Prediction of Solvatochromic Polarity Parameters for Aqueous Mixed-Solvent Systems

: Solvent polarity is important data being used in solvent selections for preliminary engineering design of chemical processes. In this work, a predictive model is proposed for estimating the solvatochromic polarity of electronic transition energy ( E T ) of Reichardt indicator for aqueous mixtures. To validate the model, the E T values of eighteen aqueous mixtures collected from the literature were used. The predictive model provided a good estimation of E T values with an overall deviation of 2.1%, compared with an ideal model (5.1%) from the mole fraction average. The linear relationship of the contribution factor of hydrogen bond donor interactions ( CF HBD ) in the predictive model with Kamlet–Taft acidity was newly proposed in order to extend the model for other aqueous mixtures. The predictive model is applicable to many aqueous mixtures and simply requires three properties of pure components as: (i) E T values, (ii) gas-phase dipole moment and (iii) Kamlet–Taft acidity. 2-pyrrolidinone; dimethylformamide dimethyl


Introduction
The use of aqueous mixtures is preferable in many chemical processes (e.g., biomass conversion [1,2], separation and fractionation [3] and processing of active pharmaceutical ingredients [4][5][6]) because of the benefit of a safe solvent. Solvent polarity is an informative data being used in solvent selections [7][8][9][10] for preliminary engineering design and understanding the solvent effects on the chemical processes [11]. Polarity [12] is generally referred to as a solvent's capability for solute dissolution and can be quantified with many physical properties of solvents (e.g., electronic transition energy, solubility parameters and dielectric constant). Among the physical properties of solvents, the electronic transition energy (E T ) is commonly used to quantify an empirical solvent scale (also known as Reichardt's polarity) because the parameter requires a simple measurement using a solvatochromic technique with an indicator [13,14].
The E T values of aqueous mixtures [15] generally tend to have a negative deviation from the ideality line (Figure 1b) due to preferential interactions of an indicator with cosolvents. There are the correlative models (preferential solvation model [16][17][18] and Jouyban-Acree model [19]) for representing a preferential trend in the aqueous mixtures, while there is no predictive model for estimating E T values of the mixture. It is supposed that an ideal model (Equation (1), Section 2.1) is most likely applied for estimations due to a simple calculation from a mole fraction average. However, the ideal model (dashed line, Figure 1b) caused a large deviation from experimental data (symbol, Figure 1b).  (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering hydrogen bond donor (HBD) contribution factor (CF HBD = 1). The ET values in part (a) were collected from the phenol blue indicator [26]. The ET values in part (b) were obtained from the Reichardt indicator [15].

Ideal Model
The ideal model is used to compare the deviation of experimental data from ideality and is defined by Equation (1):  (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering hydrogen bond donor (HBD) contribution factor (CF HBD = 1). The E T values in part (a) were collected from the phenol blue indicator [26]. The E T values in part (b) were obtained from the Reichardt indicator [15].

Ideal Model
The ideal model is used to compare the deviation of experimental data from ideality and is defined by Equation (1): where E 0 T,i is the electronic transition energy of pure component i. Component 1 denotes water and component 2 denotes hydrogen bond acceptor (HBA) cosolvent or hydrogen bond donor (HBD) cosolvent.

Predictive Model
A predictive model was originally proposed for estimating Kamlet-Taft dipolarity/polarizability (KT-π*) [22] of binary nonpolar-polar mixtures with an assumption that the gas-phase dipole moment (µ) of the polar component can quantify a trend of mixture KT-π* values at a fixed mole composition. Due to a linear relationship between KT-π* and E T values (homomorphism line, Figure 2), the predictive model for KT-π* values can directly transform into a function form for estimating E T values for nonpolar-polar mixtures reported in our previous work [20], as shown by Equation (2). In this work, the model (Equation (2)) was used to predict the E T values of aqueous mixtures due to similar negative deviation trends in both aqueous mixtures and nonpolar-polar mixtures as mentioned in the introduction (Figure 1). ∆E N T,mix = µ 2 (−x 1 ln(x 1 + 1.981x 2 ) − x 2 ln(x 2 + 0.181x 1 )) × CF HBD (2) where µ 2 is the gas-phase dipole moment of component 2. The 1.981 and 0.181 values are the universal Wilson constant parameters (Λ 12 and Λ 21 ) for predictions and were evaluated by correlating experimental data with Wilson thermodynamic excess function [22]. The ∆E N T,mix is relative normalized electronic transition energy and defined by Equations (3)- (5).
E T (kcal · mol −1 ) = 28591/λ max (nm) (5) where λ max (nm) is the maximum absorption of the wavelength of a solvatochromic indicator (Reichardt indicator) obtained from UV-Vis spectroscopy. The E T,mix and E N T,mix are the electronic transition energy and the normalized electronic transition energy of the binary mixtures. To apply the predictive model (Equation (2)) to aqueous systems studied in this work, component 1 refers to water and component 2 refers to cosolvent (HBA or HBD). According to normalization (Equation (4)), the E N T,1 and E N T,2 of pure components 1 and 2 in Equation (3) are set equal to zero and unity, respectively. The CF HBD parameter in Equation (2) is the HBD contribution factor and the parameter is only applied to an aqueous mixture of HBD cosolvents due to specific interaction of HBD solvent with indicator [20] that causes a deviation in the pure E 0 T,2 value (HBD solvent) from the homomorphism line ( Figure 2). The CF HBD values in Equation (6) can be estimated by a deviation of the actual E T value of pure HBD cosolvent (E 0 T,2 ) from the homomorphism line in Figure 2 as shown in Equation (6).
where E 0,Non T is a non-HBD bonding E T value of pure HBD cosolvents defined as a linear function of dipolarity/polarizability (KT-π*), that is the homomorphism linear line in Figure 2. The linear relationships of E 0,Non T were given in detail in our previous work [20], and the CF HBD values of nine HBD cosolvents studied in this work are given in Table 1 along with their Kamlet-Taft acidity (α).

Entry
Dashed line represents Equation (7). Symbols of HBD solvents are given in Table 1.  Table 1. Dashed line represents Equation (7). Symbols of HBD solvents are given in Table 1. Figure 4 shows a flow chart developed in this work for predicting E T values of aqueous mixtures that is divided into five steps. In step 1, pure properties of electronic transition energy (E 0 T,i ) and gas-phase dipole moment (µ 2 ) of components 1 and 2 are compiled from the literature. In step 2, solvent characteristics of component 2 (cosolvent) are determined to check HBD ability by considering their molecular structures and KT-acidity. For example, HBD cosolvents are able to donate a proton so that their KT-acidity values are relatively high (Table 1). On the other hand, HBA cosolvents lack proton donor groups and, thus, the CF HBD value of HBA cosolvent is equal to unity. In step 3, CF HBD values of HBD cosolvent are calculated by either Figure 2 or Equation (7). In steps 4 and 5, the predictive model (Equation (2)) was used to estimate the E T values of aqueous mixtures and was validated, respectively. To validate the predictive model in step 5, nine aqueous mixtures of HBA cosolvents and nine aqueous mixtures of HBD cosolvents (Table 2) were used and discussed in Appl. Sci. 2020, 10, 8480 6 of 13 Section 3. Table 2 tabulates the E 0 T,i , µ 2 [29], Hunter basicity (β H ) of component 2 [30,31] and CF HBD values of solvents used in the predictions. molecular structures and KT-acidity. For example, HBD cosolvents are able to donate a proton so that their KT-acidity values are relatively high (Table 1). On the other hand, HBA cosolvents lack proton donor groups and, thus, the CF HBD value of HBA cosolvent is equal to unity. In step 3, CF HBD values of HBD cosolvent are calculated by either Figure 2 or Equation (7). In steps 4 and 5, the predictive model (Equation (2)) was used to estimate the ET values of aqueous mixtures and was validated, respectively. To validate the predictive model in step 5, nine aqueous mixtures of HBA cosolvents and nine aqueous mixtures of HBD cosolvents (Table 2) were used and discussed in Section 3. Table  2 tabulates the 0 T,i E , µ2 [29], Hunter basicity (β H ) of component 2 [30,31] and CF HBD values of solvents used in the predictions.

Evaluation of the Frameworks
Average relative deviation (ARD) was used to evaluate a deviation between experimental ( Exp T E ) and calculated ( C al T E ) data as shown in Equation (8):  (2)) and ideal model (Equation (1) (2)) and ideal model (Equation (1)) for nine aqueous mixtures of hydrogen bond acceptor (HBA) cosolvents and nine aqueous mixtures of hydrogen bond donor (HBD) cosolvents at 25 • C that were obtained with Reichardt indicator. Average relative deviations (ARD) were calculated with Equation (8) along with dipole moment of component 2 (µ 2 ), HBD contribution factor (CF HBD ), Hunter basicity (β H ) of component 2, pure properties of electronic transition energy (E 0 T,i ) and the number of data used (N). Gray-shaded rows indicate the ARD results estimated from the calculated CF HBD (Equation (7)).

Entry
Component 2

Evaluation of the Frameworks
Average relative deviation (ARD) was used to evaluate a deviation between experimental (E Exp T ) and calculated (E Cal T ) data as shown in Equation (8): Figure 5 shows the E T values of nine aqueous mixtures of water (1)-HBA cosolvent (2) as a function of component 2 (HBA cosolvent). The E T values of all aqueous mixtures ( Figure 5) exhibited a negative deviation from the ideality (dashed lines, Figure 5), except for the aqueous mixtures of acetonitrile (ACN), tetrahydrofuran (THF) and gamma-butyrolactone (GBL) that showed a sigmoid function (Figure 5a-c). Blue solid lines ( Figure 5) show predicted E T values of the aqueous mixture using Equation (2) without CF HBD value (CF HBD = 1) that generally tended toward the experimental data (symbols, Figure 5). Table 2 shows a comparison of ARD values obtained between the ideal model (Equation (1)) and predictive model (Equation (2)) for the aqueous mixtures of HBA cosolvents (entries 1-9, Table 2) and HBD cosolvent (entries 10-18, Table 2). The predictive model (Equation (2)) generally provided a lower ARD value (2.7%, entries 1-9, Table 2) than that estimated from the ideal model (5.4%). However, the model gave a relatively high ARD value (entries 1-3, Table 2) for the aqueous mixtures that showed the sigmoid functions (Figure 5a-c). These results inferred a limitation of the predictive model and were discussed later in Section 3.4. The predictive results of aqueous mixtures of HBD cosolvents were discussed in the following section.

Prediction for Aqueous Mixtures of HBA Cosolvents
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 and HBD cosolvent (entries 10-18, Table 2). The predictive model (Equation (2)) generally provided a lower ARD value (2.7%, entries 1-9, Table 2) than that estimated from the ideal model (5.4%). However, the model gave a relatively high ARD value (entries 1-3, Table 2) for the aqueous mixtures that showed the sigmoid functions (Figure 5a-c). These results inferred a limitation of the predictive model and were discussed later in Section 3.4. The predictive results of aqueous mixtures of HBD cosolvents were discussed in the following section.  Table 2 (entries 1-9). Black dashed lines show calculations from the ideal model (Equation (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering HBD contribution factor (CF HBD = 1). Figure 6 shows the ET values of nine aqueous mixtures of water (1)-HBD solvent (2) as a function of component 2 (HBD solvent) that also exhibited a negative deviation from the ideality. A comparison in predictive model (Equation (2)) with considering CF HBD value (Table 1) and without considering CF HBD value (CF HBD = 1). Red solid lines in Figure 6 show predictions with considering CF HBD value, while the blue ones are the predictions without considering CF HBD value.

Prediction for Aqueous Mixtures of HBD Cosolvents
The predictive model with the addition of CF HBD values (red solid lines, Figure 6) could provide a better result in the calculated ET value that followed the experimental data than the result without the CF HBD value (blue solid lines, Figure 6). The ARD values obtained from the predictive model (Equation (2)) with considering CF HBD value (1.6%, entries 10-18, Table 2) are lower than those estimated from the ideal model (4.8%). Figure 7 shows parity plots of calculated ET values that are estimated from the predictive model ( Figure 7a) and the ideal model (Figure 7b). The estimated ET values from the predictive model (R 2 = 0.91, Figure 7a) are less scattered than those obtained from the ideal model (R 2 = 0.81, Figure 7b).  Table 2 (entries 1-9). Black dashed lines show calculations from the ideal model (Equation (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering HBD contribution factor (CF HBD = 1). Figure 6 shows the E T values of nine aqueous mixtures of water (1)-HBD solvent (2) as a function of component 2 (HBD solvent) that also exhibited a negative deviation from the ideality. A comparison in predictive model (Equation (2)) with considering CF HBD value (Table 1) and without considering CF HBD value (CF HBD = 1). Red solid lines in Figure 6 show predictions with considering CF HBD value, while the blue ones are the predictions without considering CF HBD value.

Prediction for Aqueous Mixtures of HBD Cosolvents
The predictive model with the addition of CF HBD values (red solid lines, Figure 6) could provide a better result in the calculated E T value that followed the experimental data than the result without the CF HBD value (blue solid lines, Figure 6). The ARD values obtained from the predictive model (Equation (2)) with considering CF HBD value (1.6%, entries 10-18, Table 2) are lower than those estimated from the ideal model (4.8%). Figure 7 shows parity plots of calculated E T values that are estimated from the predictive model ( Figure 7a) and the ideal model (Figure 7b). The estimated E T values from the predictive model (R 2 = 0.91, Figure 7a) are less scattered than those obtained from the ideal model (R 2 = 0.81, Figure 7b).  Table 2 (entries 10-18). Black dashed lines show calculations from the ideal model (Equation (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering HBD contribution factor (CF HBD = 1). Red solid lines show prediction with the predictive model (Equation (2)) with considering actual CF HBD values in Table 1. Green solid lines show prediction with the predictive model (Equation (2)) with considering calculated CF HBD values using Equation (7).

Evaluation of CF HBD Methods for Predictions
In Section 3.2, the predicted ET values of aqueous mixtures of HBD cosolvents were estimated based on the actual CF HBD values evaluated from Figure 2 and Table 1. In the section, a comparison in evaluated ET values between actual data (Table 1) and calculations with KT-acidity (Equation (7)) was made for predictions. Red solid lines ( Figure 6) show predicted ET values of the mixtures with actual CF HBD values, while the green solid lines ( Figure 6) show predictions using calculated CF HBD values from the correlation (Equation (7)). The predictive model with the calculated CF HBD values provided a similar trend in predictions with the actual CF HBD values (Figure 6), in which the ARD values from the calculated CF HBD (gray-shaded rows, Table 2) are comparable to those estimated from the actual ones (entries 10-18, Table 2).

Limitation of the Predictive Model
According to predictive results in Section 3.1, the model (Equation (2)) provided a high ARD value for aqueous mixtures having a sigmoid function (entries 1-3, Table 2). Marcus reported [33,34] that the sigmoid trends were found in aqueous mixtures when microheterogeneity occurred. It is expected that the microheterogeneity phenomenon caused a high error in predictions because this phenomenon was not considered in the development of the model as mentioned in Section 2.2. Hunter basicity of HBA solvents (β H , Table 2) can quantify the hydration shell strength in aqueous mixtures [35]. It was found that the sigmoid behaviors occurred in the aqueous mixtures that have low β H values (β H ≤ 5.3, entries 1-3, Table 2). Thus, the model (Equation (2)) is effective for predicting the aqueous mixtures having high β H values (β H ≥ 5.3).  Table 2 (entries 10-18). Black dashed lines show calculations from the ideal model (Equation (1)). Blue solid lines show predictions with the predictive model (Equation (2)) without considering HBD contribution factor (CF HBD = 1). Red solid lines show prediction with the predictive model (Equation (2)) with considering actual CF HBD values in Table 1. Green solid lines show prediction with the predictive model (Equation (2)) with considering calculated CF HBD values using Equation (7).  Table 2.

Conclusions
In this work, a predictive model is proposed for estimating the solvatochromic polarity of electronic transition energy (ET) for aqueous mixtures. The function form for ET values for aqueous mixtures can be adopted from the previous model for nonpolar-polar mixtures due to similar interactions and trends in the mixture ET values in both systems. The predictive model was validated by eighteen aqueous mixtures and was found to give a reliable ET value with an overall deviation of  Table 2.

Evaluation of CF HBD Methods for Predictions
In Section 3.2, the predicted E T values of aqueous mixtures of HBD cosolvents were estimated based on the actual CF HBD values evaluated from Figure 2 and Table 1. In the section, a comparison in evaluated E T values between actual data (Table 1) and calculations with KT-acidity (Equation (7)) was made for predictions. Red solid lines ( Figure 6) show predicted E T values of the mixtures with actual CF HBD values, while the green solid lines (Figure 6) show predictions using calculated CF HBD values from the correlation (Equation (7)). The predictive model with the calculated CF HBD values provided a similar trend in predictions with the actual CF HBD values (Figure 6), in which the ARD values from the calculated CF HBD (gray-shaded rows, Table 2) are comparable to those estimated from the actual ones (entries 10-18, Table 2).

Limitation of the Predictive Model
According to predictive results in Section 3.1, the model (Equation (2)) provided a high ARD value for aqueous mixtures having a sigmoid function (entries 1-3, Table 2). Marcus reported [33,34] that the sigmoid trends were found in aqueous mixtures when microheterogeneity occurred. It is expected that the microheterogeneity phenomenon caused a high error in predictions because this phenomenon was not considered in the development of the model as mentioned in Section 2.2. Hunter basicity of HBA solvents (β H , Table 2) can quantify the hydration shell strength in aqueous mixtures [35]. It was found that the sigmoid behaviors occurred in the aqueous mixtures that have low β H values (β H ≤ 5.3, entries 1-3, Table 2). Thus, the model (Equation (2)) is effective for predicting the aqueous mixtures having high β H values (β H ≥ 5.3).

Conclusions
In this work, a predictive model is proposed for estimating the solvatochromic polarity of electronic transition energy (E T ) for aqueous mixtures. The function form for E T values for aqueous mixtures can be adopted from the previous model for nonpolar-polar mixtures due to similar interactions and trends in the mixture E T values in both systems. The predictive model was validated by eighteen aqueous mixtures and was found to give a reliable E T value with an overall deviation of 2.1%. Three properties of pure components are basically needed for predictions: (i) E T values, (ii) gas-phase dipole moment and (iii) Kamlet-Taft acidity.