Plan
Comptes Rendus
no. 5

Glass transition temperature of ionic liquids using molecular descriptors and artificial neural networks
José O. Valderrama1, 2  ; Richard A. Campusano3, 2 ; Roberto E. Rojas4
1 Faculty of Engineering, Department of Mechanical Engineering, University of La Serena, La Serena, Chile
2 Center for Technological Information (CIT), Monseñor Subercaseaux 667, La Serena, Chile
3 Faculty of Sciences, Department of Physics and Astronomy, University of La Serena, La Serena, Chile
4 Faculty of Sciences, Department of Chemistry, University of La Serena, La Serena, Chile
Comptes Rendus. Chimie, Volume 20 (2017) no. 5, pp. 573-584.

Résumé

Glass transition temperature data of ionic liquids (ILs) are analyzed to study the capabilities of artificial neural networks to correlate and predict this property. Molecular descriptors from computational chemistry are considered as independent variables to define the characteristics of an IL molecule. Several network architectures were considered, combinations of different descriptors were analyzed, and results were compared with other values reported in the literature. The independent variables (those that could have influence on the glass transition temperature) considered for training the artificial neural networks were (1) mass connectivity index λ, (2) cation mass M(+), (3) anion mass M(−), (4) surface area SA, (5) van der Waals volume Vw, (6) connectivity index X0, and (7) number of carbon atoms nC. The mass connectivity index is a parameter previously defined by the authors and is calculated for each IL, whereas the descriptors SA, Vw, X0, and nC were determined using the software Dragon7. As a measure of the accuracy of the method, the average relative deviation and the average relative absolute deviation are evaluated. Results of this work and others indicate that appropriate selection of data, good combination of architecture, and variables can lead to acceptable correlation of data but accurate prediction is not yet possible. The lack of a clear definition of the glass transition temperature and the lack of knowledge on what are the properties that most affect liquid–solid transition are the main causes of the present incapability for accurately predicting the glass transition temperature of the IL studied in this work.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Métadonnées
Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crci.2016.11.009
Mots clés : Ionic liquids, Glass transition temperature, Neural networks, Molecular structure, Molecular descriptors

José O. Valderrama 1, 2 ; Richard A. Campusano 3, 2 ; Roberto E. Rojas 4

1 Faculty of Engineering, Department of Mechanical Engineering, University of La Serena, La Serena, Chile
2 Center for Technological Information (CIT), Monseñor Subercaseaux 667, La Serena, Chile
3 Faculty of Sciences, Department of Physics and Astronomy, University of La Serena, La Serena, Chile
4 Faculty of Sciences, Department of Chemistry, University of La Serena, La Serena, Chile 
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     author = {Jos\'e O. Valderrama and Richard A. Campusano and Roberto E. Rojas},
     title = {Glass transition temperature of ionic liquids using molecular descriptors and artificial neural networks},
     journal = {Comptes Rendus. Chimie},
     pages = {573--584},
     publisher = {Elsevier},
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     year = {2017},
     doi = {10.1016/j.crci.2016.11.009},
     language = {en},
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%A Roberto E. Rojas
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José O. Valderrama; Richard A. Campusano; Roberto E. Rojas. Glass transition temperature of ionic liquids using molecular descriptors and artificial neural networks. Comptes Rendus. Chimie, Volume 20 (2017) no. 5, pp. 573-584. doi : 10.1016/j.crci.2016.11.009. https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2016.11.009/

Version originale du texte intégral

1 Introduction

Glass transition is the change that happens from solid state to amorphous solid and knowing the temperature at which this change occurs is of interest in various applications: (1) diffusion coefficient conductivity are related to Tg [1]; (2) Tg can be used to predict the dependence of viscosity on temperature [2]; (3) Tg also serves as a cohesive energy parameter [3]; (4) Tg is one of the main criteria for the evaluation of the potential options for electrolyte applications [4]; and (5) Tg is important for phenomena in polymeric materials, amorphous pharmaceutical solids, and semiconductors [5].

Data of glass transition temperature for some ionic liquids (ILs) are available in the literature and databases such as ILs' database of the IUPAC [6], Beilstein database [7], Dortmund Data Bank [8], or the compilation by Zhang et al. [9] are available. An estimate from these sources indicates that there must be around 900 values of Tg for around 800 ILs. The total amount of data is greater than the amount of ILs because for some ILs several values, reported by different researchers, are available.

Glass transition does not occur at a specified fixed temperature, commonly denoted as Tg, but in a range of temperature or transition region [10]. However, the assignation of a single value for Tg is the common practice found in the literature for this property. In differential scanning calorimetry experiments (DSC), the usual way to determine this property is that Tg is commonly assigned to the onset point, intersection of the initial straight line and the transition region straight line, or to the midpoint of the transition region (inflection point). In this study, the highest value provided in the literature for Tg was considered as the true value. This is because the main uses of ILs are those in which the phase remains as liquid. Crystal formation or solidification is to be avoided in most applications of ILs. Therefore, by selecting or estimating a higher value for Tg the range of applicability of the IL is reduced but one can assure that crystal formation or solidification is to be avoided in most applications of ILs.

Schmelzer et al. [11] present a detailed discussion about the glass transition phenomenon and the glass transition temperature, but it is mainly dedicated to the vitreous state of glasses and not to ILs. However, the phenomenon is similar and the fundamentals are also analogous. According to the authors, “the notation glass transformation (or glass transition) temperature, proposed by Tammann, is to some extent misleading. Correct with respect to the indicated mechanism of vitrification is the proposal developed by Simon, to denote Tg as the freezing-in temperature of the glass”. In the physics of high polymers, the name temperature of vitrification is preferred and more precisely corresponds to the word Glastemperatur used in the German literature. In the area of IL research the name glass transition temperature has been preferred.

Gómez et al. [12] present a thorough analysis of the thermal behavior of pure ILs and define some of the characteristic temperatures that appear during the transition between liquid and solid (or vice versa) of an IL: melting temperature (Tm), freezing temperature (Tf), cold crystallization temperature (Tcc), solid–solid transition (Tss), and glass transition temperature (Tg). They define the glass transition temperature Tg “as the midpoint of a small heat capacity change upon heating from the amorphous glass state to a liquid state”. However, not all authors use this definition and different values for the same IL are proposed in the literature.

The solid–liquid transition temperatures of ILs are usually less than ambient temperature and can go less than −100 °C, such as in the cases of [C2mim][dca] or [C4mim][C2F5BF3] [13]. As mentioned above, the most common and efficient method for experimentally determining Tg is by DSC. The thermal behavior of ILs can be relatively complex and some peculiar and particular characteristics have been observed when cooling or heating an IL [12–14]: (1) the cooling from the liquid state may cause glass formation at low temperatures; (2) solidification kinetics is commonly slow; (3) on cooling from the liquid, the low-temperature region is not usually bounded by the phase diagram liquid line; (4) formation of metastable glasses may occur; (5) heating from the glassy state yields an exothermic transition associated with sample crystallization, followed by subsequent melting; and (6) multiple solid–solid transitions (crystal–crystal polymorphism or plastic crystal phases) may occur. The authors also provided a couple of important recommendations: (1) thermodynamic data should be collected in heating mode to obtain reproducible results; and (2) to obtain reliable transition data, long equilibration times should be allowed and small samples should be taken, to permit rapid cooling. In addition, Gómez et al. [12] state that it is not always possible to correctly identify the different transitions appearing in a thermogram using DSC and additional techniques should be included (crossed polarizing filters, X-ray diffraction, and infrared spectrometry). The authors also state that it is necessary to subject the IL to different heating and cooling rates to have a better interpretation of the thermograms and to define characteristic temperatures such as melting or glass transition. It is not unusual to have ILs with different structures presenting the same thermal behavior and ILs with similar structures presenting different behavior.

These facts may explain the great differences found in reported experimental data of Tg for the same IL. As shown in Table 1 differences up to 38 °C (20%) are found, such as the case of 1-butyl-3-methylimidazolium trifluoroacetate. Other ILs present lower differences but still of importance for modeling and analysis. For butylammonium formate the difference is 25 °C (16%) and for 1-ethyl-3-methylimidazolium-bis[(trifluoromethyl)sulfonyl]imide the difference is 20 °C (11%).

Table 1

Selected reported values of Tg showing differences between different literature sources.

Table 1
N°ILCationAnionTg (K)ΔT (K)
11-Butyl-3-methylimidazoliumTetrafluoroborate[C4mim][BF4]176.150.00
185.779.62
188.1512.00
188.8512.70
190.1514.00
192.0015.85
192.9516.80
193.5517.40
21-Butyl-3-methylimidazoliumHexafluorophosphate[C4mim][PF6]193.150.00
196.153.00
197.154.00
212.0018.85
31-Butyl-3-methylimidazoliumTrifluoroacetate[C4mim][ta]195.150.00
203.608.45
233.1538.00
41-Butyl-3-methylimidazoliumBis[(trifluoromethyl)sulfonyl]imide[C4mim][bti]169.150.00
184.1515.00
186.1517.00
187.1518.00
187.2518.10
51-Ethyl-3-methylimidazoliumDicyanamide[C2mim][dca]169.150.00
183.1514.00
61-Ethyl-3-methylimidazoliumBis[(trifluoromethyl)sulfonyl]imide[C2mim][bti]175.150.00
181.156.00
186.0010.85
186.2811.13
195.1520.00
7Alanine methyl esterThiocyanate[AlaC1][SCN]215.000.00
241.1526.15
8N-Octyl-isoquinoliniumBis[(perfluoroethane)sulfonyl]imide[C8isoq][BEI]193.750.00
218.1524.40
9ButylammoniumFormate[NHHH4][HCO2]153.050.00
178.1525.10
10EthylammoniumFormate[NHHH2][HCO2]145.650.00
167.1521.50

2 Models for Tg presented in the literature

Despite the differences between Tg data such as those shown in Table 1, some proposals have been presented in the literature for correlating and estimating the glass transition temperature. Mirkhani et al. [16] studiedquantitative structure property relationship (QSPR) models for the glass transition temperature of different types of ILs. They claim that a simple predictive model is obtained. Although the absolute average deviation was low (3.8%), deviations more than 10% were found for 10 of the 139 fluids considered in the study. Better results were obtained when the authors considered a specific type of IL such as ammonium-based ILs. In that case, average absolute deviation was 2% and maximum deviations were less than 10%.

Gharagheizi et al. [17] presented a group contribution method to correlate and predict the glass transition temperature of ILs but only for 1,3-dialkylimidazolium–type ILs. For the 190 ILs considered in the study, the authors found an average absolute deviation of 1.9% with maximum deviations of 8.2%. Mousavisafavi et al. [4] also studied the same type of ILs and the same 109 data points using a linear QSPR method and obtained average absolute deviations on the order of 2.7% and maximum deviations of 8.8%. The same group of researchers [18] proposed a nonlinear approach of the QSPR method for obtaining a model that gives average absolute deviation of 1.4% and maximum deviation of 6.7% for the same data set.

Yan et al. [19] also used the QSPR methodology using topological indexes defined by the authors. The QSPR model was applied to five types of ILs (imidazolium, pyridinium, ammonium, sulfonium, and triazolium). The average absolute deviation was 3.3% and maximum average deviation was 20%. Mokadem et al. [5] proposed an enhanced group-interaction contribution method for the prediction of Tg of ILs. A wide range of ILs (368 data points) were considered in the study obtaining average absolute deviation of 3.1% and maximum deviations of 29%. Eighteen of the 368 data points gave deviations greater than 10%.

2.1 Data available

As mentioned in Section 1, data of Tg for ILs have been presented in the literature and there are some databases, handbooks, and compilations of data for the glass transition temperature of ILs [6–9,15]. New data frequently appear in journals and monographs. However, several of the reported values of Tg for the same ILs may show great differences, as also explained above (see Table 1).

As explained in Section 1, of all data available for Tg in the open literature, the highest value of Tg within a set of data was considered for study and calculations. For instance for 1-ethyl-3-methylimidazolium-bis[(trifluoromethyl)sulfonyl]imide for which five data are available (number 6 in Table 1) the value selected as Tg is 195.15 K. Table 2 shows the available amount of data of Tg for different types of ILs (second column, total data). The third column shows the amount of data actually used in this study. In these final data only one value of Tg for each IL is included. As observed in Table 2, the imidazolium-type ILs form the group with the greatest number of data. Not only that, but within this big group of imidazolium ILs, there are some specific groups with enough data as to perform more exhaustive analyses.

Table 2

Amount of data of Tg available in the open literature.

Table 2
Type of ILTotal dataFinal data
Imidazolium353248
Ammonium193152
Triazolium4835
Pyrrolidinium4734
Piperidinium1411
Pyridinium6055
Isoquinolinium119
Sulfonium99
Guanidinium2517
Morpholinium2423
Oxazolidinium1211
Amino acids2523
Phosphonium3129

A complete study on the application of artificial neural network (ANNs) for training, testing, and predicting Tg for ILs requires that following especial aspects are analyzed: (1) the most appropriate architecture must be defined; (2) the most adequate variables that have effect of Tg must be analyzed; (3) the predictive capabilities of the ANN model must be studied; (4) the effect of cation and anion must be evaluated; (5) different types of ILs must be included in the study; and (6) comparison with literature calculations of Tg must be done.

2.2 Molecular descriptors

Todeschini and Consonni [20] define a molecular descriptor as “the final result of a logic and mathematical procedure which transforms chemical information encoded within a symbolic representation of a molecule into a useful number or the result of some standardized experiment”. Of the several possible molecular descriptors that could have effects on defining a value of the glass transition temperature the following were chosen for the study: (1) mass connectivity index λ and connectivity index of order zero X0 (to account for the type of connections); (2) cation mass M(+) and anion mass M(−) to account for the mass of each part of the molecule; (3) surface area SA and van der Waals volume Vw to account for the surface and the volume of the IL; (4) and number of carbon atoms nC to account for the alternation effect that chain families of IL properties may present. The mass connectivity index λ is a parameter defined by the authors and is calculated for each IL [21]. The descriptors SA, Vw, X0, and nC were determined using the software Dragon7 [22].

3 Artificial neural networks

An ANN is a mathematical tool that relates the values of a certain dependent function F (for instance the glass transition temperature) to values of defined independent variables x1, x2, x3… xn (for instance structural parameters of the substance). To find a relation between the function F and the variables xi, the ANN must be trained; this means that the network is provided with values of F for several values of xi to find a relation, a pattern, a dependency of the function F on the variables xi. The form in which the ANN finds the relation is inspired in the behavior of biological neurons [23].

Imaginary units that simulate neurons are organized in layers with a defined number of neurons per layer, forming what is called the “architecture” of the network. An input layer receives the information F versus x1, x2, x3… xn and makes an initial processing of such data. In fact the network assigns to each variable xi a weight and a base value (bias) specific for each neuron. With these weight and bias values the network calculates a value for the exit function F and compares the result with the known value of F. In calculating F, this function is related to the independent variables xi through specific mathematical functions named activation functions.

If the deviation between the value of the calculated property F and the experimental value is greater than a defined value, the weights and biases are reassigned and the calculations are done again. This method is known as back propagation, and the new values of the weights and biases are determined using an optimization method.

The network stores the values of weights and biases that give the lowest deviation between the calculated values and the experimental data provided as input. These values of weights and biases define the ANN model. Thus the ANN is not an explicit analytical model such as the empirical correlations commonly used in many applications but the model is a structure of weights and biases provided as matrices. Taskinen and Yliruusi [24] presented a complete review on applications of ANNs for the estimation of fluid properties. Properties of ILs were not included in that review.

The architecture of an ANN usually used for correlating and predicting properties of ILs considers a back propagation network with three or four layers: an input layer, and output layer, and one or two hidden layers [25–27]. The optimum number of neurons in each layer is usually found by trial and error because it is not possible to know in advance the appropriate number of neurons for a given application [28]. Also the disadvantages of ANN when they are used for fluid property correlation and prediction have been discussed [28–30] being the following the most important ones: (1) they require a large set of data, depending on the complexity of the relations between dependent and independent variables; (2) it is necessary to know which are the most influential variables x1, x2, x3… xn on the property F(x1, x2, x3… xn) and; (3) the network may suffer from overfitting, a situation in which the network memorizes instead of learning, losing all predictive capacity.

3.1 Application of ANN to Tg calculation

To develop an accurate model to correlate and predict the glass transition temperature in the form presented in this work, the following two files were written:

  • (1) An Excel file (Tg_data.xlsx) containing all data available for training and testing the network. The file contains several sheets including the data of glass transition temperature and the chosen independent variables, both for training and testing. For the explanation that follows, the independent variables are the following molecular descriptors: λ, M(+), M(−), SA, Vw, X0, and nC.
  • (2) A Matlab code for the ANN model (Tg_ann.m) consisting of two parts, a training section and a testing section, as presented in Table 3. In the training section, the program reads the input data (from Tg_data.xlsx), defines the architecture, trains the defined network, generates the weight and bias matrixes, and stores such data for testing. Important to notice are lines 10 and 12 in Table 3 in which the dependent and independent data are read from the Excel file Tg_data.xlsx (sheets Read_Tg_Training and Read_Variables_Training, respectively). Line 35 in Table 3 defines the storing of the correlated data and line 37 describes the storing of the weight and bias matrix (file w_Tg.mat).

Table 3

Matlab code Tg_ann.m used for training and testing an ANN with glass transition temperature data.

Table 3
1% Tg_ann.m
2%************
3%
4% This is the Matlab code for training an ANN with glass transition temperature data, using as independent
5% Variables (λ, M+, M−, Vw)
6%
7% Training section
8%*******************
9% Reading independent variables for training (λ, M+, M−, Vw) from Excel file Tg_data.xlsx
10p = xlsread('Tg_data.xlsx','Read_Variables_Training');p = p';
11%Reading the dependent variable for training (Tg listed in the file variables_training'
12t = xlsread('Tg_data.xlsx','Read_Tg_Training');t = t';
13%
14% Normalization of all data (values between −1y + 1)
15[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t);
16% Definition of ANN:(topology, activation functions, training algorithm)
17net = newff(minmax(pn), [4,4,4,1],{'tansig','tansig','tansig','purelin'},'trainlm');
18% Definition of frequency of visualization of errors during training
19net.trainParam.show = 10;
20% Definition of number of maximum iterations (epochs) and global error between iterations (goal)
21net.trainParam.epochs = 1500; net.trainParam.goal = 1e-6;
22%Network starts: reference random weights and gains
23w1 = net.IW{1,1}; w2 = net.LW{2,1}; w3 = net.LW{3,2}; w4 = net.LW{4,3};
24b1 = net.b{1}; b2 = net.b{2}; b3 = net.b{3}; b4 = net.b{4};
25%First iteration with reference values and correlation coefficient
26before_training = sim(net,pn);
27corrbefore_training = corrcoef(before_training,tn);
28%Training process and results
29[net,tr]=train(net,pn,tn);
30after_training = sim(net,pn);
31% Back-normalization of results, from values between −1y + 1 to real values
32after_training = postmnmx(after_training,mint,maxt); after_training = after_training';
33Res = sim(net,pn);
34% Saving results, correlated glass transition temperature in an Excel file
35xlswrite('Tg_data.xlsx',after_training,'Results_Correlation', 'D2');
36%Saving the network (weights and other files)
37save w_Tg.mat
38%
39%Testing Section
40%*****************
41% This is the Matlab code for testing the ANN determined above for data not used during training
42%
43%Reading weight and other characteristics of the trained ANN saved in the file W
44load w_Tg.mat
45% Reading new data not used during training from Excel file Tg_data.xlsx
46pnew = xlsread('Tg_data.xlsx','Read_Variables_Testing'); pnew = pnew';
47% Normalization of all variable (values between −1y + 1)
48pnewn = tramnmx(pnew,minp,maxp);
49anewn = sim(net,pnewn);
50% Transformation of the normalized exits (between −1y + 1) determined by the ANN to real values
51anew = postmnmx(anewn,mint,maxt); anew = anew';
52% Saving the predicted glass transition temperature in the Excel file Tg_data (sheet Results_Testing)
53xlswrite('Tg_data.xlsx',anew,'Results_Testing', 'D2');

In the testing section, the program reads the weight and bias matrixes (line 44) and the Excel sheet Read_Variables_Testing within the file Tg_data.xlsx, containing the variables for which the glass transition temperature needs to be tested (line 46). The results are stored in the same Excel file in the sheet Results_Testing (line 53). The optimum network is chosen considering the results in both the training and testing sections. It should also be mentioned that the version of the software used in the calculations is Matlab 2012.

To apply the trained network for estimating the glass transition temperature the program Tg_prediction.m including the Excel file Tg_prediction.xlxs and the weight matrix w_Tg.mat must be used. When the Matlab code is run, the variables for those cases for which the glass transition temperature is to be predicted are read from the sheet Read_Variab_Prediction in the file Tg_prediction.xlxs. The program reads the weight matrix (line 6 in Table 4) and calculates the glass transition temperature for the new variables using the ANN model defined in the matrix w_Tg.mat. The calculated glass transition temperature is stored in the file Tg_prediction.xlxs (sheet Results_Prediction). The three files needed for predicting the glass transition temperature (the Matlab code Tg_prediction.m, the Excel file Tg_prediction.xlxs, and the weight matrix w_Tg.mat) are provided as Supporting information.

Table 4

Matlab code Tg_prediction.m used for predicting the glass transition temperature for other cases using the trained ANN (w_Tg.mat).

Table 4
1% Tg_prediction.m
2%
3%This is the Matlab code for predicting Tg using the trained ANN model (w_Tg.mat)
4%
5%Reading weights and other characteristics of the trained ANN saved in the file w_Tg.mat
6load w_Tg.mat
7% Reading Excel file with new independent variables (λ, M(+), M(−) and Vv) to predict Tg
8pnew = xlsread('Tg_Prediction.xlsx','Read_Variab_Prediction'); pnew = pnew';
9% Normalization of all variables (values between −1y + 1)
10pnewn = tramnmx(pnew,minp,maxp);
11% Obtaining the properties for the variables provided by the Excel file Tg_Prediction.xlsx
12anewn = sim(net,pnewn);
13% Transformation of the normalized exits (between −1y + 1) determined by the ANN to real values
14anew = postmnmx(anewn,mint,maxt); anew = anew';
15% Saving the predicted Tg in the Excel file Tg_prediction.xlsx (sheet Results_Prediction')
16xlswrite('Tg_Prediction.xlsx',anew,'Results_Prediction', 'd3');

4 Results and discussion

As mentioned in the preceding sections, the selection of the best model considered three statistical parameters: the average relative deviation %ΔTg, the average relative absolute deviation %|ΔTg|, and the maximum relative absolute deviation |%ΔTg|max between calculated and literature data. These parameters are the most significant numbers for evaluating the accuracy and goodness of a model as discussed [31]. These deviations are defined as follows:

%ΔTg=100N∑1N[Tgcal−Tglit.Tglit.]i(1)
|%ΔTg|=100N∑1N|Tgcal−Tglit.Tglit.|i(2)
|%ΔTg|max= the maximum value of 100·|Tgcal−Tglit.Tglit.|i(3)

Three study cases were designed to analyze different aspects of the model development: case 1 in which different architectures are considered to choose the most appropriate one; case 2 in which different descriptors are considered to search for the most appropriate variables; and case 3 in which the predictive capabilities of the ANN model are analyzed by selecting three different data sets: one for training, one for testing, and one for prediction.

Following information from the study by Valderrama et al. [32] around 10% of the data was separated for testing. Thus, of the 248 data, a group of 223 data was selected for training and the remaining 25 were left for testing. In the three study cases, the selection of data for testing was randomly done. Random numbers were generated in Excel and the 25 ILs showing the lowest random numbers were chosen for testing.

4.1 Case 1: searching for the appropriate architecture

The whole set of data separated in a training set and in a testing set as explained above was considered for this case (223 for training and 25 for testing). Also, all descriptors chosen as explained in the preceding sections were included in the study (λ, M(+), M(−), SA, Vw, X0, and nC). In addition, for this study several three- and four-layer architectures were considered and analyzed. To simplify the analysis, the three-layer cases considered followed the structure (n,2n,1) and the four-layer architectures followed the structure (n,n,n,1), n being an integer number. The results for selected cases are presented in Table 5. It should be noticed that these two architectures have the same number of total neurons for a given value of n.

Table 5

Results for three- and four-layer networks using all descriptors (λ, M(+), M(−), SA, Vw, X0, and nC) for 248 data, 223 for training and 25 for testing.

Table 5
ArchitectureTrainingTesting
%ΔTg%|ΔTg|%|ΔTg|max%ΔTg%|ΔTg|%|ΔTg|max
Case 1 (a): all descriptors (three layers)
1.2.11.39.133.62.410.924.4
2.4.10.66.124.7−0.58.118.8
3.6.10.44.621.81.07.318.9
4.8.10.23.321.9−1.57.018.7
5.10.10.12.216.3−2.59.026.1
6.12.10.11.915.4−0.57.119.9
7.14.10.00.513.31.317.249.1
Case 1 (b): all descriptors (four layers)
1.1.1.11.39.033.72.410.824.1
2.2.2.10.86.636.40.68.619.8
3.3.3.10.44.921.1−0.99.320.7
4.4.4.10.33.826.2−0.56.516.9
5.5.5.10.22.812.20.68.523.0
6.6.6.10.11.611.91.311.731.8
7.7.7.10.11.413.4−1.510.136.9

A graphical representation of the result is important to see how the effects of the number of neurons in the hidden layer affect the accuracy of the model. Several authors have found that there are an optimum number of neurons that give the lowest deviation between calculated and experimental data in a given run [25,30,32]. To choose the optimum number of neurons one must decide which of the statistical parameters must be optimum. Two of these parameters could be the lowest average absolute deviation and the lowest maximum deviation. Any of this can be set during training and testing. Because low deviations during testing represent in some way the interpolating or even extrapolating capabilities of the network, we recommend using the result during testing to decide on the best architecture. Also, choosing the lowest value of the maximum absolute deviation as the target parameter would guarantee estimations below that level of error.

Fig. 1 shows the results expressed as the maximum relative absolute deviation for the three- and four-layer architectures. In the three-layer case, an architecture (4,8,1) (that means n = 4) gives the lowest deviations in the average relative absolute deviation and in the maximum relative absolute deviation. For the four-layer cases, the best architecture is found to be (4,4,4,1) (that means n = 4, and the same neurons as in the three-layer case).

Fig. 1

Maximum relative absolute deviation during testing for three- and four-layer architecture of the type (n,2n,1) and (n,n,n,1).

4.2 Case 2: searching for the appropriate descriptors

Different descriptors are considered to search for the most appropriate combination of variables. Three descriptors are fixed: the mass connectivity index, the cation mass, and the anion mass (λ, M(+), and M(−)). In other applications found in the literature these three parameters (λ, M(+), and M(−)) have been used in neural network applications and in group contribution methods [25–27]. Others descriptors are added in different combinations to analyze the effect of them in training and testing the ANN. Table 6 presents some selected results for a series of combinations of variables using the four-layer architecture (4,4,4,1).

Table 6

Results for architecture (4,4,4,1) using different types and amount of descriptors.

Table 6
CaseCase 2: Architecture: 4,4,4,1, different descriptors
Descriptors usedTrainingTesting
%ΔTg%|ΔTg|%|ΔTg|max%ΔTg%|ΔTg|%|ΔTg|max
1λ, M(+), M(−), Vw0.54.825.1−0.97.921.5
2λ, M(+), M(−), SA0.44.819.1−3.38.921.2
3λ, M(+), M(−), X00.55.328.3−4.88.827.1
4λ, M(+), M(−), SA, Vw0.34.119.62.69.424.6
5λ, M(+), M(−), Vw, X00.44.227.20.210.522.6
6λ, M(+), M(−), SA, X00.55.719.51.29.024.2
7λ, M(+), M(−), SA, Vw, X00.33.718.9−1.49.520.5
8λ, M(+), M(−), SA, Vw, nC0.34.017.8−0.86.521.1
9λ, M(+), M(−), SA, Vw, X0, nC0.33.826.2−0.56.516.9

As observed in Table 6 the inclusion of the carbon number as a parameter (last two lines) does not improve training and testing results. In the data used, just some few compounds have odd number of carbons and the alternation effect occurs for compound with odd and even numbers of carbons in the alkyl chain [33]. The addition of the connectivity index X0 (line 7) improves the results but not in an appreciable form, considering that six descriptors are used. The inclusion of the whole set of parameters (last line in Table 6) indicates that it is not necessary to include all parameters. So to balance simplicity with accuracy four descriptors are enough to get acceptable results (the three first cases in Table 6).

4.3 Case 3: analyzing the predictive capabilities of the ANN model

The set of data originally available for applying neural networks was divided into three groups: one for training, one for testing, and one for prediction. The testing and prediction sets are not directly used during training, whereas the network learns. However, results obtained during testing for each run of the ANN program are used for deciding which model is the best. This means that the chosen model is that giving low deviations during training and testing at the same time. The predicting set, however, was never used for constructing or for selecting the optimum ANN model.

For all of the imidazolium-type ILs analyzed in this work, the amount of data for each set is as follows: the total number of data is 248, the data for training are 223, the data for testing are 25, and the data for prediction are 18. The results for testing presented in Table 5 and for prediction in Table 7 represent in some way the predictive capabilities of the chosen network, because in both cases the values of Tg for the testing set and the prediction test were not used during training, but, as mentioned above, the results of the testing step were used in deciding which was the best network model. In the case shown in Table 6, the 18 values of Tg were not used in any previous step and therefore represent the true capability of the network to predict Tg for an IL of the type studied in this work.

Table 7

Predicted Tg for 18 imidazolium-type ILs using the four-layer model with the following descriptors λ, M(+), M(−), and Vw.

Table 7
CationAnionTglit.Tgcal%ΔTg%|ΔTg|max
[C1im][Br]213.15189.0−11.311.3
[C2im][Cl]216.15198.9−8.08.0
[C2mim][Br]218.00201.2−7.77.7
[C2mim][Cl]234.00198.4−15.215.2
[C3CNmim][Cl]239.05225.7−5.65.6
[C4mim][Br]223.15204.1−8.58.5
[C4mim][Br]190.15220.215.815.8
[moemim][Cl]213.15220.53.53.5
[C4mim][Cl]204.15212.74.24.2
[C3CNmmim][Cl]254.25231.0−9.19.1
[C6mim][Br]224.00217.3−3.03.0
[C6mim][Cl]198.15227.014.614.6
[C3mim][BF4]259.25189.8−26.826.8
[C8mim][Cl]210.85236.812.312.3
[neo-C5mim][BF4]221.00202.3−8.48.4
[CitronellylC1im][Br]216.15231.37.07.0
[C4mim][BF4]193.55183.7−5.15.1
[CitronellylC12im][Br]203.15203.0−0.10.1
Average−2.99.2
Max–26.8

4.4 Case 4: analyzing subfamilies of ILs

The results shown in the preceding sections were found for all imidazolium-type ILs for which data of Tg were available. Within this big family of imidazolium-type ILs, there are subfamilies such as alkyl-methylimidazolium, alkyl-dimethylimidazolium, hydroxyl-alkyl-methylimidazolium, and benzyl-imidazolium, among others. The largest of these families are the n-alkyl-3-methylimidazolium for which 105 data are available. Of these data, 95 were left for training, five for testing, and five for prediction. According to previous studies by the authors who explored the concept of homology for the study and calculation of transition properties of IL families [33] better results should be expected. In fact this was the case for the family of n-alkyl-3-methylimidazolium. Average deviations are less than 3% with maximum deviations less than 10%.

Table 8 presents some results for the 105 n-alkyl-3-methylimidazolium–type ILs. It can be observed that, as expected, simpler architectures are needed to obtain similar or better accuracy than that in the preceding cases. Networks with more layers and neurons are not recommended for the amount of data available for training.

Table 8

Results for several architectures for ILs of the type n-alkyl-3-methylimidazolium using three- and four-layer network with the following descriptors λ, M(+), M(−), and Vw.

Table 8
CaseNetworkTrainingTestingPrediction
%ΔTg%|ΔTg|%|ΔTg|max%ΔTg%|ΔTg|%|ΔTg|max%ΔTg%|ΔTg|%|ΔTg|max
1(2.4.1) (7 Neurons)0.33.919.40.21.01.86.46.419.4
2(2.2.2.1) (7 Neurons)0.34.016.61.84.26.26.16.116.6
3(3.6.1) (10 Neurons)0.12.615.00.21.53.22.53.15.0
4(3.3.3.1) (10 Neurons)0.22.618.41.32.64.94.75.618.4

Similarly, the predictive capabilities of this reduced family of n-alkyl-3-methylimidazolium ILs were also tested. The set of data used for prediction are the nine n-alkyl-3-methylimidazolium ILs shown in Table 7. In this case, the results for testing presented in Table 8 and for prediction in Table 9 also present in some way the predictive capabilities of the chosen network for this particular subfamily of n-alky-3-methylimidazolium ILs.

Table 9

Predicted Tg using a four-layer architecture (4,4,4,1) with the following descriptors λ, M(+), M(−), and Vw.

Table 9
CationAnionTglit.Tgcal%ΔTg%|ΔTg|
[C3mim][BF4]259.3195.4−24.624.6
[C4mim][BF4]193.6176.5−8.88.8
[C2mim][Br]218180.7−17.117.1
[C4mim][Br]223.2191.6−14.114.1
[C6mim][Br]224182.4−18.618.6
[C2mim][Cl]234195.5−16.416.4
[C4mim][Cl]204.2179.9−11.911.9
[C6mim][Cl]198.2195.3−1.41.4
[C8mim][Cl]210.9184.4−12.612.6

4.5 Extension to other types of ILs

The detailed analysis presented above for imidazolium-type ILs should be extended to other types of ILs to be able to draw more complete conclusions about the application of neural networks for correlating (and probably predicting) Tg for any type of IL. Of the other types of ILs (besides imidazoliums), the type of ILs with a reasonable amount of data is the ammonium-type group (152 data points). For this group, the chosen network (4,4,4,1) can be applied and a specific model for ammonium can be found. However, for other groups this type of architecture cannot be applied because the number of data is not enough to evaluate the weights and biases. One should notice that for the network (4,4,4,1) the number of parameters to be determined is 49 (36 weights and 13 biases). We established that the number of data used for training must be at least the double of the number of parameters.

The chosen architecture (4,4,4,1) was applied for ammonium-type ILs and results are as expected. This means that they are similar (and a little higher) than in the case in which all imidazolium-type ILs were studied (see Table 7). In the case of imidazolium, the maximum deviation in prediction was 26.8% (70 °C) for [C3mim][BF4], whereas for ammonium the maximum deviation in prediction was 29%. However, by observing the training and testing results, there are two acceptable models for these ammonium ILs (indicated as model 1 and model 2 in table 10, third column). With the first model (model 1) surprisingly high deviations are found (>1000%) showing again the low predictive capabilities and the erratic behavior of the model. The high number of parameters to be determined (69) compared to the number of data for training (122) could be one of the reasons for this behavior. For other types of ILs results are variable, being difficult to draw a general positive recommendation on the use of neural networks.

Table 10 presents selected results for those cases for which neural networks with simple architecture could be applied. For each type of IL results for two runs are presented. The groups with less than 30 data points (piperidinium, isoquinolinium, sulfonium, guanidinium, morpholinium, oxazolidinium, and amino acids) were not analyzed with the ANN model. This restriction is imposed because the number of parameters (weights and biases) to be calculated could be similar or greater than the number of data, a mathematical inconsistency that provides unmeaningful results. As expected, when few data are used the network can still learn and reasonable training can be obtained. However, the predictive capabilities of the network are uncertain; one could get a reasonable prediction with 12.5% deviation as the case of triazolium in Table 10 or 1137% deviation as the case of ammonium in the same Table 10.

Table 10

Results for Tg during training, testing, and prediction for various types of ILs using three- and four-layer networks with the following descriptors λ, M(+), M(−), and Vw.

Table 10
Type of ILNetworkNo. of parametersProcessNo. of data%ΔTg%|ΔTg|%|ΔTg|max
Ammonium(4,4,4,1)69 (Model 1)Training1221.15.848.9
Testing15−3.215.754.8
Prediction153.18.529.2
69 (Model 2)Training1220.44.325.4
Testing150.821.251.9
Prediction15−74.089.51136.6
Triazolium(2,4,1)19 (Model 1)Training300.01.04.6
Testing3−2.22.22.5
Prediction2−5.37.212.5
19 (Model 2)Training300.00.64.5
Testing31.61.62.6
Prediction2−6.610.517.1
19 (Model 3)Training300.01.28.6
Testing32.74.35.8
Prediction2−5.48.914.3
Pyrrolidinium(2,4,1)19 (Model 1)Training280.01.37.5
Testing31.77.79.0
Prediction3−7.511.728.8
19 (Model 2)Training280.00.52
Testing3−0.15.27.6
Prediction3−24.824.839.6
Pyridinium(2,4,1)19 (Model 1)Training450.12.79.8
Testing5−3.46.119.2
Prediction53.98.821.4
19 (Model 2)Training450.12.17.2
Testing51.88.019.0
Prediction59.120.870.3

4.6 Comparison with literature estimations of Tg

As described in Section 1, some other approaches, including QSPR models and group contribution methods, have been presented in the literature. Mousavisafavi et al. [4,18] correlated and predicted glass transition temperature of 1,3-dialkylimidazolium ILs, which could be considered as potential future electrolytes. For this purpose, QSPR method is used to finally produce satisfactory results quantified by following several statistical parameters. Yan et al. [19] also applied a QSPR method based on the general topological index to predict the glass transition temperatures of ILs. The authors considered five kinds of ILs: imidazolium (Im), pyridinium (Py), ammonium (Am), sulfonium (Su), and triazolium (Tr). Mokadem et al. [5] estimated Tg with a group-interaction contribution method. Several ILs including imidazolium-, pyridinium-, triazolium-, sulfonium-, pyrrolidinium-, piperidinium-, phosphonium-, oxazolidinium-, ammonium-, morpholinium-, guanidinium-, amino acid-, and caprolactam-based ILs were considered in the study. The authors claim that “this method represents an excellent alternative to previous approaches for the estimation of the glass transition temperature of diverse ionic liquids from the knowledge of their molecular structure”. A comparison of results from the literature and the results of this work is presented in Table 11.

Table 11

Comparison of results obtained in this work compared with result of other models proposed in the literature.

Table 11
ModelType of ILN%ΔTg%|ΔTg|maxRef.
QSPR modelsDifferent types of ILs112 (Training)1.218[34]
27 (Testing)413.7
Group contribution methodDifferent types of ILs396 (Training)3.612[17]
100 (Testing)3.712
Group contribution method1,3-Dialkylimidazolium–type ILs1091.98.6[17]
A Linear QSPR method1,3-Dialkylimidazolium–type ILs88 (Training)2.47.1[4]
21 (Testing)3.27.6
Nonlinear approach of the QSPR method1,3-Dialkylimidazolium–type ILs (109)10 (Validation)3.46.7[18]
QSPR methodology using topological indexesFive types of ILs (63 imidazolium, 17 pyridinium, 48 ammonium, 7 sulfonium, and 4 triazolium)1393.34.2[19]
ANN with molecular descriptorsImidazolium-type ILs (223 training and 25 for testing)18 (Prediction)9.226.8This work
Alkyl-3-methylimidazolium (95 for testing and 5 for training5 (Prediction)2.55This work
Ammonium-type ILs15 (Prediction)3.18.5This work
Triazolium3 (Testing)−2.22.5This work
Pyrrolidinium3 (Testing)1.77.7This work

In addition, Table 12 presents a comparison of predicted Tg for selected ILs reported in the literature and the values found in this work. As shown in the table, different variations of QSPR techniques, different forms of group contribution methods, and neural network have been explored in the literature, so comparison can be done. Deviations reported in the literature are similar for the different methods.

Table 12

Comparison of predicted Tg for selected ILs.

Table 12
Ionic LiquidsTg litTg cal%|ΔTg |maxMethodType of ILRef.
[BzimBz][dca]227.7222.42.3GC[1,3-dialkyl][mim][17]
227.7221.8−2.6ANNImidazoliumThis work
[C2OHmim][bti]194.2193.90.1GC[1,3-dialkyl][mim][17]
194.2200.73.4ANNImidazoliumThis work
[C8mim][PF6]191.2194.81.9GC[1,3-dialkyl][mim][17]
202.2196.2−2.9ANNImidazoliumThis work
[BzimBz][ta]236.0240.21.8GC[1,3-dialkyl][mim][17]
236.0235.90.0ANNImidazoliumThis work
191.2197.63.4ANNImidazoliumThis work
205.6200.2−2.6ANNImidazoliumThis work
[C10mim][PF6]202.2196.23.0GC[1,3-dialkyl][mim][17]
202.2193.3−4.4ANNImidazoliumThis work
189.2201.06.3ANNImidazoliumThis work
[PAMPBim][PF6]234.2234.20.0GC[1,3-dialkyl][mim][17]
234.2239.42.2ANNImidazoliumThis work
[C8mim][bti]189.0190.20.7GC[1,3-dialkyl][mim][17]
189.2197.44.4ANNImidazoliumThis work
[BzimBz][bti]222.9228.82.7GC[1,3-dialkyl][mim][17]
222.9218.4−2.0ANNImidazoliumThis work
[C10mim][Sac]203.9203.90.0GC[1,3-dialkyl][mim][17]
203.9218.47.1ANNImidazoliumThis work
[C9mim][bti]190.2190.90.4GC[1,3-dialkyl][mim][17]
190.2195.93.0ANNImidazoliumThis work
[DPEHIM[PF6]204.4200.32.0GC[1,3-dialkyl][mim][17]
204.4204.80.2ANNImidazoliumThis work
238.2243.02.1ANNImidazoliumThis work
[acrylateC6mim][bti]198.2202.92.4GC[1,3-dialkyl][mim][17]
198.2196.0−1.1ANNImidazoliumThis work
[DPEOIM[PF6]200.0201.60.8GC[1,3-dialkyl][mim][17]
200.0198.3−0.8ANNImidazoliumThis work
[C2im][BF4]186.2208.912.2ANNAny[16]
186.2189.21.6ANNImidazoliumThis work
[C2OHmim][PF6]201.2198.51.3GC[1,3-dialkyl][mim][17]
201.2213.86.3ANNImidazoliumThis work
[C2Omim][C2F5BF3]175.2175.20.0GC[1,3-dialkyl][mim][17]
175.2175.30.1ANNImidazoliumThis work
[DPEHIM[BF4]189.9198.94.8GC[1,3-dialkyl][mim][17]
189.9210.310.8ANNImidazoliumThis work
[DPEOIM[BF4]188.4200.36.3GC[1,3-dialkyl][mim][17]
188.4197.54.9ANNImidazoliumThis work
[DPPOIM[BF4]201.5201.50.0GC[1,3-dialkyl][mim][17]
201.5201.70.1ANNImidazoliumThis work
183.2197.98.0ANNImidazoliumThis work
[Em2im][ba]207.8201.33.1ANNAny[16]
207.8213.02.5ANNImidazoliumThis work
[C2im][ClO4]192.2213.911.3ANNAny[16]
192.2205.36.8ANNImidazoliumThis work

Results in Tables 11 and 12 are provided as relative percentage deviation with respect to literature values, the form in which most of authors present their results. For other properties (such as density or heat capacity), deviations of 10% could be acceptable because the effect of different properties has different effect in different applications [35]. However, 10% in a value of Tg could mean differences of 15–30 °C, a deviation that we find nonacceptable. Differences of 30 °C considerably reduce the probable liquidus range of the IL, range in which one wants to safely work during a process, without the threatening of crystal formation. We propose here that an acceptable margin to claim success in modeling the glass transition temperature (or any transition temperature) must be on the order of 10 °C as maximum (3%) and average deviations not greater than 3 °C (1%). Therefore, none of the available publications, or this work, can claim success yet in predicting Tg. However, the efforts of researchers such as those mentioned in this work must continue especially in the area of standardizing the definition of Tg and its experimental determination.

5 Conclusions

ANN models have been used to correlate and predict the glass transition temperature of ILs. The study and the results obtained in this work allow drawing the following main conclusions: (1) simple architectures of three or four layers with maximum of 13 neurons are sufficient to correlate the glass transition temperature of ILs; (2) the descriptors used show to be good representative parameters to distinguish data for the different ILs; (3) better training, testing, and some good predictive capabilities are observed when the family of ILs is reduced so the members of the family have minimal structural differences, such as the case of the n-alkyl-3-methylimidazolium family; (4) for reduced families, and if a reasonable amount of data is available, Tg could be predicted with deviations less than 10%; and (5) the lack of a clear definition of the glass transition temperature and the lack of knowledge on what are the properties that most affect liquid–solid transition are the main causes of the present incapability for accurately predicting the glass transition temperature of the type of ILs studied in this work.

Acknowledgments

The authors thank the National Council for Scientific and Technological Research (CONICYT) for providing support through the research grant FONDECYT 1120162 of the Center for Technological Information of La Serena-Chile for computer facilities and of the Direction of Research and Development of the University of La Serena for permanent support and for an especial graduate fellowship PT15142 for one of the authors (R.A.C.).

Notation

Symbols%

percentage

%ΔTg

average relative deviation of glass transition temperature

|%ΔTg|

average absolute relative deviation of glass transition temperature

|%ΔTg|max

maximum absolute relative deviation of glass transition temperature

F

general function or IL property

IL

ionic liquid

nC

number of carbon atoms

n

number of neurons in a network

N

number of data in a data set

M(+)

cation mass

M(−)

anion mass

Tcc

cold crystallization temperature

Tf

freezing temperature

Tg

glass transition temperature

Tm

melting temperature

Tss

solid–solid transition

Vw

van der Waals volume

X0

connectivity index of order zero

x1, x2, x3… xi,xn

general independent variables

AbbreviationsAm

ammonium

ANNs

artificial neural networks

GC

group contribution

Im

imidazolium

Py

pyridinium

QSPR

quantitative structure–activity relationship

Ref

reference

SA

surface area

Su

sulfonium

Tr

triazolium

Superscripts/subscriptscal

calculated

g

glass

lit.

literature

max

maximum

Greek lettersΔ

difference

λ

connectivity index

∑

summation

Appendix A Supporting information

The following are the supporting information related to this article:

Additional Supporting Information includes the following files: (1) A Matlab code (Tglass.m) for training and testing using a simple (4.4.4.1) architecture, that can be changed by the user; (2) an Excel file (Tg_data.xlsx) with the eight sheets described in Table 3 with one run; (3) a Matlab code (Tg_Prediction.m) for predicting Tg of any IL given values of the chosen parameters; (4) an Excel file (Tg_Prediction.xlsx) that stores the results of the predictions done with the program Tg_Prediction.m. If readers and researchers use these files for further work, this article must be cited as the source of information. This supporting information is available free of charge via the internet.

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