Effect of Temperature Dependence of Deformation Polarizability and Ionization Energy of Solvents on Surface Properties of Solid Materials

1. Introduction

In a previous paper [1], an original method was proposed to determine the surface properties of solid surfaces using the London dispersion interaction energy between model organic solvents adsorbed on solid materials. This new method led to an accurate separation between the London dispersive and polar energies of the adsorbed molecules. This allowed to the correction of the Lewis acid-base parameters of solid surfaces relatively to the classic models or methods. The new separation of London dispersive and polar energy is of great importance to predicting the various surface physicochemical properties of materials and nanomaterials [1]. Inverse gas chromatography (IGC) technique at infinite dilution was used to quantify the dispersive and polar free energies using the retention volume of adsorbed solvents on solid materials [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. The free energy of adsorption $∆G_a^0\left(T\right)$ was determined as a function of temperature. The proposed method using the London equation proved the highest superiority relative to the classic chromatographic methods based on various thermodynamic parameters such as the boiling point $T_{B.P.}$ [2], the vapor pressure $P_0$ [3,4], the London dispersive surface energy ${\gamma }_l^d$ [8], the topological index the vapor pressure $P_0$ [6,7], enthalpy of vaporization $∆H_{vap.}^0$ or the deformation polarizability ${\alpha }_{0,L}$ [5] of organic solvents.

Several previous papers [1,8,13,14,15,16,17,26,27,34] were devoted to the correction of the surface properties of solid materials such as the London dispersive energy, the polar free energy, and the Lewis acid-base constants using the thermal model that proved the temperature effect on the surface area of organic molecules.

However, the recent paper [1] introducing the London equation to separate the London dispersive and polar energy used ${\mathcal{P}}_{SX}$ as a chromatographic parameter. This thermodynamic parameter was replaced by a new parameter given by:

$${\mathcal{P}}_{SX}={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\alpha }_{0S}{\alpha }_{0X}}{{\left(4\pi {\epsilon }_0\right)}^2}}{\displaystyle \frac{{\epsilon }_S{\epsilon }_X}{\left({\epsilon }_S+{\epsilon }_X\right)}}$$

(1)

where ${\alpha }_{0S}$and ${\alpha }_{0X}$ are the respective deformation polarizabilities of solid $H$ and solvents $X$ separated by a distance $H$, and ${\epsilon }_S$ and ${\epsilon }_X$ are the ionization energies of solid and solvent. The deformation polarizabilities and ionization energies of solid and solvents were supposed independent from the temperature.

We proposed in this work to determine the surface properties of solid materials such as alumina, titania, and magnesium oxide by studying the effect of temperature dependence of deformation polarizabilities and ionization energies of solids and solvents on the London dispersive and polar energy, the Lewis acid-base parameters of the above solid surfaces, and consequently on their Lewis acid and base surface energies.

2. Materials and Methods

The solid materials and organic solvents were used in previous studies [1,16,17] applying chromatographic methods and models. The non-polar organic solvents were n-hexane, n-heptane, n-octane, and n-nonane, whereas the polar molecules were dichloromethane, chloroform, carbon tetrachloride, benzene, ethyl acetate, acetone, tetrahydrofuran, acetone, toluene, and acetonitrile. The solid materials were alumina (Al₂O₃), magnesium oxide (MgO), and titania (TiO₂) and previously characterized [1]. The net retention time of organic solvents adsorbed on the different solid surfaces was determined at different temperatures using inverse gas chromatography (IGC at infinite dilution with the help of a Focus GC gas chromatograph equipped with a flame ionization detector of high sensitivity (Sigma-Aldrich, Paris, France). A mass of 1 g of solid particles was packed into a stainless-steel column of a length of 30 cm and 2 mm internal diameter. Helium was used as carrier gas with a flow rate equal to 25 mL/min. The retention times of the different injected organic solvents were measured at infinite dilution, supposing that there is no interaction between the probe molecules themselves. The column temperatures varied from 30 to 200 °C. Average retention times and volumes were determined by repeating each solvent injection three times with a standard deviation less than 1% in all chromatographic measurements.

The IGC technique [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46] allowed to characterize the surface properties of the various solid surfaces with the help of the net retention volume of the various solvents adsorbed on the solid materials. This allowed us to obtain the free energy of adsorption $∆G_a^0$ of the adsorbed molecules by using the following fundamental equation of IGC:

$$∆G_a^0\left(T\right)=-RTlnVn+C\left(T\right)$$

(2)

Where $V_n$ is the net retention volume of a probe, $T$ the absolute temperature, $R$ the perfect gas constant, and $C\left(T\right)$ a constant depending on the temperature and the parameters of interaction between the solid and the solvent given by:

$$C\left(T\right)=RTln\left({\displaystyle \frac{sm{\pi }_0}{P_0}}\right)$$

(3)

where $m$ the mass of the solid particles, $s$ the specific surface area of solid surfaces, and $P_0$ the reference pressure and ${\pi }_0$ two-dimensional pressure given in the literature by one of the following reference states:

-

Kemball and Rideal reference state [47] given for $T_0=0$ºC by $P_0=1.013\times {10}^{5}Pa$ and ${\pi }_0=6.08\times {10}^{-5}N{m}^{-1}$.

-

De Boer et al. reference state [48] given for $T_0=0$ºC by $P_0=1.013\times {10}^{5}Pa$ and ${\pi }_0=3.38\times {10}^{-5}N{m}^{-1}$.

The total free energy of adsorption $∆G_a^0\left(T\right)$ is composed of the respective London dispersive energy $∆G_a^d\left(T\right)$ and polar energy $∆G_a^p\left(T\right)$:

$$∆G_a^0\left(T\right)=∆G_a^d\left(T\right)+∆G_a^p\left(T\right)$$

(4)

In a recent study, an original method based on the expression of the London dispersion interaction was proposed. The London dispersion equation [1] was used for the determination of the free dispersive energy $-∆G_a^d\left(T\right)$ and the fundamental equation is written as:

$$∆G_a^d\left(T\right)=-{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\alpha }_{01}{\alpha }_{02}}{{\left(4\pi {\epsilon }_0\right)}^2{H}^6}}{\displaystyle \frac{{\mathcal{N}\epsilon }_1{\epsilon }_2}{\left({\epsilon }_1+{\epsilon }_2\right)}}$$

(5)

where ${\alpha }_{01}$and ${\alpha }_{02}$ are the respective deformation polarizabilities of Molecules 1 and 2 separated by a distance $H$, ${\epsilon }_1$ and ${\epsilon }_2$ are the ionization energies of Molecules 1 and 2, and ${\nu }_1$ and ${\nu }_2$ are their characteristic electronic frequencies.

In the case of adsorption of organic solvents on solid materials, the solid molecule (Molecule 1) was denoted $S$ and the probe molecule (Molecule 2) denoted by X and combining the previous equations. The free energy of adsorption $∆G_a^0\left(T\right)$ can be written as:

$$∆G_a^0\left(T\right)=-RTlnVn+C\left(T\right)=-{\displaystyle \frac{{\alpha }_{0S}}{H^6}}\left[{\displaystyle \frac{3\mathcal{N}}{{2\left(4\pi {\epsilon }_0\right)}^2}}\left({\displaystyle \frac{{\epsilon }_S{\epsilon }_X}{\left({\epsilon }_S+{\epsilon }_X\right)}}{\alpha }_{0X}\right)\right]+∆G_a^{sp}\left(T\right)$$

(6)

A new thermodynamic parameter ${\mathcal{P}}_{SX}$ was proposed as new chromatographic indicator variable given by:

$${\mathcal{P}}_{SX}={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\alpha }_{0S}{\alpha }_{0X}}{{\left(4\pi {\epsilon }_0\right)}^2}}{\displaystyle \frac{{\epsilon }_S{\epsilon }_X}{\left({\epsilon }_S+{\epsilon }_X\right)}}$$

In the previous studies [1,26,27], the ionization energy and deformation polarizability of solid and solvents were supposed constants independent from the temperature. Even if the variations of these variables slightly vary versus the temperature, the temperature effect of these parameters on the surface properties of solid materials was investigated in this paper.

3. Results

The new method was based on the temperature effect on the chromatographic parameter ${\mathcal{P}}_{SX}$ of alumina, titania, and magnesium oxide. Table 1 gave the variations of ${\mathcal{P}}_{SX}\left(T\right)$ as a function of temperature of the adsorbed organic molecules on solid materials. It was observed a slight variation of ${\mathcal{P}}_{SX}\left(T\right)$ versus the temperature. However, there is an important variation of ${\mathcal{P}}_{SX}\left(T\right)$ depending on both solvents and solid surfaces. This was more elucidated in Table 2 giving the equations ${\mathcal{P}}_{SX}\left(T\right)$ of the different solvents with the extrapolated values of ${\mathcal{P}}_{SX}\left(0\mathrm{K}\right)$ at 0K. ${\mathcal{P}}_{SX}\left(0\mathrm{K}\right)$ largely varied from solvent to another and from solid to solid. The slope ${d\mathcal{P}}_{SX}\left(T\right)/dT$ which is equal to the derivative of ${\mathcal{P}}_{SX}$ with respect of temperature represents a thermal expansion coefficient.

The results given in Table 1 and Table 2 were used to determine the polar free energy $∆G_a^p\left(T\right)$ of the adsorbed organic solvents on solid materials. The new values of $∆G_a^p\left(T\right)$ of the various solvents adsorbed on alumina, titania, and MgO were given in Table 3 as a function of temperature.

Table 3 showed that the lowest values of free polar interaction $∆G_a^p\left(T\right)$ were obtained with the titanium dioxide, whereas MgO gave the highest $∆G_a^p\left(T\right)$. However, the values of $∆G_a^p\left(T\right)$ relative alumina are not so far from the those of magnesium oxide.

The determined free energy of adsorption of the polar solvents in Table 3 showed closest values for MgO and alumina very larger than those of titania then proving higher polar interaction for alumina and MgO.

The results showed in Table 3 were compared to those previously obtained without considering the thermal effect on the ionization energy and deformation polarizability [1]. It was observed in Table 4 an important deviation between the results of the two methods varying from 7% to 2665% (in the case of THF adsorbed on titania).

Serious consequences resulted from the above results leading to a higher disparity in the values of other surface thermodynamic parameters, particularly on the polar enthalpy and entropy of adsorption, and Lewis acid-base parameters of the solid substrates.

The polar enthalpy $\left(-∆H_a^p\right)$ and entropy $\left(-∆S_a^p\right)$ of solvents adsorbed on solid surfaces were obtained from the variations of the free energy of adsorption against the temperature using the following relation:

$$∆G_a^p\left(T\right)=∆H_a^p-T∆S_a^p$$

(7)

The values of the above thermodynamic variables were given in Table 5 compared to the previous results obtained without taking into account the thermal effect on the ionization energy and deformation polarizability of solvents.

The results in Table 5 led to the Lewis enthalpic acid–base constants $K_A$ and $K_D$ using the empirical relation (8):

$$-∆H^p=K_A\times DN+K_D\times AN$$

(8)

where $AN$ and $DN$ are, respectively, the electron donor and acceptor numbers of the polar molecule [45,46].

The values of $K_A$ and $K_D$ of solids were deduced by drawing the variations of $\left({\displaystyle \frac{-∆H^{Sp}}{AN}}\right)$ versus $\left({\displaystyle \frac{DN}{AN}}\right)$ of polar solvents using Equation (9):

$$\left(-{\displaystyle \frac{∆H^{Sp}}{AN}}\right)=K_A\left({\displaystyle \frac{DN}{AN}}\right)+K_D$$

(9)

The same procedure was used for the determination of the Lewis entropic acidic ${\omega }_A$ and basic ${\omega }_D$ constants of the various solid surfaces using Equations (10) or 11.

$$\left(-∆S_a^{sp}\right)={\omega }_A{DN}^{\text{'}}+{\omega }_D{AN}^{\text{'}}$$

(10)

$$\left({\displaystyle \frac{-∆S_a^{sp}}{{AN}^{\text{'}}}}\right)={\omega }_A\left({\displaystyle \frac{{DN}^{\text{'}}}{{AN}^{\text{'}}}}\right)+{\omega }_D$$

(11)

The Lewis enthalpic and entropic acid-base parameters were shown in Table 6 and compared to the previous results.

The results obtained showed the three solid materials exhibited an amphoteric character with higher basicity. Alumina proved the highest enthalpic and entropic Lewis basic and acidic constants followed by MgO while titania had the lowest Lewis acid and basic constants. It was observed that the Lewis acid-base parameters of MgO and alumina were very close, whereas titania presented $K_A$ and $K_D$ three times lower than those of alumina and MgO. The comparison with the previous results [1] showed comparable values of $K_A$ and $K_D$ in the case of alumina and titania but very different values for MgO surfaces. However, this deviation increased for the entropic acid-base constants ${\omega }_A$ and ${\omega }_D$ between the two methods. The new method gave more accurate quantification of the surface properties of solid materials.

4. Discussion

The temperature effect on the ionization energy and deformation polarizability of solvents adsorbed on alumina, titania, and MgO led to important variations in the surface thermodynamic properties, and especially, in polar energy and Lewis acid-base constants of solid surfaces. A correction of the surface properties of these materials relative to the previous method was carried out highlighting the effect of temperature on the thermodynamic parameters strongly affecting the Lewis acid-base properties of solid surfaces.

The original consequence of this new approach was the determination of the intermolecular distance $H\left(T\right)$ between the organic solvents and the solid materials as a function of temperature. Indeed, using Equation (4) the London dispersive free energy $-∆G_a^p\left(T\right)$ of adsorption of solvents on the different solid surfaces was obtained against the temperature and led to the values of the intermolecular distance from the following Equation:

$$\left(-∆G_a^d\left(T\right)\right)={\displaystyle \frac{{\mathcal{P}}_{SX}\left(T\right)}{{H\left(T\right)}^6}}$$

(12)

And $H\left(T\right)$ was determined as a function of temperature from Equation (13):

$$H\left(T\right)={\left[{\displaystyle \frac{{\mathcal{P}}_{SX}\left(T\right)}{\left(-∆G_a^d\left(T\right)\right)}}\right]}^{1/6}$$

(13)

The variations of $H\left(T\right)$ between the solvents and the solid substrates shown in Table 7 highlighted an effect of temperature on the intermolecular distance with increasing tendency of $H\left(T\right)$ as the temperature increased for n-alkanes while a decrease of $H\left(T\right)$ was observed for polar solvents. The results in Table 7 also showed large differences of the values of the intermolecular distance strongly depending on the polarity of solid surfaces. Indeed, $H\left(T\right)$ of different solvents was the lowest with alumina followed by MgO, while the highest values were observed with titania then confirming the results obtained with the Lewis acid-base properties of solid materials where alumina was proved to have the highest acid-base constants leading to lowest values of $H\left(T\right)$ due to the strong van der Waals interaction.

5. Conclusions

The surface properties of solid surfaces such as alumina, titania, and magnesium oxide were determined using a new method based on the effect temperature on ionization energy and deformation polarizability of solvents and consequently on the different surface thermodynamic parameters of solid materials. Even if a slight variation of ionization energy and deformation polarizability of organic molecules was observed, however, an important difference in the surface properties of oxides was shown leading to different Lewis acid-base constants between the new thermal method and the previous method which supposed a neglected temperature effect on the polar and dispersive energy of adsorption of solvents on the solid surfaces. The large differences proved between the values of the intermolecular distance between the solvents and the various solid substrates confirmed the superiority of the new method.

These findings highlight the critical need to account for temperature-dependent electronic and polarizability effects when evaluating surface reactivity, adhesion, and interfacial interactions. From a broader materials science perspective, this work establishes a foundational link between molecular-scale properties of probe molecules and macroscopic surface behaviors, offering new insights for the rational design of functional materials, surface coatings, and nanostructured interfaces with tailored thermodynamic and interfacial properties.