*3.2. Volumetric Properties*

## 3.2.1. Excess Molar Volume

The excess molar volumes (*VE*) were calculated using the experimental density data according to the following equation

$$V^{E} = \frac{\mathbf{x}\_1 M\_1 + \mathbf{x}\_2 M\_2}{\rho} - \frac{\mathbf{x}\_1 M\_1}{\rho\_1} - \frac{\mathbf{x}\_2 M\_2}{\rho\_2} \tag{5}$$

where *d* is the density of the mixture and *xi, Mi*, and *ρ<sup>i</sup>* are: the mole fraction, the molar mass and density of DES (*i* = 1) and water (*i* = 2), respectively.

The obtained values of *VE* are presented in Tables S1–S4 and plotted as a function of the DES mole fraction, *x*1, in Figure 2. Figure 2a shows the plots of *V<sup>E</sup>* against mole fraction of DES for all studied mixtures at 298.15 K and Figure 2b depicts the temperature dependence of excess molar volumes for the system (TBAB:BAE + water) as an example. It can be observed in these figures that the curves of *VE* are asymmetrical and their values are negative over the whole composition and temperature range. The minimum was found between mol fraction of 0.35 and 0.4 of DES equal to −0.98 for DES1, −1.00 for DES2, −1.13 for DES3, −1.02 for DES4. The asymmetry of the curves is due to the difference between the molar volumes of the components mixture.

**Figure 2.** Dependence of the excess molar volume of aqueous solutions of DESs on molar fraction of deep eutectic solvent: (**a**) at 298.15 K for - DES1; • DES2; DES3; DES4; (**b**) for DES1 at 298.15 K (-); 298.15 K (•); 303.15 K (); 308.15 K (); 313.15 K (); —, Equation (6).

In Figure 2, the dashed lines represent the correlated values according to the Redlich– Kister polynomial [34]

$$V^E = \mathbf{x}\_1 \mathbf{x}\_2 \sum\_{i=0}^2 A\_i (\mathbf{x}\_1 - \mathbf{x}\_2)^i \tag{6}$$

where the *Ai* values are adjustable parameters.

As it can be seen, the calculated values agree very well with the experimental data. The *Ai* values were determined using the least squares method and they are listed in Tables S6–S9, along with their RMSD. For all systems, excess molar volumes were correlated using three-parameter Redlich–Kister polynomial equation.

The negative values of excess molar volumes can be explained based on the strength of the specific interactions, size, and shape of molecules. When DES is added to water, the intra-molecular interactions between DES or water molecules are disrupted and new hydrogen bonding interactions between water and chloride/bromide anion of HBA and between water and -OH group and -NH2 group or –NH of amino alcohol are forming. Moreover, water molecules—as much smaller than the deep eutectic solvent one—may

fit into the interstices of the DES. Therefore, the filling effect of water in the interstices of DES, and the strong hydrogen bonding interactions between the unlike components of the systems, all lead to the negative values of the excess molar volumes.

The temperature dependence of the excess molar volumes can determine what kind of effect—i.e., the packing phenomenon or the strong forces between the components is responsible for the negative values of *VE.* In general, as temperature increases, the specific interactions break down and due to the increased thermal fluctuation, more holes of sufficient size for the accommodation of the unlike component are formed. These effects influence the excess molar volume in a reverse manner. A decrease of specific interactions causes an increase in *V<sup>E</sup>* values, while a loosening of the DES structure leads to a decrease of excess molar volume with temperature. Thus, the observed increase of *V<sup>E</sup>* with rising temperature for all systems investigated suggests that specific interactions determine the volumetric behavior of aqueous solutions of deep eutectic solvents based on alcohol amine. A similar phenomenon was observed by other researchers for aqueous solutions of DES based on choline chloride [18,21,22,24,35–38] or allyltriphenylphosphonium bromide [19,20]. What is interesting, for the (DES + alcohol) systems, due to the decrease in the excess molar volume with temperature, the dominance of the packing effect was postulated [39,40].

The dominance of specific interactions in the aqueous solutions of the DESs studied can be confirmed by the Prigogine–Flory–Patterson (PFP) theory [41–45]. This theory has been originally used in interpreting the values of the excess molar volumes of binary systems formed by polar compounds which do not form strong electrostatic or hydrogen bond interactions. Over time, however, it has emerged that the use of the Flory formalism can still provide an interesting correlation between the excess volumes of more complex mixtures. So far, the PFP theory has been successfully applied to predict and model the excess molar volumes of many mixtures containing ionic liquids [46–50] and some aqueous systems with deep eutectic solvents [18,51,52].

According to the PFP theory, the excess molar volume contains three contributions: an interactional contribution, a free volume contribution, and a pressure contribution. The expression for *V<sup>E</sup>* is given as

$$\frac{V^{E}}{\mathbf{x}\_{1}V\_{1}^{\*} + \mathbf{x}\_{2}V\_{2}^{\*}} = \frac{\left(\tilde{V}^{1/3} - 1\right)\tilde{V}^{2/3}\psi\_{1}\Theta\_{2}\chi\_{12}}{\left[\left(4/3\right)\tilde{V}^{-1/3} - 1\right]P\_{1}^{\*}} + \frac{-\left(\tilde{V}\_{1} - \tilde{V}\_{2}\right)^{2}\left[\left(14/9\right)\tilde{V}^{-1/3} - 1\right]\psi\_{1}\Psi\_{2}}{\left[\left(4/3\right)\tilde{V}^{-1/3} - 1\right]\tilde{V}} + \frac{\left(\tilde{V}\_{1} - \tilde{V}\_{2}\right)\left(P\_{1}^{\*} - P\_{2}^{\*}\right)\psi\_{1}\Psi\_{2}}{P\_{2}^{\*}\psi\_{1} + P\_{1}^{\*}\psi\_{2}}\tag{7}$$

where *VE* is excess molar volume, *x* mole fraction, *V*\* characteristic volume, *P*\* characteristic pressure, *<sup>ψ</sup>* molecular contact energy fraction, *<sup>θ</sup>* molecular surface fraction, *<sup>V</sup>*\$ reduced volume, and *χ*<sup>12</sup> interactional parameter.

The reduced volume for pure substance *i* is defined in terms of the thermal expansion coefficients, *αi*, as

$$\mathcal{V}\_{i} = \left(\frac{1 + \frac{4}{3}a\_{i}T}{1 + a\_{i}T}\right)^{3} \tag{8}$$

The reduced volume of mixture, *<sup>V</sup>*\$, is calculated from

$$
\widetilde{V} = \psi\_1 \widetilde{V\_1} + \psi\_2 \widetilde{V\_2} \tag{9}
$$

where the molecular contact energy fraction, *<sup>ψ</sup>*, is expressed by: *<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>ψ</sup>*<sup>2</sup> <sup>=</sup> *<sup>φ</sup>*<sup>1</sup> *<sup>p</sup>*<sup>∗</sup> 1 *φ*<sup>1</sup> *p*∗ <sup>1</sup>+*φ*<sup>2</sup> *p*<sup>∗</sup> 2 with the hardcore volume fraction, *<sup>φ</sup>*, calculated from *<sup>φ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*1*V*<sup>∗</sup> 1 *x*1*V*∗ <sup>1</sup> +*x*2*V*<sup>∗</sup> .

2 The characteristic volume, *V*∗ *<sup>i</sup>* , is calculated from the molar volume from the expression *V*∗ *<sup>i</sup>* <sup>=</sup> *<sup>V</sup>*<sup>0</sup> *i V*\$*i* and the characteristic pressure is expressed by

$$p\_i^\* = \frac{\mathfrak{a}\_i}{\mathfrak{K}\_{T\bar{i}}} T \tilde{V}\_i^2 \tag{10}$$

where *κTi* is the isothermal compressibility obtained from the isentropic compressibility from the thermodynamic relation

$$\kappa\_{Ti} = \kappa\_{Si} + \frac{V\_i^0 \alpha\_i^2 T}{\mathcal{C}\_{pi}} \tag{11}$$

with the isobaric heat capacity, *Cpi*.

The molecular surface fraction of component 2 is given by: <sup>Θ</sup><sup>2</sup> = *<sup>φ</sup>*<sup>2</sup> *φ*1 *s*1 *<sup>s</sup>*<sup>2</sup> <sup>+</sup>*φ*<sup>2</sup> , in which the ratio of the surface contact sites per segment is given by

$$\frac{s\_1}{s\_2} = \left(\frac{v\_2^\*}{v\_1^\*}\right)^{\frac{1}{3}}\tag{12}$$

In present study, the thermal expansion coefficient, *αp*, defined as: *α<sup>p</sup>* = <sup>1</sup> *V* -*∂V ∂T P* , was calculated by use the temperature dependence of density, which was found to be the second-order polynomial equation.

The isobaric heat capacity of DESs was determined experimentally. Table S10 shows, for all pure DESs used in this work, the thermal expansion coefficient, the isobaric heat capacity and the Flory parameters necessary for the application of the PFP theory. The results of our experiments compare the Cp of DES1–DES4. Quite noticeable differences are observed. Generally, the *Cp* values [J·mol−<sup>1</sup> <sup>K</sup>−1] are arranged in order: DES 3 < DES 2 < DES 1 < DES 4 at set temperature points.

According to the PFP theory, for the separation of the values of excess molar volume into three contributions, the interactional parameter *χ*<sup>12</sup> must be found. In the present study, it has been done by minimalization of the objective function, considering deviations in the prediction of the excess volume, defined as

$$OF = \sum\_{i=1}^{n} \left( V\_{exp}^{E} - V\_{calc}^{E} \right)^{2} \tag{13}$$

In calculations, the value of the interactional parameter *χ*<sup>12</sup> was assumed to be independent on the composition of mixture. Figure 3 shows the composition dependence of calculated excess molar volume, together with the three contributions (*V<sup>E</sup> int*, *<sup>V</sup><sup>E</sup> f v*, *<sup>V</sup><sup>E</sup> <sup>P</sup>*<sup>∗</sup> ), compared with the experimental *VE* data for each system studied at 298.15 K.

Moreover, Table 3 reports the adjusted values of interactional parameter *χ*<sup>12</sup> and calculated three contributions to excess molar volume for the binary mixtures of deep eutectic solvents with water at all temperatures investigated and at *x*<sup>1</sup> = 0.4 together with RMSD.

Study of the data presented in Table 3 as well as an analysis of Figure 3 reveals that the interactional contribution is always negative and it seems to be the most important to explain the values of the excess molar volume. It decides about the sign and magnitude of the *VE* due to its greater value compared to the other two contributions for all investigated systems at all temperatures. The free volume contribution, which is a measure of geometrical accommodation, is negative but its magnitude is much smaller than for the interactional contributions. Therefore, it can be said that the PFP model confirms the conclusions resulting from the dependence of excess molar volume on the temperature, postulating little significance of the packing effect for the systems studied. The third contribution is the result of differences in internal pressure and in the reduced volumes of the components. It is positive, its magnitude is smaller than the interactional contribution and decreases distinctly with temperature.

**Figure 3.** The dependence of the excess molar volume of aqueous solutions of DESs on molar fraction of deep eutectic solvent at 298.15 K: (**a**) for DES1; (**b**) for DES2; (**c**) for DES3; (**d**) for DES4; (-) experimental data; (**—**) calculated using the PFP model; ( **... ... ...** ) interactional contribution (*V<sup>E</sup> int*); (**–––**), free volume contribution (*V<sup>E</sup> f v*); (**– –**), characteristic pressure contribution (*V<sup>E</sup> <sup>P</sup>*<sup>∗</sup> ).

**Table 3.** Calculated values of the interactional parameter χ12, root mean square deviation of fit RMSD, and the three contributions (*V<sup>E</sup> int*, *<sup>V</sup><sup>E</sup> f v*, *<sup>V</sup><sup>E</sup> <sup>P</sup>*<sup>∗</sup> ) from the PFP theory to the excess molar volumes for the binary mixtures of deep eutectic solvents with water at x1 = 0.4 and T = (293.15 − 313.15) K.



**Table 3.** *Cont.*

For deeper analysis of the obtained results, the percentage of the three contributions in excess molar volume was calculated. Figure 4 presents the obtained results at 298.15 K.

**Figure 4.** Percentages of the three contributions in excess molar volume of aqueous solutions of DESs for *x*<sup>1</sup> = 0.4 at 298.15 K.

As can be seen, the interactional contribution and the characteristic pressure contribution determine the order of the excess molar volume observed for the studied systems, which is as follows: TBAB:MAE (DES 3) < TBAB:BAE (DES 4) < TBAC:AP (DES 2) ≈ TBAB:AP (DES 1). The free volume fraction has practically no effect on the excess molar volume, and its absolute value increases with the length of the alkyl chain in the amino alcohol.

Moreover, the results show that the anion of the salt in DES does not practically effect on the value of excess molar volume. As depicted in Figures 3 and 4, almost identical interactional contribution, free volume contribution and characteristic pressure contribution are observed for aqueous solutions of TBAC:AP (DES 2) and TBAB:AP (DES 1).

Further analysis of Table 3 shows that the interactional contribution is mainly responsible for the increase of excess molar volume with increasing temperature. The decrease in its absolute value is greater than the decrease of the positive characteristic pressure contribution, and consequently the excess volumes of the studied systems increase with temperature. The percentage of free volume contribution increases with increasing temperature, but due to the low absolute values *V<sup>E</sup> int*, it has no influence on the excess molar volume or its dependence on temperature.

Summing up, it is evident from Figure 3 that the PFP theory predicts the experimental data satisfactorily. Thus, while the PFP theory does not take into account the strong interactions between components—such as electrostatic, hydrogen bonding, and complex formation—we can infer that the PFP model reproduces the main characteristics of the experimental data by using only one fitted parameter to describe excess molar volume.

## 3.2.2. Excess of Thermal Expansion

As the temperature dependence of density was found to be second order polynomial, type: ln(*ρ*) = *at*<sup>2</sup> + *bt* + *c*, the isobaric thermal expansion coefficients at different temperatures were derived according to the equation

$$\alpha\_p = \frac{1}{V} \left(\frac{\delta V}{\delta T}\right)\_P = -\frac{1}{\rho} \left(\frac{\delta \rho}{\delta T}\right)\_P = -\left(\frac{\delta ln\rho}{\delta T}\right)\_P = -(2a+b) \tag{14}$$

Then, excess thermal expansion, Δ*αp*, was calculated using the equation

$$
\Delta \alpha\_p = \alpha\_p - \sum\_{i=1}^n \Phi\_i \alpha\_{p,i} \tag{15}
$$

where Φ*<sup>i</sup>* is the volume fraction of pure component *i*, defined as *φ<sup>i</sup>* = *xiVi*/ ∑*<sup>i</sup> xiVi*,. The values of *α<sup>p</sup>* and Δ*α<sup>p</sup>* are given in Tables S1–S6 in Supporting Material and variation of the excess thermal expansion with DES mole fraction, *x*1, is plotted in Figure 5.

**Figure 5.** Dependence of the excess thermal expansion of aqueous solutions of DESs on molar fraction of deep eutectic solvent: (**a**) at 298.15 K for - DES1; • DES2; DES3; DES4; (**b**) for DES1 at 293.15 K (-); 298.15 K (•); 303.15 K (); 308.15 K (); 313.15 K (); —, Equation (6).

It can be seen that the values of excess thermal expansion are positive in the entire composition range for all systems studied, regardless of temperature. Since positive Δ*α<sup>p</sup>* are typical for the systems containing molecules capable to self-associate, the obtained results confirm strong hydrogen bonds between water molecules or between molecules of deep eutectic solvents, the strength of which decreases with temperature, as indicated by a reduction in excess thermal expansion with temperature [53]. Moreover, the less positive values of excess thermal expansion obtained for the (TBAB:BAE (DES 4) + water) system indicate the weakest hydrogen bond interactions between molecules of this DES compared to the others.
