This study selected 41 dyes with different structures, including azo dyes (such as methyl red), anthraquinone dyes (such as carmine), arylmethane dyes (such as crystal violet), and indigo dyes (such as indigo) (Supplementary Table S1). Among them, Disperse Red 60, Vat Blue 4, and Vat Violet 1 were used to test the optimal QSAR model. The other 38 dyes were used to construct the training set for the QSAR model.

All concentrations (50 mg/L) of dye solutions were stirred and dissolved at 298 ± 1 K and initial pH values of 4.0 ± 0.1, 7.0 ± 0.1, and 10.0 ± 0.1. Then, 100 mg/L Al³⁺ solution was added to each reactor at a molar ratio of coagulant to tested dyes of 20:1. After stirring for 15 min and standing for 30 min, the supernatant was filtered through 0.45-μm microporous membrane and examined by UV spectrophotometry (UV-1600, Shanghai Mapada Instruments Co., Ltd.) at their maximum absorption wavelengths. All dyes and inorganic compounds were of analytical grade and purchased from Sinopharm Chemical Reagent Co., Ltd. or Aladdin Industrial Corporation.

Four kinds of molecular parameters were selected for calculation: geometric and physicochemical, electrical, molecular frontier orbital energy, and other quantum chemical parameters. All dye structures were initially optimized using HyperChem Release 8.0 and then deep optimized by density functional theory (DFT) (B3LYP/6-311G (d, p) basis set) in Gaussian 09 software. A total of 38 molecular parameters (Supplementary Table S3) were calculated using HyperChem 8.0, Materials Studio 7.0, and Gaussian 09. The other eight combination descriptors (E _GAP , E _GAP ² , E _SUM , ESP _0.25, ESP _0.75, ∑q(O + N), ∑q(H)/N _H, and ∑q(−)/N _C) (Supplementary Table S1) were obtained by simple mathematical calculations.

Seven dye parameters were calculated in HyperChem 8.0. Three geometric parameters, surface area approximation (SAA), molecular volume (V), and molecular weight (MW), were basic geometric parameters that directly affect the structure and properties of substances. According to the calculated results, dyes with the largest SAA values, such as Acid Blue 93, Acid Green 1, Direct Red 28, Ponceau S, and Reactive Yellow 3, also had the largest MW values among all the dyes, indicating a significant positive correlation between SAA and MW (Supplementary Table S5; Figure 2C). In addition, four physicochemical parameters, hydration energy (HE), refractivity (R), polarizability (Po), and oil–water partition coefficient (Log P), were selected as the macro-descriptors of dyes.

Five descriptors describing the charge of a single atom on dye molecules, qH ⁺, q(CH ⁺)_n, q(CH ⁺)_x, q(C ⁻)_n, and q(C ⁻)_x, and seven descriptors describing the charge of whole dye molecules, ∑q(O), ∑q(N), ∑q(O + N), ∑q(H), ∑q(H)/NH, ∑q (−), and ∑q(−)/N _C, were electrical parameters and obtained using Gaussian 09 software. The electrical parameters also included the electrostatic potential of the molecular surface (ESP _MAX, ESP _MEAN, ESP _MIN, ESP _0.25, and ESP _0.75), which were calculated using Materials Studio 7.0. The average ESP _MAX value of all dyes was 82.9850 a.u. Acid Blue 93 had the largest ESP _MAX value (209.1500 a.u), while Mordant Black 17 had the smallest ESP _MAX value (30.6940 a.u), a nearly seven-fold difference (Supplementary Table S4-1). The electrostatic potential of the molecular surface (ESP), which is closely related to the electron density distribution, is an effective molecular descriptor to predict favorable sites of electrophilic and nucleophilic reactions or reveal preferred sites of electrostatically dominated non-covalent interactions (Tao et al., 2016; Scheiner, 2017). ESP can also reflect the attack sites of electrostatic adsorption (Tanzifi et al., 2020). Disperse Red 16 showed the largest q(C ⁻)_x (0.852 e) and the smallest q(C ⁻)_n (−0.692 e) (Supplementary Table S4-1).

The molecular frontier orbital energy parameters, including E _HOMO, E _LUMO, E _SUM, E _GAP, E _GAP ², I, A, η, S, χ, CP, and ω, were calculated in Gaussian 09. The energy of the highest occupied molecular orbital (E _HOMO) and energy of the lowest unoccupied molecular orbital (E _LUMO) were closely related to the activity of the chemical reaction (Lamya et al., 2021; Amri et al., 2023; Bensalah et al., 2023; Lebkiri et al., 2023). The higher the E _HOMO, the stronger the electron-donating capacity of the molecule and vice versa. E _LUMO represents the electronic affinity, in which the smaller the value, the stronger the electron-accepting ability of the molecule (Fukui, 1982; Regti et al., 2016). The gap in E _LUMO and E _HOMO (E _GAP) was used to describe the chemical and kinetic stability of the molecules. Molecules with a small E_GAP were generally reactive, while molecules with a large E _GAP were stable and unreactive (Zhang & Musgrave, 2007).

In this study, other quantum chemical parameters were selected for analysis using Gaussian 09 and Materials Studio 7.0, including the total energy of a molecule (E _B3LYP), dipole moment (μ), bond order (BO _n/BO _x), and Fukui indices (f(+)_n/f(+)_x, f(−)_n/f(−)_x, and f(0) _n/f(0) _x). These quantum chemical parameters were important indicators of the physicochemical properties of organic matter. They are useful to reveal the degradation mechanism of organic matter in oxidation and coagulation processes (Jia et al., 2015; Su et al., 2016; Cheng et al., 2018b; Cheng et al., 2018c). The total energy (E _B3LYP) directly reflects the degree of difficulty of the reactions between molecules. The average E _B3LYP value of all dyes was −2679.8717 kcal/mol, while the highest and lowest values were observed for isatin (−513.19 kcal/mol) and bromocresol green (−11879.78 kcal/mol), respectively, a 23-fold difference, demonstrating the large structural diversity of the dyes selected in this study (Supplementary Table S4-1). The dipole moment (μ) was used to investigate intermolecular interactions. The greater the dipole moment, the stronger the interaction force between molecules (Johnson & Otero-de-la-Roza, 2012; Kumar et al., 2014). Bond order (BO) reflects the stability of a chemical bond. Usually, the BO is <4. The larger the BO, the more stable the molecule, and vice versa (Mayer, 1984; Su et al., 2016).

According to the previous research (Cheng et al., 2018b; Tan et al., 2021a; Tan et al., 2021b), the 38 dyes in modeling were divided into training and test sets in a ratio of approximately 4:1. First, IBM SPSS Statistics for Windows, version 25.0 was used to analyze the correlations between color removal rates (R _exp ) and 46 parameters and identify the molecular parameters with the great correlations. Then, the stepwise linear regression (MLR) method was applied to develop the QSAR models with R _exp as the dependent variable and the molecular parameters as the independent variables. The squared regression coefficient (R ² ), standard deviation (SD), root mean square error value (RMSE), significance testing (P), Akaike information criterion (AIC), t-test, and Fisher’s tests were used to evaluate the quality and performance of the constructed models (Ma et al., 2010; Tropsha, 2010). The variation inflation factor (VIF) test was applied to check for multicollinearity among the selected descriptors for each model (Gupta & Basant, 2016). When the values of leave-one-out internal validation (Q ² _INT) and external validation (Q ² _EXT) using MATLAB were both >0.5, the developed model indicated robust and predictive (Gramatica, 2007). Finally, Y-randomization and applicability domain tests were performed for the optimal model to verify its stability and credibility (Melagraki & Afantitis, 2014; Gupta & Basant, 2017; Ortiz et al., 2017).

Figure 1 shows the color removal rates (R _exp) of 38 dyes by the AlCl₃ coagulant at pH values of 4, 7, and 10. The average R _exp values were 0.1756 ± 0.0165, 0.2403 ± 0.0218, and 0.4311 ± 0.0230, respectively. Overall, the R _exp values under alkaline conditions were higher than those under neutral and acidic conditions. A pH = 4, Eriochrome Black T (EBT), Direct Red 28 (DR28), and Disperse Red 16 (DR16) had R _exp values of almost 100%, while most of the other dyes had R _exp values <30%. The results under neutral conditions were nearly the same. Only the R _exp value of Acid Orange 7 (AO7) exceeded 90%. At pH = 10, the R_exp values of most dyes are high overall. Therefore, AlCl₃ coagulation is suitable for the removal of dyes under alkaline conditions.

AlCl₃ generally hydrolyzes to form an aluminum hydroxide colloid in an aqueous solution. However, under acidic conditions, many H⁺ easily react with this colloid, destabilizing the colloid and weakening dye coagulation and adsorption, leading to low color removal rates. The zeta potential of the AlCl₃ in Table 1 shows that the zeta potential is the largest under alkaline conditions. Therefore, the colloid stability is highest under alkaline conditions. In acidic conditions, the zeta potential of AlCl₃ is close to zero, indicating that the colloid is unstable and the coagulation effect is poor.

Under acidic conditions, only the oil–water partition coefficient (Log P) was significantly correlated with R _exp (Figure 2A). The experimental results of R _exp values of dye by AlCl₃ showed that the removal effect of AlCl₃ was unsatisfactory and some dyes were not removed at all. This led to a weak regularity of correlation analysis under acidic conditions.

Under neutral conditions, no strong correlation between AlCl₃ and molecular parameters was found due to its low removal effects on most kinds of dye. Only the parameters BO _n, Log P, and E _HOMO showed significant negative correlations with R _exp (p < 0.01). Additionally, the correlation coefficient was −0.413, indicating that it has a great effect on the R _exp of dye (Figure 2B). Moreover, parameter I was significantly positively correlated with R _exp; because I = -E _HOMO , it showed the opposite correlation to E _HOMO .

However, more molecular parameters were significantly correlated with R _exp under alkaline conditions than under acidic and neutral conditions. The parameters with significant negative correlations were Log P, ∑q(O + N), ∑q(O), ∑q(−), E _B3LYP, CP, E _SUM, E _LUMO, ∑q(−)/NC, f(0) _n, and f(−)n. The parameters with significant positive correlations were SAA, MW, ESP _MEAN, V, q(C ⁻)_x, ∑q(H)/N _H, R, χ, q(CH ⁺)_x, ESP _0.25, A, and P. Surface area (approx.) (SAA) and molecular weight (MW) showed the largest correlation coefficients (0.665 and 0.625, respectively) and had the greatest impact on the removal effect of dyes by AlCl₃ at pH = 10 (Figure 2C).

Generally speaking, the larger the MW, the larger the molecular surface area and the easier it is for the molecule to be removed by coagulant through electrostatic adsorption or catching–sweeping in an aqueous solution. The absolute values of the significant correlation parameters were ordered as follows: SAA > MW > Log P > ∑q(O + N) > ∑q(O)> ∑q(−) > V > q(C-)_x = E _B3LYP > ∑q(H)/N _H > R > χ = E _SUM = CP > q(CH ⁺)_x > ESP _0.25 > A = E _LUMO > Po > ∑q(−)/NC > f(0) _n > f(−)_n. The geometric and physicochemical parameters SAA and MW mainly affected R _exp under alkaline conditions, followed by the electrical parameters.

QSAR models are constructed by using multiple linear regression (MLR). All models must satisfy the conditions of R ² > 0.6, Q ² _INT >0.5, Q ² _EXT >0.5, and p < 0.01 (Cheng et al., 2018a; Cheng et al., 2018b; Liu et al., 2020). No QSAR models meeting all validation were constructed under acidic or neutral conditions in this study. These findings are consistent with the experimental results that the positively charged colloid generated by AlCl₃ is easily destabilized under acidic and neutral conditions, resulting in low adsorption capacity and poor removal effect. In alkaline conditions, the positively charged colloids generated by AlCl₃ are relatively stable and can easily combine with the negatively charged anionic dyes in the solution through electrostatic forces. Four QSAR models at pH = 10 are shown in Table 2. In general, QSAR models with high R² and Q² _INT met the other validation conditions and were selected as the optimal model (Dearden et al., 2009).

Model 1 has two positive variables (SAA and q(C ⁻)_x) and one constant, while model 2 has two positive variables (SAA and q(C ⁻)_x), one negative variable (ESP _MAX ), and one constant. For models 1 and 2, the internal validation (Q ² _INT) values are only 0.5162 and 0.5754, which meet the criterion. Model 3 is the optimal one model, with R _pre = −0.503 + 0.001 SAA + 0.700 q(C ⁻ )_x - 0.003 ESP _MAX - 3.932E - 5 E _B3LYP. Although the R ² of model 4 is 0.8128, higher than that of model 3 (0.7962), its external validation (Q ² _EXT ) is 0.2864 (<0.5), indicating that it is not an acceptable model.

As shown in Table 4, the independent variable ω in model 4 does not meet the t-test and significance test. The statistical indexes, internal/external validation, and statistical tests of the optimal model are shown in Tables 3, 4. The R ², Q ² _INT, and Q ² _EXT of the optimal model are 0.7962, 0.6664, and 0.7230, respectively, which were far greater than the criteria of the acceptable models. This model also has larger F and small SD and RMSE values, indicating that model 3 is reliable, predictive, and acceptable.

According to the aforementioned models, four groups of R _pre values are predicted, as shown in Figure 3. The green and red triangles represent the training and test sets, respectively. The blue solid line represents the 1:1 regression line. The closer the data samples are to the regression line, the more accurate the predicted R _pre of the model. The blue dotted line represents an error range of ±0.2 to the predicted value. The scattered points in model 1 were slightly scattered, with three points far from the regression line (Figure 3). The scatter-point distributions in models 2 and 4 are close to the regression line; while the scatter degree decreases, two points are still far from the regression line. In model 3, the scattered-point distribution is concentrated along the regression line except for indigo in the test set, indicating that its prediction ability is relatively accurate.

Table 5 shows that 31 dyes had absolute values of Diff. that are <0.2 in model 3, compared to 28, 30, and 29 dyes, respectively, in models 1, 2, and 4. Meanwhile, 16 absolute values of Diff. are <0.1 in model 3, compared to 12, 13, and 15 in models 1, 2, and 4, respectively (Table 5). Therefore, model 3 should be the most satisfactory and acceptable QSAR model of dye removal by coagulation of aluminum chloride under alkaline conditions.

The Y-randomization and applicability domain tests were performed for the optimal QSAR model under pH = 10. First, the R _exp values were randomly rearranged, while the 46 molecular parameters are kept in the same order to obtain a new matrix. Then, a new QSAR was established. According to the test criteria, the R ² and Q ² _INT of the new model were smaller than those of the original model. In this study, the aforementioned steps were repeated 20 times to obtain the R ² and Q ² _INT values of the new models. The Y-randomization test results are shown in Figure 4A. The R ² and Q ² _INT values of the 20 new models were far lower than those of the original model. Therefore, the stability of the original model was not due to a chance correlation or structural dependency of the training set.

Except for indigo, the sample data of all dyes fall within the applicability domain (Figure 4B). This is due to the large differentials between its predicted R_pre and experimental R_exp, but the standardized residual (σ) is 3.5, not exceeding the critical value by too much (Figure 4B). Therefore, when the optimal model is used to predict the R _pre of AlCl₃ coagulant for other organic dyes under alkaline conditions, the four parameters of the dye molecule, SAA, q(C ⁻)_x, ESP _MAX, and E _B3LYP, should be calculated before determining the leverage value and standardized residual of the dyes. If they fall within the applicability domain of Figure 4B, the predicted R _pre of dye is accurate. Otherwise, the predicted R _pre is uncertain.

To further verify the predictive ability of the optimal QSAR model, Disperse Red 60, Vat Blue 4, and Vat Violet 1 were selected for coagulation experiments on dye color removal by AlCl₃ under the same experimental conditions. The experimental data (R _exp) were compared with the predicted value (R _pre ) of the optimal QSAR model, which are listed in Table 6. As shown in Table 6, except for the absolute value of the Diff. of Disperse Red 60, which is slightly more than 0.2, the R _exp and the R _pre of the other two vat dyes show a good fit. In general, the optimal QSAR model of dye color removal rate by AlCl₃ coagulant under alkaline conditions has good predictive ability.

In this study, the optimal model equation was as follows: R _pre = −0.503 + 0.001 SAA + 0.700 q(C ⁻)_x - 0.003 ESP _MAX - 3.932E - 5 E _B3LYP. It contains four variables, among which the coefficients of SAA and q(C ⁻ )_x are positive and those of ESP _MAX and E _B3LYP are negative. Thus, the larger the SAA and q(C ⁻)_x or the smaller the ESP _MAX and E _B3LYP , the higher the R _pre . SAA is the parameter with the greatest effects. The larger the SAA of the molecule, the larger its contact surface area in an aqueous solution and the greater the chance of contact with the aluminum hydroxide colloid. This increases the possibility of hydrogen-bonding adsorption between the adsorption site of dyes and the surface of the coagulant. In addition to hydrogen-bonding interactions, adsorption bridging between dyes and colloids also plays a role in dye removal. The larger the SAA of dye molecules, the more apparent the adsorption bridging effect. Similarly, dyes with the largest R _exp under alkaline conditions have higher SAA values compared to other dyes. For example, the SAA values of Acid Green 1, Direct Red 28, Ponceau S, Reactive Yellow 3, and Disperse Red 16 were 1324.87, 1248.24, 1210.02, 1108.74, and 1105.93, respectively. The R _exp values of isatin, methylene blue trihydrate, and azure B were low for their small SAA values (Supplementary Table S4-4).

The q(C ⁻)_x values are the maximum of the negative partial charge of carbon atoms of dye molecules. These may be the active sites of electrostatic adsorption with the coagulant. Compared with the dipole moment (μ), which reflects the overall polarity of the molecule, the electrostatic potential can better reflect the partial polarity of the dye molecule. ESP _MAX represents the maximum electrostatic potential of the molecular surface. The larger the ESP _MAX, the weaker the hydrophobicity and the stronger the ability of banding water. This weakens the ability to combine with the aluminum hydroxide colloid and reduces the color removal effect. The coefficient between R _exp values and E _B3LYP is negative and weakly significant, which has only mathematical fitting significance but no physical significance in the model. In summary, the internal factors affecting the color removal of dyes by AlCl₃ coagulant under alkaline conditions include molecular size (SAA and E _B3LYP) and charge distribution (q(C ⁻)_x and ESP _MAX). Therefore, the coagulation mechanism of AlCl₃ under alkaline conditions is through electrostatic and hydrogen-bonding forces.

The removal mechanism of DR16 is shown in Figure 5, as an example. In the ESP distribution map, the positions of the C=O, N=O, and N=N double bonds are blue. The negative charge near them makes it easy to generate electrostatic adsorption with the colloid produced after the hydrolysis of AlCl₃ under alkaline conditions. The position of q(C ⁻)_x also proves this finding. In addition, N=N, N=O, and C=O double bonds are also potential active sites for hydrogen-bonding adsorption.