Ionic surfactants critical micelle concentration prediction in water/organic solvent mixtures by artificial neural network

1 Introduction

Surfactants are organic compounds of an amphiphilic nature consisting of two different polarity zones with a unique structure. They have a hydrophilic group, which has affinity for the polar phase and is located at the head of the compound, and a hydrophobic group, located at the tail, which has an affinity for the non-polar phase. ^{1 , 2} These compounds can be produced chemically (synthetic surfactants) or being based on biological materials (biosurfactants). ³ According to Hussain et al., ⁴ surfactants can be classified into four groups based on the charge of the head group: nonionic, zwitterionic, anionic and cationic, playing an important role in transitions and aggregation structures depending on that. ⁵ Thanks to these characteristics and their propensity to self-assemble allow them to form micelles. ⁶ Surfactants have been widely used to reduce the surface tension between the two immiscible phases. ^{7 , 8} Their applications are related to different purposes such a pharmaceutical and cosmetic uses, ^{9 , 10} detergents, ^{11 , 12} wetting agents ¹³ and food additives ¹⁴ among others.

Nowadays, self-assembly is an important phenomenon in daily life and science where molecules are organized into stable aggregates. An example of this phenomenon of molecular aggregation in biochemical systems is the formation of micelles. ¹⁵ Micelles are aggregates of oriented surfactants in an aqueous solution, where the interior is hydrophobic and the exterior is hydrophilic part. ¹⁶ According to Schork, ¹⁷ the micelles contain about 50–100 monomeric units, and they can improve the solubility of hydrophobic substances. ¹⁸ The critical micelle concentration (CMC) is the minimum concentration at which the micelle begins to form. ¹⁹ Each surfactant molecule presents a certain CMC value that depends on factors including temperature and electrolyte concentration. ^{20 , 21} When the concentration of surfactants is higher than the CMC, they tend to aggregate forming thermodynamically stable micelles. ²² The study of CMC is based on drastic changes in macroscopic parameters. ²³ For this reason, the physicochemical methods are used for measuring and determination of CMC, such as surface tension, conductivity, osmotic pressure, density and light dispersion. ^{24 , 25} Finally, the kinetics of micellization have an important role in many applications. Therefore, understanding the physicochemical properties of the surfactant’s solution is crucial from both a technological and environmental perspective. ²⁶

The temperature over the CMC of surfactants in water does not follow a linear trend. The CMC initially decreases with increasing temperature until reaching a minimum, and then increases at higher temperatures. It is known that higher temperatures lead to the formation of micelles more easily due to the lower hydration of the hydrophilic group of the surfactant, but also to the destruction of micelles due to the rupture of the structured water molecules that surround the hydrophobic part of the surfactant. ²⁷ Furthermore, the addition of salt to anionic surfactant solutions generally conduct to a reduction in surface tension, being more significant at higher salt concentrations. This effect can be explained by electrostatic interactions that facilitate the migration of surfactant monomers towards the interface. ^{27 , 28} According to Niraula et al., ²⁹ when adding an electrolyte, micelles will generally form at a concentration lower than the CMC of the pure surfactant. Physical-chemical properties such as the degree of the ionization, reaction rates and phase separation are affected when incorporating the additive to a set of surfactants. ³⁰ On the other hand, the presence of alcohols in water leads to a decomposition of the water structure and the reduction of its dielectric constant, which contributes to the increase of CMC. ³¹

Because the self-aggregation of this surfactant in solution is controlled by CMC, accurately determining CMC values is of great scientific interest. One way to reliably obtain these values is experimental measurements. However, this method is considered challenging, particularly at high temperatures and pressures, due to the time-consuming and expensive nature of these procedures. ²⁷ For this reason, the application of prediction methods and mathematical models can be a good practical alternative in this area. ²⁷ In this sense, QSPR models using different linear regression techniques such as multiple linear regression (MLR) and partial least squares (PLS) that are widely used for CMC predictions. ^{32 , 33 , 34 , 35}

On the other hand, algorithms based on machine learning (ML), very useful today, are also used for the study of surfactants due to their potential to overcome the challenges associated with experimental and simulation approaches. ³⁶ In fact, using an machine learning (ML) model within QSPR (ML-QSPR) has the advantage that it allows finding those most revealing molecular descriptors for the target variables based solely on the analysis of the relationships between the input (descriptors) and the output (target variables). ³⁷ Among ML models, artificial neural network (ANN) model can be applied to QSPR studies as a nonlinear regression model. ^{35 , 38 , 39 , 40} Although ANNs provide robust predictable capabilities compared to alternative methods, their application on surfactants represents an innovative approach. ⁴¹

According to Abooali et al., ²⁷ most studies that use QSPR models to estimate CMC are developed from chemical descriptors at constant room temperature (20–25 °C) and use pure water as a solvent without salinity. However, as has been shown, the temperature, the presence of additives and the solvent composition influence the CMC. Furthermore, CMC is significantly affected by other factors such as the chemical structure of the surfactant, pressure, and pH of the solution. ⁴² For these reasons, it is interesting to approach this research considering these multiple variables.

The aim of this study is to develop ANN models to estimate the CMC for 10 ionic surfactants (six cationic and four anionic). The experimental data were collected from various scientific publications in the literature. The effective parameters on CMC such as temperature, composition of the surfactant chemical formula and the nature of the solution were incorporated as variables in prediction models. Furthermore, the alcohol properties such as log P (log K _WO), molecular weight and molar fraction were also included. Indeed, log P is the ratio of that substance’s concentrations in a two-phase mixture made up of two immiscible solvents in equilibrium – n-octanol and water. It calculates the differential solubility of a solute in these two solvents. It enables the establishment of a substance’s hydrophobicity scale.

2 Materials and methods

2.2 Artificial neural networks

Artificial neural network (ANN) is a class of machine learning model that mimics the structure and function of biological neural networks present in the human brain. ⁵⁹ This prediction algorithm is widely used for complex pattern modelling and regression problems. ⁶⁰ Furthermore, ANN can find and learn hidden patterns between input data and target sets. ⁶¹

There are different classes of ANNs according to its structure, where the multiplayer perceptron (MLP) is the simplest and used for different fields of prediction and classifications. ⁶² The MPL is a fully connected multilayer neural network composed of neurons located in three areas of the network: an input layer, one or more hidden/intermediate layers, and an output layer. ^{63 , 64} The input layer receives information and is responsible for transferring it to the hidden layer, which then transfers this information to the output layer. ⁶⁵ This transmission between different layers is controlled by the propagation function. ⁶⁶

The operation of ANN can be described as follows: random weights are assigned to neurons located in the different interconnected layers to solve mathematical operations. ⁶⁷ These neurons recognize the relationship between target values and input data sets. ⁶⁸ The input information received by these neurons is structured in different layers to convert them into a single output. ⁶⁹ This output is then sent to the other layer to perform the same sequence and generate the output. ⁷⁰ The output of each neuron is produced from a series of operations performed by the activation function. ⁶⁶ The main activation functions are linear, logarithmic and tan sigmoid. ⁷⁰

According to Gue et al., ⁷¹ the backpropagation algorithm is the most used learning algorithm. It minimizes the least squares error between the network output signal with the signal propagating forward and the desired response. ⁶⁴ This algorithm corrects the weights between different neurons according to the error output. This process continues until the desired value is achieved. ⁶⁶

In this research, we used a multilayer neural network algorithm with one hidden layer to collect data from different training cases. The ANN models were constructed using various topologies, with the number of hidden neurons varied between 1 to 2n+1 (where n is the number of input variables). According to number of input variables used in this study, the number of hidden neurons varied between one and 25 (Figure 1). The backpropagation algorithm was used as a learning algorithm. In addition, the sigmoid function was used as an activation function.

Figure 1:

Scheme of ANN with a topology of 12-25-1 (12, 25 and one neurons in the input, hidden and output layer, respectively) to model CMC. n is the number of neurons.

For all ANN models developed, the input variables included molar fraction, molecular weight, log P, temperature and number of atoms of each element present in the chemical formula of each ionic surfactant. Finally, log CMC was the output variable (the variable to forecast).

The hyper-parameters used to develop the ANN models included training cycles (ranging between 1 and 524,288) in 19 steps, in linear (designed without subscript) or logarithm scale (designed with a subscript L) and decay (true or false). The hyper-parameters were determined using the trial-and-error procedure. Using these two hyper-parameter combinations, six different ANN approaches were developed (ANN, ANN _L , ANN _R , ANN _R-L , ANN _Z , and ANN _Z-L ). Four ANN models were normalized to a determined range because the data showed different scales and units. The main aim of the normalization process is to prevent any of the variables from having a disproportionate effect on the training of the model. Two normalization methods, that were used to input and output variables, were applied: range transformation, designed with a subscript R, varying between −1 and 1 (ANN _R ) and Z-transformation, designed with a subscript Z (ANN _R ). These methods were applied to all data groups in the following order: first to the training data and then to the validation and test data. The results were then de-normalized to compare between them. This transformation is necessary since it allows the predictions obtained to be interpreted in their original context.

2.4 Software

The different machine learning models were developed using the RapidMiner Studio Educational 10.2.000 version (RapidMiner GmbH). The computational equipment used were an Intel^® Core(TM) i9-10900K CPU at 3.70 GHz processor with 64 GB RAM installed and with Windows 11 Pro. Figure 1 was made with Microsoft PowerPoint 2016 (Microsoft) and the scatter plots (Figure 2) was made with SigmaPlot 14.0 (Systat Software Inc.).

Figure 2:

Scatter plots showing the real and predicted values of log CMC real (axis x), and log CMC predicted (axis y) for the ANN _Z model. The red dashed line corresponds to the line with slope one.

3 Results and discussion

In this research, six different models were developed using ANN algorithm to predict the CMC value for ionic surfactants. To evaluate the performance of each ANN model, three statistical parameters were utilized. Table 3 shows the six ANN models alongside their corresponding statistical parameters values.

According to Table 3, the best adjustments for the training and validation phases were observed in the ANN models that were normalized on both linear and logarithmic scales. Significant differences in all statistical parameter values can be observed between the unnormalized and normalized ANN models. This discrepancy is due to the normalization process, which ensures that data measured on different scales do not disproportionately influence the model, especially since the variables are measured in different units. The four normalized models showed similar RMSE values in the validation phase (between 0.040 M and 0.050 M, being the lowest value for ANN _Z model). In terms of MAPE, the values for these four models showed very close values, ranging from 2.7 % and 3.0 % (being the lowest value for ANN_Z model), and the linear square correlation coefficient (R ²) values also varied slightly between 0.992 and 0.995. Regarding to the training phase, the RMSE values also showed low oscillations. RMSE values varied between 0.025 M and 0.030 M, being the lowest value for the ANN _Z model. In the case of MAPE, the ANN _Z model presented the lowest value (1.5 %), while ANN _Z-L showed the highest value (2.0 %). Finally, the R ² parameter exhibited similar values in all models (around 0.997).

Hence, it is demonstrated that the normalized models showed excellent fits for both the training and validation phases. That is, these models trained well with the internal data. However, it is crucial to also verify this performance with external data, i.e. data that was not used to train the model (testing phase) to avoid overfitting problems.

Regarding the test phase, the models exhibited slightly worse adjustments of RMSE, MAPE and R ² compared to those to the training and validation phases (Table 3). Specifically, RMSE values ranged between 0.061 M (for ANN _R-L model) and 0.069 M (for ANN _Z model), while MAPE values ranged between 2.9 % (for ANN _R model) and 3.5 % (for ANN _Z model). Finally, R ² values were below 0.982 for the four normalized models.

A good model should be able to perform good adjustments in the training and validation phases, but also in the testing phase. Based on the results in Table 3, all normalized ANN models presented good adjustments in the three phases, without problems of model overfitting and with high model generalization capacity.

The best ANN model from the four normalized models was then selected for further study of its performance. The lowest RMSE value in the validation phase was used as the criterion for selecting the best model. According to this, the ANN _Z model was the best model with an RMSE value of 0.040 M.

Figure 2 shows scatter plots (predicted vs. observed values) of the ANN_Z model across the three phases.

Figure 2A illustrates the dispersion between the CMC values predicted by the ANN _Z model compared to those observed for the internal data (training and validation phase). Generally, the points are close to the pointed red dashed line (line with slope 1), although it should be noted that there are some points that present significant deviations from the line. For example, the training phase (−1.31, −1.49) and validation phase (−1.90, −1.78) cases, both located in the lower middle part of the graph, presented the highest absolute error values (0.174 M and 0.120 M, respectively). Regarding the training case, the observed value was −1.31 M, while the model predicted a value of −1.49 M, leading to an absolute percentage error of 13.2 % (overestimation). Regarding the validation case, it showed an absolute percentage error of 6.3 % (underestimation). On the other hand, it should be noted that there are also cases, especially in the lower part of the graph, which are not so deviated from the slope line 1 (low absolute errors), but present significant values of absolute percentage errors. For example, the validation cases (−0.733, −0.819) and (−0.848, −0.941), showed absolute percentage errors of 11.8 % and 11.0 %, respectively. If those cases that contain absolute percentage errors greater than 10 % were excluded, the MAPE value is reduced to 1.58 %.

Figure 2B shows the dispersion for the external data (test phase). Generally, it can be observed that the points are close to the dashed red line with slope 1. However, there is an important case, very deviated from the straight line, which showed important errors. This test case (−1.82, −2.09) is in the low part of the graph, in which it presents the highest values of absolute error and absolute percentage error. Particularly, the model predicted a CMC value of −2.09 M versus −1.82 M, leading to an overestimation of 14.4 %, while the absolute error was 0.263 M. If this case was excluded, the MAPE value is reduced to 3.0 %. It is worth mentioning another case, that is not so easily visible, but which presents a relatively high absolute error. This is the case (−1.95, −1.83), which is located in the lower part of the graph, in which the absolute error is 12.9 M.

There are numerous studies in the literature that have developed quantitative structure-property relationship (QSPR) models to establish relationships between molecular structures and the critical micellar concentration surfactants. Notably, most of these studies focus on a single class of surfactants, particularly anionic. For example, Yuan et al. ⁷⁵ used a QSPR model with multiple linear regression to predict the CMC of 37 anionic surfactants in aqueous solution at 40 °C. In their study, the octanol/water partition coefficient (log P), the dipole moment of the molecule and the area of the molecule were the selected descriptors, achieving an R² greater than 0.980. Roberts ⁷⁶ used QSPR analysis to predict the CMC of anionic surfactants, including ether sulphates and ester sulfonates, using the hydrophobic parameter ðh for the hydrophobic domain of the surfactant. This study achieved good results, with an R ² of 0.976 for 133 compounds. Another study carried out by Li et al. ⁷⁷ estimated the CMC of 98 anionic surfactants at 40 °C, using three descriptors: total atom number of the hydrophobic-hydrophilic segment, the dipole moment of the segment, and the maximum net atomic charge on the carbon atom in the segment. Their QSPR model, developed using multiple linear regression techniques, showed an R ² of 0.980, demonstrating excellent robustness. Roy et al. ⁷⁸ developed QSPR models using multiple linear regression to model the CMC of 37 anionic surfactants measured at 25 °C in water without additives. This study used extended topochemical atom (ETA) indices along with computed hydrophobicity descriptors. According to the results, the ETA models provided better adjustments (R ² = 0.923) compared to models without ETA topological descriptors along with hydrophobicity terms (R ² = 0.885). All these results suggest that the use of the QSPR models is appropriate for predicting the CMC of anionic surfactants.

Machine learning models can also be incorporated in QSPR models. According to Thacker et al., ³⁶ the use of machine learning models for CMC prediction can overcome the existing limits with experimental and simulation approaches. Several studies in the literature have employed machine learning models for this purpose. For example, Rahal et al. ³⁵ developed QSPR models using three regression methods to predict the CMC of 50 anionic surfactants. According to the results, the best adjustments were obtained using multilayer perceptron-artificial neural networks (MLP/ANN). Brozos et al. ⁷⁹ used the graph neural network (GNN) algorithm together with the QSPR model to estimate, among other properties, the CMCs of different surfactants. According to the results, the GNN predicted quite accurate results for CMC. ⁷⁹ In another study, Qin et al. ⁸⁰ used graph convolutional neural network (GCN) to predict CMC from the molecular structure of surfactant. The results concluded that GCNs are suitable and effective for high-throughput screening of surfactants. ⁸⁰

All of these studies mentioned used mathematical correlations to predict experimental CMC values based on chemical descriptors under fixed temperatures, pure water conditions and without additives. However, as mentioned in this research, physical-chemical properties such as temperature, pH, pressure, salinity of the solution and the type of organic solvent water mixture strongly influence the CMC. To date, there have been few studies where these properties were used as input variables for developing CMC prediction models. In fact, the estimation of CMC at different temperatures is considered novel and innovative. Abooali & Soleimani ²⁷ used different nonlinear models employing two machine learning approaches, stochastic gradient boosting trees (SGB) and genetic programming (GP), to model the CMC of 111 anionic surfactants. Their study used five chemical factors (Lop, CIC2, EEig12x, BEHp2, and G3s) and three physical parameters (temperature, with a range between 273.15 K and 363.15 K, pH and salinity). The results showed that the stochastic gradient boosting (SGB) model had better statistical parameter values (RMSE = 0.019 and R ² = 0.999) compared to the genetic programming (GP) model (RMSE = 0.166 and R ² = 0.954). Another study conducted by Boukelkal et al. ³⁸ used QSPR methodology to establish a relationship between the molecular structure of 593 surfactants of different classes (anionic, cationic, nonionic, zwitterionic and Gemini) and the negative logarithmic value of CMC. They used different machine learning models to build QSPR models with 14 molecular descriptors along with temperature, varying from 10 °C to 60 °C. The authors concluded that the support vector regression model with Dragonfly hyperparameter optimization (SVR-DA) provided the best adjustments for predicting pCMC values (RMSE = 0.205 and R ² = 0.974). ³⁸ Finally, our study is another example in which physical-chemical properties are used as modelling variables for estimating CMC. However, comparing the current study with those existing in the literature is challenging due to the differences in the size of the data set, in the division of the data, in the variables used for the development of the models, and in the modelling methods used to predict CMCs. Despite these differences, ANN_Z model (RMSE = 0.035 and R ² = 0.995), together with the SGB model, showed the best adjustments for the global phase. Therefore, we can conclude that our ANN models have the potential to provide excellent prediction capabilities.

4 Conclusions

CMC estimation is one of the most interesting aspects for both academic and industrial applications involving surfactants. Machine learning-based approaches, such as those performed in this study, have demonstrated to be useful as a cost-effective and time-efficient alternative to traditional laboratory measurements. Our ANN models for predicting the CMC of 10 ionic surfactants were developed using physico-chemical properties of the organic solvent-water mixture and chemical properties of the surfactant. The research concluded that the normalized ANN models were highly accurate based on three statistical parameters. Particularly, the ANN_Z model presented the best adjustments, with an RMSE of 0.040 M (R ² = 0.995) for the validation phase and of 0.069 M (R ² = 0.970) in the test case. The mean absolute percent errors were 2.7 % and 3.5 %, respectively. In this sense, this implemented algorithm is reliable and effective for predicting CMC values of ionic surfactants. Overall, ANN models show excellent potential as tools for accurate and efficient CMC prediction.

However, we must add that although ANNs are a powerful tool in machine learning and have been shown to be effective in predicting CMC values both in the present work and in previous works by our research group, ⁶⁶ there are alternatives that may be more appropriate in certain contexts or for specific types of problems. This is the case of support vector machines (SVM) – a type of supervised model whose objective is to find a hyperplane that maximizes the separation between data classes and that can be used for both classification and regression – or random forests (RF) – which are built as a combination of several decision trees and the prediction is made by majority vote of the individual trees. We must point out that these alternatives have been successfully applied by our research group for the prediction and modelling of various issues. ^{81 , 82 , 83 , 84} A future line of research would be to check the validity of these alternatives applied to the prediction of CMC values. In fact, some authors have already successfully incorporated some of these techniques for modelling the behaviour of different surfactants. ⁸⁵