The IKBI approach depends on obtaining the Kirkwood-Buff integrals as follows ^[17-22]

Here G _1,3 and G _2,3 are the Kirkwood-Buff integrals (in cm³ mol^-1) as obtained from the thermodynamic data: the isothermal compressibility of the solvent mixtures, κ _τ (in GPa^-1), and standard partial molar volume of solute in this mixture, V₃ (in cm³ mol^-1). The functions.

D and Q (in kJ mol^-1, as is RT) are given in equations (5) and (6), and depend on the first derivative of standard molar Gibbs energies of transfer, ∆_tr G ^o, and the second derivative of the excess Gibbs energy of mixing of the two solvents, G ^E ₁₊₂ , with respect to the composition ^[23,24].

A final quantity that is required is the correlation volume around 3, V _œr , within which preferential solvation takes place and the local mole fraction, x^L _1,3, is defined ^[18,25].

This volume involves one solvation shell and depends on the size of the solute molecule (its radius, r₃ in nm, equation 8) plus a distance of one mean solvent diameter from the surface of the solute.

The resulting correlation volume, in cm³ mol^-1, the numerical coefficients of which relate sizes to volumes, requires iteration, since it involves the local mole fractions of the two solvents.

Finally, the resulting preferential solvation parameter is then:

However, the correlation volume requires iteration, because it depends on the local mole fractions ^[19,26,27].

Standard molar Gibbs energy of transfer of tricin from neat water to {ethanol (1) + water (2)} mixtures is calculated and correlated to a non-regular polynomial from the drug solubility data by using equation (10). Figure 3 shows the Gibbs energy of transfer behavior at 323.15 K. Tus, the coefficients of the polynomials are shown in Table 2.

Table 2

Figure 3

Thus D values are calculated from the first derivative of polynomial models (Equation 11) solved according to the co-solvent mixtures composition. This procedure was done varying by 0.05 in mole fraction of methanol but in the following tables the respective values are reported varying only by 0.10.

Q and RT κ _τ values for {ethanol (1) + water (2)} binary mixtures, as well as the partial molar volumes of ethanol (1) and water (2), at the three temperatures considered here, were taken from the literature ^[22,38,39].

Otherwise, partial molar volumes of nonelectrolyte drugs are not frequently reported in the literature. This is because of the large uncertainty obtained in its determination due to their low solubilities, in particular in aqueous media. For this reason, in the first approach, the molar volume of tricin was considered as independent of co-solvent composition and temperature, as they are calculated according to the groups contribution method proposed by Fedors ^[29]. On the other hand, the radius of the drug molecule was calculated by using equation 8, as r₃ = 0.407 nm.

In order to apply the IKBI method, the correlation volume was iterated three times by using the equations (1), (7) and (9) to obtain the final values reported in Table 3. This property is almost independent on temperature in water-rich mixtures but increases to some extent in ethanol-rich mixtures. This would be expectable according to the variation of the respective molar expansibilities with the mixtures composition ^[40].

Table 3

X1	293.15 K	298.15 K	303.15 K	308.15 K	313.15 K
0.00	790.74	791.26	791.60	792.09	792.87
0.10	814.11	810.66	808.86	812.33	813.15
0.20	938.40	939.33	941.50	944.50	946.85
0.30	1033.69	1036.60	1039.18	1041.04	1043.50
0.40	1115.76	1118.16	1119.75	1122.04	1124.45
0.50	1196.96	1197.75	1198.42	1202.63	1205.53
0.60	1281.26	1280.71	1281.36	1288.19	1292.30
0.70	1363.04	1364.41	1367.61	1376.55	1383.36
0.80	1428.86	1435.71	1443.43	1451.91	1462.05
0.90	1485.75	1493.95	1501.84	1507.52	1516.46
1.00	1560.82	1566.06	1571.22	1576.68	1582.68

The preferential solvation parameter, δx _1,3 for ethanol (1) around tricin (3) is displayed in Fig. 4 at five temperatures. The values of δx ₁₃ vary non-linearly with the proportion of ethanol (1) in the alcoholic mixtures (figure 4). The addition of ethanol to water causes a negative change in δx _1,3 from pure water (2) up to the 0.20 in molar fraction of ethanol (1) reaching minimum values near to -0.08 at 0.05 in molar fraction of ethanol at 303.15 K. In this composition, water is preferred over ethanol around the tricin (3), this because possibly the structuring of water molecules around the non-polar groups of this drug leading to hydrophobic hydration of the aromatic and methyl groups (Fig.1), contributes to lowering of the net δx _1,3 to negative values in these water-rich mixtures. This behavior is observed at all study temperatures as can be seen in Table 4. Although, a clear influence of the temperature in water-rich mixtures is not perceived, in ethanol-rich mixtures the solvation by ethanol increases with the increase in temperature what matches the solubility profile.

Figure 4

Table 4

X 1	293.15 K	298.15 K	303.15 K	308.15 K	313.15 K
0.00	0.000	0.000	0.000	0.000	0.000
0.10	-0.070	-0.075	-0.078	-0.075	-0.075
0.20	-0.012	-0.013	-0.013	-0.012	-0.012
0.30	0.010	0.011	0.010	0.009	0.009
0.40	0.016	0.015	0.013	0.012	0.011
0.50	0.022	0.019	0.015	0.016	0.015
0.60	0.034	0.027	0.023	0.028	0.027
0.70	0.043	0.038	0.037	0.044	0.046
0.80	0.032	0.034	0.039	0.044	0.051
0.90	0.009	0.014	0.018	0.019	0.023
1.00	0.000	0.000	0.000	0.000	0.000

In the mixtures with composition 0.25 < x₁ < 1.00, the local mole fraction of ethanol is greater than the one in the bulk and it increases with the temperature increasing. In this way, the co-solvent action may be related to the breaking of the ordered structure of water (hydrogen bonds) around the nonpolar moieties of the drug which increases the solvation of the tricin and has a maximum value near to x₁ = 0.75, i.e. (δx _1,3 = 0.052. On the other hand, although, as mentioned above, the maximum solubility was not achieved in a cosolvent mixture, whit would be expected theoretically, the maximum solvation is achieved in mixtures with polarities similar to those of tricin (3), this factor could also be involved in these behaviors.

The tricin (3) could act in solution as Lewis acids due to the hydrogen atom in their -OH groups in order to establish hydrogen bonds with proton-acceptor functional groups in EtOH and water (oxygen atom in -OH). In addition, these drugs could act as Lewis bases due to free electron pairs in oxygen atoms of hydroxyl and alkoxy groups to interact with hydrogen atoms present in both solvents. In this context, tricin has three hydrogen-bonding donor and seven hydrogen-bonding acceptor groups.

According to the preferential solvation results, it is conjecturable that in intermediate composition mixtures and ethanol-rich mixtures, tricin (3) is acting as Lewis acids with the ethanol molecules because this co-solvent is more basic than water as indicated by the Kamlet-Taft hydrogen bond acceptor parameters (β), i.e. 0.75 for ethanol (1) and 0.47 for water (2) ^[41,42]. In this way, tricin would prefer ethanol (1) instead of water. This behavior is similar to that presented by other drugs in aqueous cosolvent systems ^[27,43,44].