 1991a, Ortega, 1991b, Ortega, 1992, Ortega et al., 1990a, Ortega et al., 1990b, Ortega et al., 1991, Ortega et al., 1992a, Ortega et al., 1992b, Ortega et al., 1999a, Ortega et al., 1999b, Ortega et al., 1999c, Ortega et al., 2015, Ortega and Gonzalez, 1993a, Ortega and Gonzalez, 1993b, Pérez et al., 2016, Toledo et al., 2000, Vidal et al., 1997), which will lead to a more rigorous study. Another is because it is relevant that the experimental values of the chosen homologous series show a monotonous change with the chemical structure of the constituent compounds and the proposed framework should assume the phenomenology of the mixing process, which is a working desideratum. All this must be able to establish a formal qualitative relationship between the parameters of the thermodynamic model and the descriptors and molecular interactions. Lastly, the chosen set is gaining interest for its involvement in different biotechnological processes, as the enzymatic synthesis of esters (Martins et al., 2011), in food technology (Herrera et al., 2019), and also in chemical engineering as the biofuels production (Nabi et al., 2006).