Estimating the Temperature-Dependent Surface ... - ACS Publications

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Estimating the Temperature-Dependent Surface Tension of Ionic Liquids Using a Neural Network-Based Group Contribution Method Juan A. Lazzús,* Fernando Cuturrufo, Geraldo Pulgar-Villarroel, Ignacio Salfate, and Pedro Vega Departamento de Física y Astronomía, Universidad de La Serena, Casilla 554, La Serena 1700000, Chile S Supporting Information *

ABSTRACT: A neural network-based group contribution method was developed in order to estimate the temperature-dependent surface tension of pure ionic liquids. A metaheuristic algorithm called gravitational search algorithm was employed in substitution of the traditional backpropagation learning algorithm to optimize the update weights of our neural network model. A total of 2307 experimental data points from 229 data sets of 162 different ionic liquid types, such as imidazolium, ammonium, phosphonium, pyridinium, pyrrolidinium, piperidinium, and sulfonium, were collected from the specialized literature. In this database, a wide temperature range from 263 to 533 K, and a wide surface tension range from 0.015 to 0.062 N·m−1, were covered. The input parameters contained the following properties: absolute temperature, the molecular weight of the ionic liquid, and 46 structural groups that composed the molecule. The accuracy of the proposed method was checked using the mean absolute percentage error (MAPE) and the correlation coefficient (R) between the calculated and experimental values. The results show that, for the training phase, our method presents a MAPE = 1.17% and R= 0.998, while for the prediction phase, the method shows a MAPE = 1.29% and R = 0.991. In addition, the relative contribution of each input parameter was calculated from the optimal weights of the network. Also, the effects of the temperature, molecular weight, and cation and anion types on the estimation of the surface tension were analyzed. Finally, the proposed method was compared with other methods available in the literature. All results demonstrated the high accuracy of our method to estimate the temperature-dependent surface tension for several ionic liquid types.

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INTRODUCTION Ionic liquids (ILs) are salts commonly composed of an organic cation and an inorganic anion, which have the particularity of being present in a liquid state over a wide temperature range (generally below 100 °C) in which other salts are found in solid state.1 The properties of the ionic liquids will depend on the specific combination of the cation and anion.2 It is estimated that there are 106 combinations of structures (cation−anion) that can be formed to synthesize an ionic liquid.3 By having this large number of cations and anions to design the ionic liquids, the most appropriate combination can be chosen so that it takes specific properties for a particular application. Thus, an ionic liquid designer has the main component responsible for the chemical behavior (i.e., cation) and another component that contributes most of the physical properties (i.e., anion).2 The most important properties to consider in the design of ionic liquids are a negligible vapor pressure at moderate pressures and temperatures, high thermal stability, large liquidus range, high ionic conductivity, high solvating capacity, nonvolatility, and nonflammability, mainly for use as solvents.4 The second group of specific properties of these substances is their volumetric properties (density, molar volume), transport properties (viscosity, electrical conductivity, thermal conductivity, self-diffusion coefficient), thermal properties (heat © XXXX American Chemical Society

capacity, thermal decomposition temperature, melting point temperature), surface tension, etc. Note that the latter group of properties has particular importance for the application of ionic liquids in organic synthesis, catalysis, electrochemistry, chemical separation, metal extraction, and nanoparticle formation,5 and among these properties the surface tension of ionic liquids is one of those required for their use in the fields of physical chemistry and engineering sciences.1 In physical chemistry, surface tension is caused by the cohesive forces among liquid molecules due to the fact that each molecule is pulled equally in every direction by neighboring liquid molecules, obtaining a net force equal to zero. As the molecules at the surface of liquid do not have the same molecules on all sides of them, they are pulled inward, creating an internal pressure, forcing at liquid surface to contract in a minimal area.6 In other words, surface tension is the energy that must be supplied to increase the surface area by one unit by performing like an energy necessary to create surfaces. Thus, surface tension can manifest itself in forms of both surface force and surface energy.7 For the engineering Received: March 24, 2017 Revised: May 6, 2017 Accepted: May 23, 2017

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DOI: 10.1021/acs.iecr.7b01233 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Table 1. Some Methods Proposed in the Literature To Estimate the Surface Tension of Pure Ionic Liquidsa method

no. of data

no. of ILs

ΔT/K

9 38 6 60 30 105 51 48 115 59

298 268−393 293−353 283−393

10

parachor parachor11 parachor12 parachor13 corresponding states theory method14 corresponding states theory method15 group contribution method16 quantitative structure−property relationship17 quantitative structure−property relationship18 neural network model19

361 209 385 1224 920 930 930

268−533 268−744 268−744 268−532

Δσ/N·m−1

MAPE/%

R 0.830

5.75

0.015−0.076 0.024−0.065 0.023−0.064

8.50 2.98 4.95 3.60 1.05 3.43 4.59

0.919 0.991 0.952 0.960

In this table, ΔT is the temperature range, Δσ is the surface tension, MAPE is the mean absolute percentage error, and R is the correlation coefficient.

a

different artificial neural network models have been proposed for estimating several physicochemical properties of organic and inorganic compounds,25 and most recently, some neural network models have been proposed in the literature to estimate physicochemical properties of ionic liquids, e.g., density,26 melting point temperature,27,28 viscosity,29,30 and thermal conductivity,31 among others. For the case of the surface tension of ionic liquids, Atashrouz et al.19 proposed a neural network model based group method of data handling, called GMDH-NN, for predicting the surface tension of 59 ionic liquids over a temperature range of 268 to 532 K using as input parameters the molar density, reduced boiling temperature, reduced temperature, reduced pressure, acentric factor, and critical compressibility factor. On the other hand, Lashkarbolooki32 applied a neural network model in order to estimate the surface tension of 32 binary ionic liquids in ILs/ non-ILs systems using as input data the melting point and molecular weight for the ionic liquids and non-IL component, and the mole fraction of the ionic liquid. Note that the main disadvantage of these two methods is the use of additional physicochemical properties to obtain the surface tension of an ionic liquid. In order to solve this disadvantage, improved methods based on the molecular structural characteristics of ionic liquids are required. In this study, we proposed a neural network-based group contribution method for estimating the temperature-dependent surface tension of ionic liquids over a wide temperature range and for several ionic liquid types (that includes: imidazolium, ammonium, phosphonium, pyridinium, pyrrolidinium, piperidinium, and sulfonium), only from the knowledge of its molecular structure. One of the innovations of this method is the use of another metaheuristic algorithm called gravitational search algorithm33 in substitution of the traditional backpropagation learning algorithm34 in order to optimize the update weights of our neural network model. Note that traditional optimization techniques such as backpropagation learning algorithm can also determine the number of network parameters, such as network connection weightings; however, they are not able to control parameter optimizations in the absence of gradient information, while the gravitational search algorithm is able to solve this problem.35 In addition, the group contribution method implemented was designed and divided respecting the ionic nature of those substances into cation part and anion part. To the best of the authors’ knowledge, there are not any methods to estimate the temperature-dependent surface tension of ionic liquids using a hybrid technique of group contribution method plus artificial neural networks plus

sciences, surface tension is required in the manufacture of many industrial products such as coatings, paints, detergents, cosmetics, and agrochemicals, and it also affects some production processes such as catalysis, adsorption, distillation, and extraction.8 Thus, the knowledge of surface tension of a substance is highly relevant, but in particular, for ionic liquids, surface tension is required to describe the intrinsic energies that are involved in the interactions between ions9 (among others uses), because it is a physical property that is closely related to the molecular composition and structure. Commonly, the temperature-dependent surface tension of ionic liquids is experimentally measured, however, lack of resources and the increasing number of ionic liquids make its determination impractical. Hence, new estimation methods of the temperature-dependent surface tension of ionic liquids based on simple calculation and properties are required. Nowadays, some estimation methods of temperature-dependent surface tension of ionic liquids can be available in the scientific literature such as parachor models,10−13 corresponding states theory methods,14,15 group contribution methods,16 and quantitative structure−property relationship methods.17,18 Table 1 shows a comparison between some available methods in the literature for estimating the temperature-dependent surface tension of pure ionic liquids. As shown, some of these methods may be complex and require additional thermophysical properties and simplified assumptions to complete the calculation. One way to improve an accuracy of the estimation of the surface tension of ionic liquids is the use of heuristic/ metaheuristic algorithms. Heuristic/metaheuristic algorithms are powerful computational techniques designed to provide a sufficiently good solution to a complex problem, specially with limited information.20 This algorithm may make few assumptions about the problem being solved, and so they may be used for a variety of problems.21 Here, we do not discuss a precise definition of these algorithms and their complexities. The interested reader can review more information in recent books.20,21 From this wide spectrum of computational algorithms, artificial neural network is one of the most used in several scientific fields.22 An artificial neural network (ANN) is a computational bioinspired algorithm based on the basic structure and functions of biological neurons.23 This heuristic/metaheuristic algorithm processes information that flows through its structure (i.e., layers) via processing units (i.e., neurons) so the network can learn by updating the synaptic connections (i.e., weights) between input and output patterns.24 Over the past decades, B


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algorithm, called gravitational search algorithm,33 was implemented to train the network by replacing the backpropagation algorithm.34 Note that, nowadays, the artificial neural networks are one of the most commonly used heuristic/metaheuristic algorithms and are also the most studied.22 For this reason, we will not go deeper into its mathematical formulation and computational algorithm construction. But we will give more attention to the other algorithms involved in this study that have been less developed in the literature. For interested readers, there are several books related to a general or specific description of artificial neural networks and their application fields.38 Gravitational Search Algorithm. Gravitational search algorithm (GSA) is a recently introduced heuristic/metaheuristic algorithm33 based on Newton’s laws of gravity and of motion.39 Thus, gravitational search algorithm can be considered as a system composed of agents, called masses, that obey the fundamental principles of the physical laws of gravitation and masses, where a particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.40 Then the masses (i.e., agents) are attracted between them due to the gravity force by causing a global movement of all objects toward the objects with heavier masses.33 In this way, the algorithm agents (masses) cooperate by means of the gravitational force, using it as a form of communication, where the heavy masses, representing the good solutions of the algorithm, are moving more slowly than lighter ones in order to guarantee the good exploitation of the algorithm solutions through a search space.35 Additionally, each mass has four characteristic parameters, such as position, inertial mass, active gravitational mass, and passive gravitational mass,33 where the position of the mass corresponds to a solution of the problem, and its gravitational and inertial masses are adjusted using a fitness function.33,35 Thus, following the procedure described by Rashedi et al.,33 the gravitational search algorithm initializes by randomly placing L masses (i.e., agents) into an l-dimensional search space, where the position of each mass of this system Zn = (z1n, ..., zhn, ..., zln) with n = 1, 2, ..., L, represents a possible solution for the problem; here zhn denotes the position for the nth mass in the hth dimension. At time t, the action of the gravitational forces from mass m on mass n can be calculated as

gravitational search algorithm that be similar to the one presented in this study.

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COMPUTATIONAL METHOD Artificial Neural Network. Artificial neural networks (ANN) are heuristic/metaheuristic algorithms based on the biological neuronal system.23 This algorithm processes the information through using processing units called neurons that are organized in layers and that are interconnected between them through synaptic coefficients called weights. From this interconnection, a network can correlate input and output patterns by adjusting the values of its weights, in a process called training.36 Usually, to train the network employs a supervised learning method in order to optimize the network weights, such as backpropagation learning algorithm.34 In this learning algorithm, the network output is calculated using the network inputs multiplied by the network weights that are summed and fed forward through the network layers. During the training, these interconnections are optimized in order to minimize the prediction errors and so the network reaches a specified accuracy level. At this point, the error is fed backward through the network layers, and the final error adjusts the network weights for all the neuron connections. Once the network has learned thought of the training process, it can be given new input data not used during the learning phase to predict a new output.23 Table 2 shows the statistical significance of the common terms in artificial neural networks. Table 2. Statistical Significance of Terms Used in Artificial Neural Network Theory ANN

statistics

input parameter output parameter connection weights bias weight training validation pattern error

independent variable dependent variable regression coefficients intercept parameter parameter estimation interpolation/extrapolation observation residuals

A disadvantage that can occur during the network training is known as overfitting. In this case, the error obtained by the training set is minimized to a very small value, but when new data input are presented to the artificial neural network, the error is very large. Then, the artificial neural network only has memorized the training examples, but it has not learned to generalize to new situations.24 Different methods have been proposed to prevent overfitting or restricting the generalization capacity of the artificial neural network, such as by limiting the number of neurons in the hidden layers, by adding a penalty term of large weights to the objective function, and by limiting the training epochs using the cross-validation technique.23 Other disadvantages are due to the backpropagation learning algorithm’s being based on error gradients, so it is essentially a hill-climbing algorithm, and it is susceptible to premature convergence of local minima. On the other hand, another disadvantage is its dependence on the starting point of the search, i.e., the initial weight values in the case of artificial neural networks.37 In order to avoid the disadvantages generated in the use of the artificial neural networks, a new heuristic/metaheuristic

h (t ) = G (t ) Fnm

M pn(t ) × Mam(t ) R nm(t ) + ϵ

(zmh(t ) − znh(t ))

(1)

where G(t) denotes the gravitational constant at time t; Mpn is the passive gravitational mass associated with mass n; Mam is the active gravitational mass associated with mass m; ϵ corresponds to a small costant value; and Rnm denotes the Euclidian distance between the masses n and m calculated as R nm = ∥Zn(t),Zm(t)∥2.35 The total action of the gravitational forces now acting on mass n in dimension h are calculated as follows: L

Fnh(t ) =

∑ m = 1, m ≠ n

h rmFnm (t )

(2)

where rm denotes a random value into an interval [0,1]. Applying the laws of motion on the system, the acceleration of the mass n in h dimension at time t is computed as C


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Industrial & Engineering Chemistry Research anh(t ) =

Fnh(t ) M in(t )

H

(3)

i=1

where Min denotes the inertial mass of the mass n. This acceleration value is used to update the velocity and position for the mass n, as follows: vnh(t + 1) = rn × vnh(t ) + anh(t )

(4)

znh(t + 1) = znh + vnh(t + 1)

(5)

fit n(t ) − worst(t ) best(t ) − worst(t )

M̃ (t ) Mn(t ) = L n ∑ M̃ m(t ) m=1

(9)

i ,max

i ,min

the minimum and maximum data input values, respectively; wij are the weights that interconnect the input layer and the hidden layer, and wik are the weights that interconnect the hidden layer and the output layer; bj and bk are the biases of the hidden and output layer, respectively; f j is the activation function for the hidden layer, and f k is the activation function for the output layer; and yi denotes the normalized output vector. For interested readers, a full description of the feedforward neural network processing can be reviewed in a famous technical book.23 Step 2. At this point, the network generates random values for their initial weights and biases and calculates an output response. To evaluate this response, the network compares the difference between target outputs and network outputs using an objective function defined as the root-mean-square error (RMSE). Thus, for given input−output vectors, RMSE is calculated as

(6)

where G0 denotes the initial gravitational constant value, α is the descending coefficient, and t and tmax are the current and maximun iteration, respectively.35 The gravitational masses Mpn, Mam and inertial masses Min, Mn are calculated by the fitness evaluation. Assuming that there is equality between them (Mpn = Mam = Min = Mn with n = 1, 2, ..., L), they are then computed via a map of fitness. Thus, the gravitational and inertial masses are updated as follows: M̃ n(t ) =

i=1

where xi denotes the normalized input vector calculated as 2 xi = (Xi − Xi ,min) X − X − 1; here Xmin and Xmax denote

where rn is the random element in the interval [0,1]. During this process, the gravitational constant is updated as follows: G(t ) = G0 × exp( −α × t /tmax )

I

yi = fk (∑ wikf j (∑ wijxi + bj) + bk )

D

(7)

RMSE =

∑d = 1 (Ycalc − Yreal)d 2 D

(10)

where Ycalc denotes the neural network output response that is obtained by denormalization of the output vector yn; Yreal is the given output value; D is the total data point, and d denotes a particular data point. If the obtained RMSE value complies with the established error minimization condition (RMSE ≤ ξ), the algorithm finalizes its operation without the need of use the GSA part. If not, the algorithm continues its process integrating the GSA to continue with the adjustment of the network weights and biases. Step 3. In order to update the initial weights and biases generated by the feedforward neural network, the gravitational search algorithm is applied. Here the network training parameters are assigned into the vector Zn = [wij,bj,wik,bk], and then the training based GSA consists of identifying the optimal training parameters through a fitness function ξ(Zn). In other words, each mass (agent) moves through l-dimensional search space under the actions of the laws of gravitation and of motion to reach the optimal goals by using the procedure described in eqs 1 to 8. Note that the size of the l-dimensional search space corresponds to the total elements contained in the vector Zn. This training process is repeated with each iteration until ANN + GSA complies with the established error minimization condition (RMSE ≤ ξ), and then the hybrid algorithm finalizes its operation successfully. For interested readers, some recent applications that integrate artificial neural networks with the gravitational search algorithm can be reviewed in the literature.35,43

(8)

where fitn(t) denotes the fitness value obtained for the mass n at time t; worst(t) represent the weakest mass in the population defined for a minimization problem as worst(t) = max fitm(t) with m ∈ {1, ..., L}; best(t) is the strongest mass in the population defined also for a minimization problem as best(t) = min fitm(t) with m ∈ {1, ..., L}; and Mn(t) operates as a normalization for the calculated masses. Then for each iteration the velocity and position of all masses will be updated using eqs 4 and 5, while the gravitational constant and masses will be updated using eqs 6 and 7. This process will be repeated until the algorithm stops by meeting an ending criterion. In recent years, the accuracy and capabilities of the gravitational search algorithm to solve optimization problems have been proved by some authors.41 But most recently, the gravitational search algorithm has been proposed to train neural networks,35 due to the fact that the main advantage of this algorithm is to reduce the problems of trapping of local minima during the convergence process. For interested readers, a full description of gravitational search algorithm and their applications can be reviewed in a recent book.42 Hybrid ANN + GSA. The proposed hybrid algorithm (ANN + GSA) integrates two powerful heuristic/metaheuristic techniques whose operation may be difficult to understand. In order to clarify this fact, the step-to-step procedure for the operation of ANN + GSA is presented, as follows: Step 1. A fully interconnected feedforward neural network is implemented to calculate the output vector from the input vector of the network through three processing layers. One input layer contains I input neurons associated with each input parameter selected, one hidden layer contains H hidden processing neurons, and one output layer contains O output neurons associated with each active output. The fully input− output processing can be computed as

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ESTIMATING THE SURFACE TENSION OF ILS VIA ANN + GSA Database Used. In order to estimate the temperaturedependent surface tension σ(T) for several ionic liquid types, a heterogeneous set of ionic liquids such as imidazolium, ammonium, phosphonium, pyridinium, pyrrolidinium, piperidinium, and sulfonium was covered. A total of 2307 σ(T) experimental data points from 229 data sets of 162 ILs were D


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Industrial & Engineering Chemistry Research Table 3. Input Parameters Considered in the Proposed Method

set. Table S1 shows the experimental values of σ(T) considered in this study. In the data sets, an acceptable range of σ(T) properties were covered, such as a wide temperature range from 263 to 533 K, and a wide surface tension range from 0.015 to 0.062 N·m−1. Furthermore, the ionic liquids included in this database have very different physicochemical characteristics, e.g., low molecular weight substances such as propylammonium formate (MW = 105.14),46 or high molecular weight substances such as trihexyl(tetradecyl)phosphonium tris(pentafluoroethyl)trifluorophosphate (MW = 928.88),47 and low surface tension value for 1-tetradecyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (σ = 0.01550 N·m−1 at 512.9

collected from the specialized literature,44 by considering that experimental protocol measurements were clearly described (as is suggested in another study).45 A leave-25%-out cross-validation/prediction method was used to evaluate the prediction capabilities of our ANN + GSA model. Thus, the total data set was divided into a training set with 1727 experimental data points (this set includes 1150 experimental data points used for the learning process and 577 experimental data points used for the validation process), and a prediction set containing 580 experimental data points to test our designed ANN. In our strategy of dividing the database, all data points for one ionic liquid were assigned to one same data E


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Industrial & Engineering Chemistry Research Table 4. An Example of Making the Input Vector

K),48 and high surface tension value for 1-ethyl-3-methylimidazolium dicyanamide (σ = 0.06196 N·m−1 at 278.16 K).49 On the other hand, several anion types were also covered such as iodide, bromide, chloride, formate, nitrate, acetate, trifluoroacetate, tosylate, tetrafluoroborate, hexafluorophosphate, dicyanamide, trifluoromethanesulfonate, tris(pentauoroethyl)trifluorophosphate, bis[(trifluoromethyl)sulfonyl]imide, and amino acid, among others. In addition, the database contains symmetric cations, e.g., 1,3-dimethylimidazolium, and asymmetric cations, e.g., 1-tetradecyl-3-methylimidazolium or trihexyl(tetradecyl)phosphonium or triethylhexylammonium; also it contains symmetric anions, e.g., tetrafluoroborate or hexafluorophosphate, and asymmetric anions, e.g., heptadecafluorooctanesulfonate or p-toluenesulfonate. Note that other subclassifications of ionic liquids were also covered such as aprotic ionic liquids,50 e.g., 1-ethyl-3methylimidazolium tetrafluoroborate,51 or 1-butyl-3-methylimidazolium hexafluorophosphate,51 and protic ionic liquids,52 e.g., propylammonium formate46 or propylammonium acetate.46 This analysis shows that the database used is the most complete so far in order to estimate the temperature-dependent surface tension of ionic liquids (see Table 1 for a comparison with other methods). Input Parameters. A vector containing the following properties was used as input parameter: absolute temperature in kelvins, the molecular weight of the ionic liquid, and the structural groups that compose the molecule. In this last point, the additive-constitutive structural group basic principles were employed in order to develop a group contribution method by respecting the ionic nature of these substances. Thus, 17 structural groups were generated for the cation part of the ionic liquids (g+c,1, ..., g+c,17), and another 29 structural groups were also generated for its anion part (g−a,1, ..., g−a,29). Specifically, the cation groups were constituted by 7 cation head groups (imidazolium, ammonium, phosphonium, pyridinium, pyrrolidinium, piperidinium, and sulfonium) and 10 other substitute groups, while the 29 anion groups were formed by the non-ring, ring, halogen, oxygen, nitrogen, and sulfur groups. Thus, for a specific ionic liquid, the occurrence of a structural group is defined as 0 when the group does not appear in the substance and n when the group appears n times in the substance.45 Table 3 shows the input parameters applied in the ANN + GSA model. In addition, Table 3 also contains very important information related to the development of the input vector of our application. Thus, Xmin and Xmax are necessary data for the normalization of the input vector (see description of eq 9); mwg is the molecular weight associated with each structural group of the group contribution method, and it can be used in order to calculate the total molecular weight MW of a specific

ionic liquids as the sum of the atomic weights of all the atoms contained in the molecule.53 Also, Table 3 contains the overall occurrence of each structural group in the training and prediction sets. As is evidenced, the data sets were selected randomly, but considering that, in the group contribution method, the molecules were decomposed into fragments and that all fragments were represented with a suitable frequency in the training set (that includes learning and validation sets).26,31 Table 4 shows an example of the making of the input vector using the developed group contribution method. Once the input vector is generated from the developed group contribution method Xd = [T, MW, g+c,1, ..., g+c,17, g−a,1, ..., g−a,29], this is used in order to estimate the temperature-dependent surface tension of ionic liquids via the ANN + GSA model, in an integrated ANN + GSA based group contribution method denominated GCM + ANN + GSA. Training Phase. Note that the inner parameters of the hybrid algorithm must be tuned according to the problem to be solved. In order to estimate the temperature-dependent surface tension of ionic liquids using the GCM + ANN + GSA, we tuned the ANN + GSA parameters via a trial-and-error procedure. Table 5 shows the optimal ANN + GSA parameters used for estimating σ(T). In order to complete the design of the optimum topology of the GCM + ANN + GSA for estimating σ(T) of ionic liquids, a different number of neurons in the hidden layer were tested to Table 5. Specifications for Estimating σ(T) via GCM + ANN + GSA section

parameter

value

ANN

NN type no. of hidden layers max no. of hidden neurons normalization range δx wt range δw bias range δb activation function (hidden layer) f j activation function (output layer) f k max no. of iterations tmax objective function min error criterion no. of initial agents L gravitational constant G0 descending coeff α max no. of iterations tmax fitness function min error criterion

feedforward 1 30 [−1, 1] [−10, 10] [−5, 5] hyperbolic tangent linear function 1500 RMSE 0.0001 50 100 20 1500 RMSE ≤ ξ(Zn) 0.0001

GSA

F


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architecture selected of 48-16-1. Note that the weight values represent the parameters that provide the solution to the given problem.36 The accuracy of the best network selected was checked by employing a complementary analysis by using the absolute percentage error (individual APE and mean MAPE) and the correlation coefficient R between the calculated values (calc) and the real data (real). These statistical parameters were determined as follows:

select the most accurate network topology. Note that the number of neurons in each layer forms the so-called network topology or architecture.23 Thus, to guarantee that the information contained in the input data is correctly represented by an optimum learning process, we must provide a sufficient number of neurons in the hidden layer to ensure a correct training.36 Determining the number of hidden processing neurons is not a trivial problem, and it has a huge influence on the output value, so this task must be carefully considered. If small number of hidden neurons are used, there will be an underfitting, which occurs when there are an insufficient number of neurons to process the information on the data set. But if a large number of neurons are used in the hidden layer, there will be an overfitting, which occurs when the network has so much information processing capacity that the data contained are not able to train all of the hidden processing neurons.23 There are not any specific methods for determining the optimum number of neurons in the hidden layer H; because of this fact many alternate combinations may be possible.26,31 We opted to determine this number of neurons by adding neurons in a systematic form from 1 to 30 units and by evaluating the RMSE (eq 10) for each neural network architecture during the training phase.36 Figure 1 shows the RMSE as a function of the number of hidden processing neurons H in the hidden layer. As observed,

APE = 100·

MAPE =

Ycalc − Yreal Yreal

100 D

D

∑ d=1

Ycalc − Yreal Yreal

(11)

d

(12)

D

R=

∑d = 1 (Ycalc − Ycalc ̅ )(Yreal − Yreal ̅ )d D

D

2 2 ∑d = 1 (Ycalc − Ycalc ̅ )d ∑d = 1 (Yreal − Yreal ̅ )d

(13)

where Ycalc is the network output value, Yreal is the observed output value, Y̅real is the mean of the observed output values, D is the total data points, and d denotes a particular data.

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RESULTS AND DISCUSSION Prediction Phase. At the end of the training phase, employing the best found architecture for the network and with the determined optimal weights, the results obtained by the proposed method to estimate the surface tension of ionic liquids can be analyzed. The results obtained by the GCM + ANN + GSA method from the selected architecture 48-16-1 by during the training phase are present in Figure 2. Figure 2a shows the correlation between calculated and experimental values for σ(T) of ILs during the training phase. In this panel, the solid line represents the experimental value of σ(T), while the blue and green dots denote the calculated values for the learning and validation sets, respectively. As shown, both sets present a lower discrepancy between the calculated and expected values with correlation coefficients of Rlea = Rval = 0.998 for the learning and validation sets. Figure 2b shows a representation of the previous results as a function of the absolute temperature for all ionic liquids included in the training set, with an estimation accuracy of mean absolute percentage errors of MAPElea = 1.00% and MAPEval = 1.34% for the learning and validation sets, respectively. All results present in Figure 2 show that the ANN-based group contribution method can be correctly trained by using the gravitational search algorithm to estimate the temperature-dependent surface tension of ionic liquids with an acceptable accuracy. Once the accuracy of the training phase was validated, then the prediction capabilities of the proposed method were checked. Here a testing set with data not employed during the training phase was used. Figure 3 shows the results obtained for the testing set. Figure 3a shows the correlation obtained between the calculated and experimental values of σ(T) of ILs during the prediction process, with an Rpre = 0.991. Figure 3b shows the calculated values of σ(T) for all ionic liquids included in the prediction set, with MAPEpre = 1.29%. In general, Figures 2 and 3 show that our GCM + ANN + GSA method is a powerful technique for estimating σ(T) of several IL types.

Figure 1. Convergence graphics during the training phase. RMSE as a function of the number of hidden neurons H in the hidden layer. Blue denotes the results obtained for the learning set, and green denotes the results obtained for the validation set.

the optimum number of neurons in the hidden layer which gives less RMSE with no underfitting or overfitting is emplaced between 14 and 17 units. Thus, the network giving the lowest deviation during the training phase was the one with 48 neurons in the input layer, 16 neurons in the hidden layer, and one neuron in the output layer (48-16-1), where the RMSE of this architecture was 0.00053 for the learning set and 0.00063 for the validation set. Once the best architecture was determined, then the optimum weights required to carry out the estimation of the temperature-dependent surface tension of ionic liquids were obtained by using steps 1 to 3 described in a previous section. Table 6 shows the optimum weights for the correct estimation of σ(T) via GCM + ANN + GSA by employing the best G


2

−0.6072 −0.6355 −0.0278 0.5266 −0.1839 −0.4609 −0.3921 −0.0831 0.2030 0.1261 0.1191 −0.9700 −0.3975 −0.2656 0.0771 0.2716 −0.1671 0.3871 0.0597 0.4254 0.1165 0.2288 −0.0788 0.2983 0.1064 −0.1107 0.0469 0.1792 0.0761 −0.6114 −0.0729 −0.3688 0.6353 0.0560 −0.2127 0.4778 0.3411 0.1961 0.2283 0.3515 −0.3855 −0.3199 0.4600

1

−0.1619 0.8011 −0.3181 −0.2374 −0.0338 −0.3871 0.0833 0.2524 0.0096 0.2247 0.2578 1.2520 −0.1229 0.2336 0.2080 −0.1686 0.1562 −0.2570 −0.0658 −0.0893 0.1688 0.2363 −0.1455 0.1509 −0.2564 −0.3737 0.2418 −0.2612 −0.1294 −0.0612 −0.1027 −0.1812 −0.4106 0.1315 0.0864 0.0245 −0.2824 0.0677 0.0674 −0.3415 −0.3121 0.1730 0.1153

w′ij

p1 p2 g+c,1 g+c,2 g+c,3 g+c,4 g+c,5 g+c,6 g+c,7 g+c,8 g+c,9 g+c,10 g+c,11 g+c,12 g+c,13 g+c,14 g+c,15 g+c,16 g+c,17 g−a,1 g−a,2 g−a,3 g−a,4 g−a,5 g−a,6 g−a,7 g−a,8 g−a,9 g−a,10 g−a,11 g−a,12 g−a,13 g−a,14 g−a,15 g−a,16 g−a,17 g−a,18 g−a,19 g−a,20 g−a,21 g−a,22 g−a,23 g−a,24

−0.0224 0.2206 −0.0509 −0.1250 0.2363 0.2117 0.3155 −0.0439 −0.1825 −0.0676 0.3480 −0.6392 −0.2256 0.0806 −0.0347 −0.0967 −0.1162 0.3059 0.0579 −0.3777 −0.0709 0.2596 −0.4427 −0.1998 0.0014 −0.1451 0.1083 −0.3570 −0.0617 0.2351 −0.1727 −0.0107 0.3312 0.3217 −0.1440 0.0598 0.0385 0.1614 0.1187 0.1394 −0.0678 0.2476 −0.4776

3 0.2024 −1.1256 0.4405 −0.7421 −0.0137 0.5869 0.1624 0.1715 0.2270 0.3710 −0.0815 −2.2657 0.3153 −0.1005 −0.2282 0.0125 0.3164 0.0445 0.2357 0.4259 −0.7258 0.0210 −0.4037 −0.0761 −0.0937 0.5286 0.1725 −0.1977 0.2162 0.3304 −0.4919 0.0310 0.3762 0.1556 −0.1438 −0.1021 0.1814 0.4394 0.2272 0.4023 0.0864 −0.1821 0.3354

4 1.0122 −0.1197 −0.1644 −0.1951 0.1736 0.0455 −0.1732 0.2519 −0.0048 0.2508 0.2205 −0.6113 −0.3397 0.3154 0.4424 0.3172 −0.4278 0.1566 −0.2928 −0.1574 0.3107 −0.3389 −0.2908 −0.0910 0.4351 0.0099 −0.1633 −0.0298 0.3753 −0.4595 −0.1538 0.2927 −0.1132 0.8023 0.4108 −0.0188 0.3822 −0.2532 −0.1814 0.8619 0.1920 −0.2744 −0.1926

5 −0.4486 0.5549 0.0416 0.2293 0.2177 −0.3520 −0.0739 0.0240 −0.2609 0.3938 0.3077 0.7326 −0.2818 −0.0776 0.0487 0.3700 0.0917 0.1471 −0.3637 −0.3746 0.1267 −0.0984 −0.2842 −0.3100 0.1202 −0.0569 0.4175 0.0715 −0.3882 0.5497 −0.0879 −0.4807 0.5522 −0.4391 −0.4498 −0.1055 0.1176 −0.2267 −0.1244 0.2751 −0.0525 −0.2717 −0.1734

6 −0.2055 −0.9421 0.0721 −0.6121 1.0560 −0.0047 0.5499 0.4648 0.4103 0.3202 0.1345 −2.0663 0.3142 −0.2766 −0.0693 0.4713 −0.2937 −0.0168 0.2380 −0.6691 −0.8859 0.2905 −0.0522 −0.3188 0.3049 −0.0738 0.2105 0.2292 −0.0121 1.3429 −0.1018 −0.1343 −0.5215 −0.2907 −0.1004 0.2553 −0.3364 0.1204 0.0111 0.3533 0.8258 0.4468 0.1269

7 0.2039 −0.8001 0.1408 0.3474 0.1450 −0.2118 0.0248 0.2354 0.0947 0.6428 −0.2896 −0.5588 −0.1161 −0.2663 0.2478 0.0702 0.2811 −0.1519 −0.0091 0.0371 0.8419 −0.0503 0.0107 −0.0513 0.1066 0.2751 0.0171 0.0640 0.2199 0.2803 −0.1229 −0.2573 −0.0418 0.2563 0.1483 0.1275 0.3601 0.1584 −0.2381 −0.1594 −0.5598 0.2903 0.6133

8 −0.1882 −1.8550 1.0123 −0.0954 0.3941 −0.2078 −0.1126 0.5106 0.3490 −0.3364 0.3774 −3.7043 0.2887 0.4603 0.0564 0.3558 0.7853 0.5579 0.3612 −0.0973 0.5803 −0.0737 −0.2087 0.0343 0.2318 0.7300 −0.2972 0.2464 0.1701 0.1259 0.3524 0.1405 0.3474 0.9089 0.1832 0.1800 −0.0320 0.3357 0.5189 −0.1583 −0.5513 −0.0643 −0.4426

9 0.4981 0.1799 −0.2442 0.1277 −0.2521 −0.2506 −0.1521 −0.1947 −0.2759 0.3334 0.0551 0.5108 −0.3111 0.1809 −0.2185 0.0002 0.3251 0.1145 −0.1308 −0.2324 0.3042 −0.1419 0.0735 −0.1898 0.1582 −0.2438 −0.0651 0.1099 0.2333 −0.1505 0.2772 0.0590 −0.2339 −0.0358 0.2414 −0.2909 0.1257 −0.0781 0.1731 −0.0186 0.2721 −0.4219 0.0426

10

Table 6. Optimal Weights and Biases Obtained by GCM + ANN + GSA for the Network Model 48-16-1 0.0701 −0.1751 −0.3772 0.1145 0.3877 −0.1757 −0.0934 −0.3060 0.3341 −0.3265 −0.0623 0.3255 0.2793 0.3197 −0.0552 0.0381 −0.2548 0.2762 0.3603 0.3285 0.1364 −0.1289 −0.1963 −0.2462 0.1368 −0.1941 −0.2970 0.2074 0.2316 −0.3426 0.0958 0.5197 0.3310 −0.1518 −0.1081 0.0096 −0.1611 −0.1075 −0.2722 0.0194 0.0308 −0.2122 0.2316

11 −0.0982 −0.1096 0.0723 0.3040 0.2565 0.2520 −0.0807 0.2907 0.1174 −0.1316 0.1530 0.2849 0.2881 0.0861 0.1303 −0.0697 0.2976 0.3460 −0.0533 0.0463 −0.0529 −0.0700 −0.0853 0.0926 −0.3445 0.2778 −0.0606 0.1522 −0.2323 0.3151 0.1590 −0.3285 −0.0401 −0.1752 0.2693 0.1864 0.2484 −0.2765 0.2147 0.2408 0.1137 −0.2854 0.3137

12 −0.3103 0.1682 0.0944 0.1860 −0.3708 −0.6427 −0.1388 −0.1323 −0.2129 −0.0688 0.2679 0.7417 −0.1942 −0.1959 0.2119 −0.3877 −0.0818 0.2773 −0.2954 −0.0116 −0.4877 0.1781 −0.4207 0.1263 −0.0806 0.2904 0.2982 0.2515 −0.3750 0.3683 0.3776 −0.5389 0.6066 −0.5844 −0.4634 0.0747 0.0632 −0.0163 0.1272 −0.2406 −0.2155 −0.7614 0.7321

13 0.2992 0.2673 −0.1632 0.0231 0.1363 0.3125 0.0432 −0.2253 −0.3107 −0.3478 −0.2030 0.1077 −0.0730 0.4204 0.1152 0.3039 0.1063 0.0280 0.0425 −0.2099 −0.0264 −0.0788 −0.0062 −0.2668 −0.1823 −0.1815 0.1861 −0.2757 −0.0674 −0.3202 −0.3827 −0.2579 −0.3473 −0.3211 0.2890 0.2014 0.1976 −0.0124 −0.0311 0.0361 −0.3792 0.0931 −0.1686

14 −0.3732 0.1968 0.0786 −0.2133 0.0788 0.3133 −0.0092 −0.2672 −0.2567 −0.9000 0.1103 0.1223 0.1530 −0.2758 −0.1415 −0.2898 −0.0409 0.1543 0.0422 −0.0504 −0.0635 0.1123 −0.0760 0.1960 −0.4564 0.3303 −0.1479 −0.1603 0.0130 0.0387 −0.1632 −0.2217 0.4656 0.1114 −0.1202 0.4362 0.3233 −0.0025 0.3103 −0.1182 −0.0499 −0.0529 0.0766

15 0.1099 0.0706 0.2200 −0.1715 0.2936 0.0868 −0.2905 0.0946 0.1583 −0.7335 0.0703 −0.4609 −0.2797 0.1207 0.1011 −0.0226 −0.1309 0.2582 −0.1859 0.3503 0.1176 −0.0521 0.0833 −0.1143 −0.3727 0.2464 0.0081 0.0221 −0.2776 0.1873 0.3646 −0.6967 −0.7017 −0.2145 −0.0017 0.0574 −0.4393 0.0460 −0.2305 0.2710 0.5860 0.1354 0.4380

16

0.41 0.62 0.28 0.35 0.34 0.40 0.22 0.30 0.29 0.50 0.26 1.12 0.34 0.32 0.21 0.26 0.31 0.30 0.22 0.33 0.38 0.20 0.24 0.25 0.30 0.34 0.23 0.25 0.27 0.46 0.30 0.40 0.52 0.40 0.29 0.23 0.32 0.20 0.26 0.32 0.38 0.36 0.42

RC

Industrial & Engineering Chemistry Research Correlation

H


Correlation

−0.0633 0.1676 0.2640 0.1110 0.3516 1.0125 −1.3504

−0.1709 0.1981 0.4066 −0.2031 0.0279 −0.9151 1.5347

0.24 0.26 0.25 0.30 0.21

Table S1 contain all results obtained during the training and prediction phases. Table 7 summarizes the statistical parameters (eqs 11 to 13) obtained by the GCM + ANN + GSA method for both phases. It is necessary to emphasize the low deviations obtained by the proposed method in the estimation of σ(T) of ILs during the training and prediction phases. Note that, for the training, GCM + ANN + GSA presents a MAPE = 1.17%, an APEmax = 12.11% with APEmax > 5% for only 31 data points (∼1.8% of the overall training set) and APEmax > 10% for only 1 data point, while for the prediction set, our proposed method shows a MAPE = 1.29%, an APEmax = 11.40% with APEmax > 5% for only 39 data points (∼6.7% of the overall prediction set), and APEmax > 10% for only 8 data points (∼1.4% of the overall prediction set). All results have shown the high accuracy of GCM + ANN + GSA to estimate the temperature-dependent surface tension for several ionic liquid types. Relative Contribution of Inputs. In order to quantify the network interpretation of the input parameters based on group contribution method, we used the determined optimal weights during the training phase to calculate the relative contribution RC of each input via Garson’s algorithm,54 as follows:

−0.3266 0.0730 −0.1348 0.3180 0.0510 −1.1217 −0.8334 −0.0908 −0.1723 0.3185 0.2690 0.3237 −0.5255 −0.6929 0.3017 −0.1232 0.2676 0.2501 0.1180 −0.5379 −0.5267

0.2404 −0.1334 −0.1214 −0.0965 −0.1546 0.1867 1.0892

13 12 11

14

15

16

RC


−0.2598 −0.0119 0.0916 0.2258 0.1615 0.4511 0.3445 −0.0423 0.5008 −0.2498 0.3266 0.0128 2.4594 −0.3227 −0.2022 −0.3052 −0.1271 0.0702 −0.0214 −1.2762 −0.1397 −0.2276 −0.2258 −0.2259 0.0998 −0.2499 0.5078 0.1586 −0.1330 −0.1592 −0.0354 0.0929 −0.2274 0.6539 −0.4955 −0.2128 0.1102 −0.0936 0.1523 −0.0028 −1.2945 −0.7514 −0.0823 0.3871 −0.1751 0.1871 0.2901 1.0283 −1.0110 0.2061 −0.2258 0.0474 0.4253 0.1908 −0.6567 1.0579 −0.0394 0.2907 −0.0525 0.2740 −0.0556 0.6109 1.2908 bj wik′ bk

10 9 8 7 6 5 4 3 2 1

⎛

∑ ⎜⎜

|wijwik|

(14)

where RC is the relative contribution of each input X; |wijwik| is the absolute value of the product of the connection weights from input neurons to hidden neurons (wij) and the connections from hidden neurons to output neurons (wik); and ∑i I= 1|wijwik| is the sum of the product |wijwik| . Table 6 (column 18) shows the RC obtained for each input used in the designed group contribution method. From these values, the higher relative contributions were contributed by g+c,10 (−CH2−; RC = 1.12), p2 (MW; RC = 0.62), g−a,14 (>N− [−N−][>N< ]; RC = 0.52), g+c,8 (−H; RC = 0.50), g−a,11 (−CF3; RC = 0.46), g−a,24 (−P[>P< ]; RC = 0.42), and p1 (T; RC = 0.41). Note that the special importance assigned by the network to these input paremeters has several theoretical bases, such as, σ decreases with the absolute temperature, since the cohesion forces decrease with increasing the thermal agitation, resulting in a lower effective intensity of intermolecular forces;9 σ decreases with increasing the alkyl chain length,55 since the number of occurrences of the group g+c,10 (−CH2−) into the IL molecule increases; also, the cation groups g+c,8 (−H) and g+c,10 (−CH2−) are strongly associated with the asymmetric cation effect, since the σ for the asymmetric cation is always higher than that for the other ILs due to the stronger effect of their polar interaction;55 in addition, the anion groups g−a,11 (−CF3), g−a,14 (>N−[−N−][>N< ]) and g−a,24 (−P[>P< ]) are associated with the anion size (e.g., bis[(trifluoromethyl)sulfonyl]imide and hexafluorophosphate anions), since several authors have suggested that σ decreases with increasing the anion size;9,56,57 also the size of the IL molecule (associated with MW) increases while σ decreases due to the ion charge dispersions and the reduction on the hydrogen bond strength between the cation and anion.57 On the other hand, the relative contributions for the cation head groups have the following order: g+c,4 (pyridinium; RC = 0.40) > g+c,2 (ammonium; RC = 0.35) > g+c,3 (phosphonium; RC = 0.34) > g+c,6 (piperidinium; RC = 0.30) > g+c,7 (sulfonium; RC = 0.29) > g+c,1 (imidazolium; RC = 0.28) > g+c,5 (pyrrolidinium; RC = 0.22). Note that the RC values granted by the network

0.0913 −0.1152 0.2246 −0.2847 −0.0100 1.3336 1.5500 0.5708 g−a,25 g−a,26 g−a,27 g−a,28 g−a,29

w′ij

Table 6. continued

⎞ ⎟ I ⎟ ∑ | | w w j=1 ⎝ i = 1 ij ik ⎠ H

RC =

I


Correlation


Figure 2. (a) Correlation between experimental (solid line) and calculated values (color dots) of σ(T) of ILs during the training phase. Here, blue dots denote the learning set, and green dots denote the validation set. (b) Comparison between experimental (black diamonds) and calculated values (color dots) of σ(T) of ILs during the training phase. Also, blue dots denote the calculated values for the learning set, and green dots denote the calculated values for the validation set.

Figure 3. (a) Correlation between experimental (solid line) and calculated values (color dots) of σ(T) of ILs during the prediction phase. Here, red dots denote the calculated values for the testing set. (b) Comparison between experimental (black diamonds) and calculated values (color dots) of σ(T) of ILs during the prediction phase. Also, red dots denote the calculated values for the testing set.

produce large variations on the properties of IL.56,58 Note that the comprehension of the RC values can offer a better understanding of the effects caused by the input parameters that govern our group contribution method on the estimation of σ(T) of ILs. Then we directly analyze each one of them. Temperature Effects. As was mentioned, the surface tension of an ionic liquid decreases with increasing the temperature in a linear manner.9 Satisfactorily our proposed method can reproduce this tendency for all ILs used as shown in Figures 2b and 3b. Note that our method can estimate the surface tension over an extended temperature range from 263

emphasize the differences between the cation nature of the seven head groups, such as the different cyclic aliphatic rings (e.g., piperidinium and pyrrolidinium) or other heterocyclic rings (e.g., pyridinium or imidazolium), and the branching of the alkyl chains from a different ion (e.g., N+, P+, S+). Additionally, the relative contributions of the anion constitutive groups have the following order: rings < oxygens < halogens < non-rings < nitrates < sulfates < borates < phosphates. For this case, the RC values assigned by the network can also distinguish the variations between these species, and quantify the fact that little differences in the anion structures can J


Correlation

Industrial & Engineering Chemistry Research Table 7. Summary of Statistical Parameters Obtained in the Estimation of σ(T) of ILs Using the GCM + ANN + GSA Method training

prediction

statistical param

learning

validation

testing

total

no. of data no. of ILs no. of sets APEmin/% APEmax/% APE > 5% APE > 10% MAPE/% R

1150 64 104 0 7.55 19 0 1.00 0.998

578 44 59 0 12.11 12 1 1.34 0.998

579 54 66 0 11.40 39 8 1.29 0.991

2307 162 229 0 12.11 70 9 1.18 0.996

Figure 4. Reproduction of the effect of MW on σ(T) of 1-alkyl-3methylimidazolium tetrafluoroborate at T = 298 K via the GCM + ANN + GSA method. Dots denote experimental σ values, and red asterisks denote the calculated values. In increasing alkyl order: (○) ethyl at T = 298.10 K,51 (□) butyl at T = 298.00 K,56 (▽) pentyl at T = 298.15 K,65 (△) hexyl at T = 298.07 K,51 and (◇) octyl at T = 298.40 K.66

to 533 K; this fact enables the application of the Eötvös59 or Guggenheim60 equations in order to correlate the surface tension data with temperature and their use for the estimation of the hypothetical critical and normal boiling temperatures of ILs.61 On the other hand, the estimation of surface thermodynamic properties such as surface entropy62 (Sσ = −(dσ/dT)) and surface enthalpy62 (Hσ = σ − T(dσ/dT)) can be derived through a correct estimation of the surface tension variation with temperature, and an important aspect to consider for this fact is that for the ILs considered in this study their slopes (dσ/ dT) vary in a range from −0.003 to −0.363 mJ·m−2·K−1, but our method can cover all these slopes with good accuracy as is evidenced in Figures 2b and 3b. Molecular Weight Effects. From our results, molecular weight has one of the largest RC values associated with the surface tension of ILs, and this fact can evidence the strongest dependence between both properties. For other substances, several authors have observed that the surface tension presents a linear dependence with the molecular weight.63,64 In ILs, the importance and dependence of MW with other properties have been also evidenced.26,31 But, related to σ, it decreases with increasing MW due to a stronger van der Waals force being exhibited by the greater molecules, and this stronger intermolecular force causes a dispersion of the ionic charge and also a diminution of the hydrogen bond strength.57 Note that our proposed method can reproduce this effect on σ of ILs with a good accuracy. Figure 4 shows the molecular weight effect on 1-alkyl-3-methylimidazolium tetrafluoroborate. Here, dots denote the experimental σ values at T = 298 K for the alkyl series: ethyl,51 butyl,56 pentyl,65 hexyl,51 and octyl;66 and red asterisks denote the obtained values by GCM + ANN + GSA. As shown, σ decreases when MW is increasing. On the effect of MW on the estimation accuracy of σ(T) of ILs, our results show a good accuracy during the training and prediction phases. Figure 5 illustrates the accuracy in the estimation of σ(T) according to MW of ILs by using APE (eq 11). Note that, for the extreme values of MW, the corresponding values of σ(T) were calculated with low deviations by the proposed method. For example, propylammonium formate (MW = 105.138) obtains a MAPE = 0.11% with an APEmax = 0.34% for a data set of 9 data points,46 while trihexyl(tetradecyl)phosphonium tris(pentafluoroethyl)trifluorophosphate (MW = 928.883) obtains a MAPE = 1.86% with an APEmax = 2.48% for a data set that contains 9 data points.47 In general, our method can estimate the σ(T) for

Figure 5. Accuracy in the estimation of σ(T) according to MW of ILs vs the absolute percentage error obtained during the training and prediction phases: (blue dots) learning set, (green dots) validation set, and (red dots) testing set.

ionic liquids on a wide molecular weight range with low deviations. Cation Effects. Table 8 summarizes the results obtained by CGM + ANN + GSA for the different cation types used. The results obtained indicate the following MAPE ranges during the training phase: ammonium (2.42%) > pyrrolidinium (1.48%) > imidazolium (1.11%) > phosphonium (0.82%) > pyridinium (0.76%) > sulfonium (0.30%) > piperidinium (0.22%). For the prediction phase the MAPE ranges have the following order: pyrrolidinium (2.88%) > pyridinium (2.49%) > ammonium (1.11%) > phosphonium (0.93%) > imidazolium (0.48%) > sulfonium (0.35%) > piperidinium (0.31%). In particular, ammonium and pyrrolidinium cations exhibit a MAPE > 1.00% in both phases. This statistical analysis shows that the proposed method can estimate σ(T) of ILs with low deviations for several K


Correlation


Table 8. Statistical Parameters Obtained by GCM + ANN + GSA in the Estimation of σ(T) of ILs According to the Cation Types learning set

validation set

testing set

cation type

no. of data

MAPE

APEmax

R

no. of data

MAPE

APEmax

R

no. of data

MAPE

APEmax

R

imidazolium ammonium phosphonium pyridinium pyrrolidinium piperidinium sulfonium

752 104 93 137 35 22 7

0.94 2.15 0.90 0.60 1.10 0.37 0.30

6.14 7.55 3.16 3.01 2.67 0.58 0.43

0.999 0.974 0.996 0.999 0.994 0.999 0.999

383 49 17 93 29 7

1.28 2.70 0.74 0.92 1.85 0.08

12.11 6.02 2.26 6.37 5.13 0.19

0.998 0.958 0.979 0.997 0.997 0.999

338 72 22 118 15 7 7

1.11 2.88 0.48 0.93 2.49 0.35 0.31

11.40 8.62 1.45 5.07 6.76 0.87 0.50

0.990 0.953 0.999 0.997 0.695 0.999 0.999

alkyl chain lengths such as butyl ([−CH2−]3) present higher σ value than the larger alkyl chain lengths such as nonyl ([−CH2−]8). This figure confirms the discussion presented above and shows that our proposed method can reproduce the effect of the n-alkyl chain length on σ(T) of ILs with a good accuracy. Another important effect associated with the cation is the impact of the symmetry/asymmetry itself. Some authors have observed that the σ(T) values of asymmetrical cations are higher than for the symmetrical cations.55 However, we find no clear experimental evidence on this fact in the database collected for our study. Figure 7 presents the temperature-

cation types. Note that the impact of the cation type on the σ(T) of ILs has been evidenced by several authors,9,12,55,57,67 however, our method can characterize the cation type via the RC in order to reproduce the cation effects with good accuracy. An important effect associated with the cation is that σ(T) decreases with increasing the alkyl chain length.9,55 This behavior has been attributed to a weakening of the Coulomb interaction force68 and to a strong interaction of the van der Waals intermolecular forces that depend of the alkyl chain length.67,69 Different studies have suggested that, according to the Langmuir principle,70 the main contribution to the surface tension value of an ionic liquid is only attributed to the parts of the molecule located on their outer surface.67,71 In other words, the effect of the alkyl chain length on the σ(T) of ILs is associated with the number of occurrences of the substructure that contain one carbon atom bonded to two hydrogen atoms (−CH2−). Note that the importance of the cation substructure −CH2− (g+c,10) was correctly evidenced by the GCM + ANN + GSA since it was quantified with the largest RC value of our method (see, Table 6). Figure 6 shows the capability of the GCM + ANN + GSA for the reproduction of the effect of the n-alkyl chain length [−CH2−]n on the surface tension of 1alkyl-3-methylimidazolium hexafluorophosphate at T = 298 K. As shown, at the same temperature (T = 298 K) the smaller

Figure 7. Reproduction of the effect of symmetrical/asymmetrical cation on σ(T) of ILs based on bis[(trifluoromethyl)sulfonyl]imide anion via the GCM + ANN + GSA method. Dots denote experimental σ values, and red asterisks denote the calculated values. Here, (◀) 1,3diethylimidazolium,12 (▶) triethylsulfonium,74 (○) 1-butyl-1-methylpiperidinium,75 (□) 1-butyl-1-methylpyrrolidinium,76 (▽) 1-butyl-3methylimidazolium,77 (△) 1-butyl-3-methylpyridinium,78 and (◇) butyltrimethylammonium.74

dependent surface tension of ionic liquids composed of symmetrical/asymmetrical cations and bis[(trifluoromethyl)sulfonyl]imide anion. Note that these data sets were randomly selected from our database. In contrast, as shown in Figure 7, symmetrical cations such as 1,3-diethylimidazolium12 and triethylsulfonium74 have higher σ(T) values than the asymmetrical cations such as 1-butyl-1-methylpiperidinium,75 1butyl-1-methylpyrrolidinium,76 1-butyl-3-methylimidazolium,77 1-butyl-3-methylpyridinium,78 and butyltrimethylammonium.74 On the other hand, the effect of cation headgroup on the σ(T) of ILs can also be analyzed in Figure 7. For this case, the

Figure 6. Reproduction of the effect of n-alkyl chain length [−CH2−]n on σ(T) of 1-alkyl-3-methylimidazolium hexafluorophosphate at T = 298 K via the GCM + ANN + GSA method. Dots denote experimental σ values, and red asterisks denote the calculated values. In increasing alkyl order: (○) butyl at T = 298.00 K,72 (□) pentyl at T = 298.19 K,73 (▽) hexyl at T = 298.97 K,72 (△) heptyl at T = 298.15 K,73 (◇) octyl at T = 298.13 K,72 and (☆) nonyl at T = 298.16 K.73 L


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Table 9. Statistical Parameters Obtained by GCM + ANN + GSA in the Estimation of σ(T) of ILs According to the Anion Types learning set anion type acetate bis[(trifluoromethyl)sulfonyl]imide bromide chloride dicyanamide diethylphosphate formate hexafluorophosphate iodide lactate perfluorobutanesulfonate propionate tetracyanoborate tetrafluoroborate thiocyanate tricyanomethanide trifluoromethanesulfonate tris(pentafluoroethyl) trifluorophosphate others

validation set

no. of data

MAPE

APEmax

R

21 361 11 8 81 16 9 166 58 15 6 9 12 177 10 12 7 72

0.51 1.57 0.30 0.16 0.83 1.84 0.11 0.98 0.48 1.89 0.18 0.67 0.33 0.51 0.70 0.63 1.12 0.58

0.99 7.55 0.61 0.23 3.16 3.61 0.34 3.40 1.39 3.81 0.34 2.34 0.94 2.68 1.14 1.40 2.00 2.48

0.999 0.989 0.999 0.999 0.999 0.913 0.999 0.998 0.998 0.999 0.999 0.985 0.996 0.999 0.965 0.999 0.990 0.997

99

0.85

3.70

0.999

no. of data

MAPE

APEmax

testing set R

213 11

2.19 0.46

6.42 1.99

0.988 0.996

53 6

0.37 1.26

1.24 2.26

0.999 0.986

86 27 9

0.62 0.15 0.57

1.73 0.49 1.34

0.999 0.996 0.997

no. of data

MAPE

APEmax

R

6 138 4 12 31

0.32 2.20 1.25 0.23 1.08

0.95 8.62 1.45 0.43 5.07

0.994 0.943 0.999 0.999 0.997

6 26

0.30 3.86

0.80 11.40

0.993 0.302

5 6

0.67 0.48

1.23 0.78

0.998 0.984

1.37 2.49 0.94 0.41 0.50

3.51 6.76 0.35 2.01 1.84

0.997 0.695 0.995 0.998 0.984

1.07

8.26

0.997

6 45 20 11

0.43 1.77 1.73 2.89

0.70 12.11 6.37 4.33

0.999 0.965 0.942 0.600

32

0.58

1.80

0.992

49 15 6 123 53

59

0.78

3.54

0.999

99

increases80 or decreases57 according to the anion size, it being difficult to establish a general dependency between them even if they are divided into subgroups of chemical similarity.9,66,67 Some authors explain the different values of σ(T) by considering the anion dependency on the orientation of the ions at the molecular surface, and on the interaction strength67 (where the Coulomb forces decrease with increasing the dispersive forces associated with the anion size).9,67,81 However, our method was able to describe all trends observed experimentally (from the collected data sets) between the surface tension and the anion types with a good accuracy (see Table 9). Figure 8 illustrates the capabilities of GCM + ANN + GSA to reproduce with low deviations some observed effects of the 1-butyl-3-methylimidazolium series with different anion types on the temperature-dependent surface tension. For this data set with 132 experimental data points, MAPE = 0.32% and R = 0.999. Comparative Analysis. Several authors have proposed different methods to estimate the temperature-dependent surface tension of pure ionic liquids (see Table 1). Mainly these method are based on techniques such as parachor models,10−13 corresponding states theory methods,14,15 group contribution methods,16 quantitative structure−property relationship methods,17,18 and neural network modeling.19 Depending on the complexity of these methods, the molecule and properties of an ionic liquid can be described in many different manners.90 Some authors have proposed to calculate σ(T) making use of other thermophysical properties, such as Gardas−Coutinho’s method11,12 that employs the density; Ghatee’s method91 that uses the viscosity; Mousazadeh− Faramarzi’s method14 that uses the melting and boiling temperatures; Wu’s method15 that makes use of the boiling and critical temperatures; Ghasemian−Zobeydi’s method92 that employs the acentric factor and the critical temperature and pressure; Lemraski−Zobeydi’s method13 that also employs the density, but it is calculated from the critical properties; and Atashrouz’s ANN19 that makes use of the density, boiling

evidence observed by several authors shows that there are remarkable differences between different cation heads with the same alkyl homologous series,66,67 due to the fact that the Coulomb interaction forces can be very similar in the homologous series while the van der Waals interaction forces are very different for the cation head groups.67,79 Here, the GCM + ANN + GSA can reproduce the effects of cation headgroup on σ(T) of ILs with a high accuracy. Anion Effects. Table 9 summarizes the results obtained by our proposed method for the most common anion types employed. As shown, the MAPE obtained during the training phase present the following order: formate < chloride < perfluorobutanesulfonate < iodide < bromide < tetracyanoborate < 0.5% < acetate < tris(pentafluoroethyl)trifluorophosphate < dicyanamide < propionate < hexafluorophosphate < others < 1.0% < trifluoromethanesulfonate < tetrafluoroborate < thiocyanate < lactate < 1.5% < diethylphosphate < tricyanomethanide < bis[(trifluoromethyl)sulfonyl]imide < 1.9%. During the prediction phase the MAPE obtained have the following order: chloride < formate < acetate < trifluoromethanesulfonate < propionate < tris(pentafluoroethyl)trifluorophosphate < 0.5% < perfluorobutanesulfonate < tricyanomethanide < 1.0% < others < dicyanamide < bromide < tetrafluoroborate < 1.5% < bis[(trifluoromethyl)sulfonyl]imide < thiocyanate < hexafluorophosphate. These results show that our method can estimate σ(T) of ILs with low deviations for different anion types. For small size anions (e.g., halides: bromide, chloride, and iodide), our method obtains lower deviations than for large size anions (e.g., bis[(trifluoromethyl)sulfonyl]imide). This fact demonstrates that GCM + ANN + GSA was able to quantify with good accuracy the effect of the type, complexity, and size of the anion on σ(T) of ILs through the RC values assigned to each structure of them. Here, the remarkable structural differences between the anions employed in this study, such as shape and size besides the coordination strength of anion−cation,67 cause that σ(T) M


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temperature, critical temperature and pressure, acentric factor, and critical compressibility factor. But the main disadvantage of these methods is precisely the use of other properties for estimating σ(T), since these annexed properties (i) are hypothetical properties (e.g., boiling point, acentric factor, and critical properties of ILs), or (ii) are not available for most ILs (e.g., density, viscosity, melting temperature). Note that, to obtain the hypothetical properties, these methods13−15,19,91,92 make use of the Valderrama’s GCM,93 which can be used with good accuracy in calculations related to phase equilibria of systems containing ionic liquids,94 however misusing these hypothetical properties can generate wrong values in the estimation of the thermophysical properties of ILs.95 Note that, from a theoretical aspect, it is possible to find great errors for the hypothetical properties of the Valderrama’s GCM when obtaining them by an inverse process, i.e., from the surface tension of an ionic liquid to estimate their critical properties.12,56,57 On the other hand, recent reviews have shown the limitations and accuracy to obtain thermophysical properties of ILs.96 It has been studied that melting temperature is one of the most difficult properties to estimate, considering that only for organic compounds this property can be calculated with an accuracy higher than 7.5% for several methods reported in the literature, and with an accuracy higher than 12.0% for three other commercial software available,97 while for ILs, it can be calculated with an accuracy of ∼5.0% with maximum deviations higher than 30% for some methods.96 Other properties such as

Figure 8. Reproduction of the effect of the 1-butyl-3-methylimidazolium [Xanion]− on σ(T) of ILs via the GCM + ANN + GSA method. Dots denote experimental σ values, and red asterisks denote the calculated values. Here, anions used in the series are (●) acetate,82 (■) bis[(trifluoromethyl)sulfonyl]imide,77 (◆) dicyanamide,49 (▲) glycine,83 (▼) glycolate,83 (▶) hexafluorophosphate,72 (◀) hydrogen sulfate,84 (☆) iodide,85 (○) lactate,83 (□) methylsulfate,86 (◇) octyl sulfate,77 (△) tetrafluoroborate,87 (▽) thiocyanate,88 (◁) trifluoromethanesulfonate, 51 and (▷) tris(pentafluoroethyl)trifluorophosphate.89

Table 10. Comparison between Gharagheizi’s GCM16 and the Proposed GCM + ANN + GSA for Estimating σ(T) of ILs MAPE/% ionic liquid

T/K

σ/N·m−1

Gharagheizi’s GCM16

this work

1-butyl-1-methylpyrrolidinium dicyanamide66 1-butyl-1-methylpyrrolidinium thiocyanate66 1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide87 1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide57 1-butyl-3-methylimidazolium octyl sulfate77 1-butyl-3-methylimidazolium tetrafluoroborate87 1-butyl-3-methylimidazolium tetrafluoroborate66 1-butyl-3-methylimidazolium tetrafluoroborate57 1-butyl-3-methylimidazolium trifluoromethanesulfonate51 1-butyl-3-methylimidazolium trifluoromethanesulfonate51 1-butyl-3-methylimidazolium trifluoromethanesulfonate51 1-butyl-3-methylimidazolium trifluoromethanesulfonate57 1-butyl-4-methylpyridinium thiocyanate66 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide87 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide77 1-ethyl-3-methylimidazolium dicyanamide49 1-ethyl-3-methylimidazolium dicyanamide49 1-ethyl-3-methylimidazolium tetrafluoroborate51 1-ethyl-3-methylimidazolium tetrafluoroborate51 1-ethyl-3-methylimidazolium trifluoromethanesulfonate51 1-ethyl-3-methylimidazolium trifluoromethanesulfonate51 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide87 1-hexyl-3-methylimidazolium tetrafluoroborate51 1-hexyl-3-methylimidazolium tetrafluoroborate51 1-methyl-3-octylimidazolium tetrafluoroborate66 1-octyl-3-methylimidazolium hexafluorophosphate72 1-octyl-3-methylimidazolium hexafluorophosphate72 hexyltrimethylammonium bis[(trifluoromethyl)sulfonyl]imide98 N-butyl-3-methylpyridinium dicyanamide66 trihexyl(tetradecyl)phosphonium tris(pentafluoroethyl)trifluorophosphate47

293−353 303−344 284−352 293−343 279−328 284−351 294−352 293−341 292−293 293−356 292−353 293−343 303−342 283−352 279−328 278−356 278−348 288−356 288−356 268−356 268−356 283−351 269−356 269−356 298−362 284−354 284−354 300−333 293−350 299−343

0.051−0.056 0.045−0.050 0.031−0.033 0.031−0.034 0.025−0.028 0.041−0.045 0.041−0.045 0.042−0.045 0.033−0.035 0.033−0.035 0.033−0.035 0.033−0.036 0.038−0.048 0.033−0.037 0.034−0.037 0.056−0.062 0.056−0.062 0.050−0.055 0.050−0.055 0.038−0.041 0.038−0.041 0.029−0.032 0.035−0.040 0.035−0.040 0.029−0.032 0.031−0.035 0.031−0.035 0.034−0.036 0.036−0.044 0.027−0.029

1.12 2.96 2.09 2.66 0.31 1.50 1.65 1.17 1.68 1.80 1.31 1.75 10.12 6.20 7.01 13.03 13.69 5.89 6.11 3.78 3.35 15.74 1.94 1.90 2.00 3.21 4.19 0.11 11.49 0.36

0.46 2.49 1.36 2.00 0.68 0.25 0.38 0.42 0.19 0.24 0.11 0.34 2.73 0.47 0.63 0.62 0.29 1.10 0.86 0.42 0.44 2.60 0.32 0.35 0.60 1.00 1.04 0.41 1.48 1.86

N


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tional search algorithm in order to obtain a good performance of the network. This full methodology was called GCM + ANN + GSA. Based on the results presented in this study, the following conclusions were obtained: (i) The GCM + ANN + GSA method can estimate with good accuracy (MAPE = 1.29% and R = 0.991) the surface tension of any pure ionic liquid that can be decomposed among the 48 proposed structural groups, and for wide ranges of temperature (263−533 K) and surface tension (0.015−0.062 N·m−1). Note that the incorporation of 7 cation head groups (such as imidazolium, ammonium, phosphonium, pyridinium, pyrrolidinium, piperidinium, and sulfonium) allows the method to be available for a considerable number of ionic liquids. (ii) The incorporation of the gravitational search algorithm to train our network was advantageous for reducing the problems of trapping of local minima caused during the convergence process. (iii) The remarkable differences in the physical properties and the molecular structure of the ionic liquids employed in this study added further difficulties to the given problem that the proposed method has been able to solve. In order to understand how the network can interpret all these differences, the relative contribution RC of the inputs parameters was calculated. Note that the special importance assigned by the network to the input parameters was according to the developed theoretical basis for these substances. (iv) Recognized effects of the temperature, molecular weight, and cation and anion types on the surface tension of ionic liquids can be reproduced with a good accuracy by using the proposed method. (v) The results show that the proposed method can estimate the temperature-dependent surface tension of pure ionic liquids with a better accuracy than other methods proposed in the literature.

density and viscosity of ILs present similar problems with the accuracy of their estimation.26,90,96 Thus, the reliability of the annexed properties cannot be established, and the interpretation of deviations of the methods that use them11,12,14,91 and the experimental values of σ(T) of ILs is impossible to be made. Other authors have proposed to estimate σ(T) of ILs making use of structure-based methods, such as Wu’s method15 based on a corresponding-states GCM; Gharagheizi’s method16 based on an additive GCM; Mirkhani’s method17 based on a quantitative structure−property relationship model; Shang’s method18 also based on a quantitative structure−property relationship model; and Lemraski−Zobeydi’s method13 based on a modified GCM. The main disadvantages of these methods are (i) being based on methods that do not respect the ionic nature of these substances13,15 and (ii) employing complex molecular descriptors in their calculations.17,18 All methods analyzed show high deviations from 2.98% to 8.50% for small or large data sets. We attributed these high deviations to the problems mentioned above. However, Mirkhani’s method17 can estimate σ(T) of ILs with a MAPE = 1.05%, but this method employs 7 molecular descriptors (such as connectivity indices, 3D-MoRSE, GETAWAY, constitutional descriptors, Geary 2D autocorrelations, and information indices) by using a quantitative structure−property relationship based on least-squared support vector machine method. Then, Mirkhani’s method17 can be used only in ILs that have all these molecular descriptor values. In order to evaluate the estimation accuracy and capability of the GCM + ANN + GSA, we compare our results with other methods reported in the literature for estimating σ(T) of ILs. Gharagheizi et al.16 have introduced a traditional group contribution method in order to estimate σ(T) of ILs making use of 920 data points of 51 diverse ILs taken from the literature. They calculated the contribution of 19 substructures of ILs for predicting σ(T) with a MAPE = 3.6% between experimental and calculated values, while our proposed method was developed making use of 2307 experimental data points collected for 162 diverse ionic liquids. We employed 46 structural groups in order to estimate σ(T) with a MAPE = 1.2% from experimental data. It should be noted that both techniques acquired their results employing different data sets, and using different correlation and testing sets (or training and prediction sets), and making use of different methodologies; thus these results cannot be compared directly with one another. Here, we chose to make a comparison by using some common data sets for both techniques. Table 10 presents the comparison between the Gharagheizi’s GCM16 and the proposed GCM + ANN + GSA for estimating the temperature-dependent surface tension of pure ionic liquids by using 30 data sets of ILs in common for both methods.47,49,51,57,66,72,77,87,98 From this comparison, the capabilities and accuracy of the proposed GCM + ANN + GSA method are quite evident. Here, Gharagheizi’s GCM16 shows an overall MAPE = 4.3% and R = 0.912, while our proposed GCM shows an overall MAPE = 0.9% with an R = 0.999. But in general, the proposed GCM + ANN + GSA method presented a higher accuracy and more advantages than the other available methods for predicting σ(T) of ILs (see Table 1).

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b01233. Table S1: all data sets and results obtain by the proposed method (GCM + ANN + GSA) during the training and prediction phases (PDF) Table S2: program code for estimating the surface tension of ILs by using the trained method (GCM + ANN + GSA) (TXT) Please note that the code is not made for a specific commercial software, and can be used with any calculation software. Using this code, you may be able to independently reproduce the results present in this study. If readers and researchers use these files for further work, this paper must be cited.

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AUTHOR INFORMATION

Corresponding Author

CONCLUSIONS In this study, the temperature-dependent surface tension of pure ionic liquids was estimated employing a neural networkbased group contribution method optimized with a gravita-

*E-mail: [email protected]. ORCID

Juan A. Lazzús: 0000-0003-1136-3395 O


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Industrial & Engineering Chemistry Research Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Direction of Research and Development of the University of La Serena (DIDULS) through the Research Projects PEQ16141, P116141, and PI15141, and the Department of Physics and Astronomy of the University of La Serena (DFULS) for the special support that made possible the preparation of this paper.

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