New Physicochemical Methodology for the Determination of the Surface Thermodynamic Properties of Solid Particles

1. Introduction

In most industrial processes, such as, adhesion, adsorption, polymer synthesis, dispersion, food manufacturing, pharmaceutical drugs, biomedicine, clays, composites, materials and nanomaterials, it is necessary to determine the physicochemical, surfaces and interfaces properties of solid materials. One of the most famous techniques that give information on the surface properties of materials and nanomaterials is inverse gas chromatography (IGC). This technique had a large success to determine the surface physicochemical properties of materials such as the dispersive surface energy, the specific free energy of adsorption and the Lewis-acid base parameters. Using IGC technique allowed the observation of the interactions between oxides, polymers or polymers adsorbed on oxides and organic solvent systems [1]. It was proved that this technique is very precise, sensitive, and more competitive to study the heterogeneous surfaces of solid surfaces, their physicochemical properties [2], and to determine dispersive surface energy of powdered materials [3,4]. This attractive technique was used since 1970 to quantify the specific interactions and the surface properties and glass transition of polymers, copolymers, their blends, polymer films [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. It was also used to determine the surface energy, the physicochemical properties and the Lewis’s acid base of metals and metal oxides, minerals, clay minerals [19,20,21,22,23,24,25,26,27,28,29,30,31], silicas and porous materials [32,33,34,35,36,37,38,39], nanomaterials [40,41,42,43,44,45,46], pharmaceuticals and food products [47,48,49,50,51,52,53,54], and other materials [55,56,57,58,59,60,61,62,63,64,65,66,67,68].

In this paper we developed a new methodology for the determination of the physicochemical, dispersive and polar properties of alumina particles by using IGC technique at infinite dilution. We used all classical IGC methods and proposed to apply the new Hamieh’s thermal model [69] that was proved to be more accurate than the other models to determine the specific free enthalpy and enthalpy of adsorption and the acid-base constants of the different materials. Eight molecular models of organic molecules were also used to calculate the dispersive component of the surface energy of solid particles.

2. Methods and Models

2.1. Classical methods

Many methods were proposed in literature and used during the last fifty years. At the beginning, Sawyer and Brookman [70] found in 1968 an excellent linearity of the logarithm of the net retention volume $Vn$ of an adsorbed solvent on a solid, as a function of the boiling point $T_{B.P.}$ of n-alkanes $lnVn=f\left(T_{B.P.}\right)$. The separation method of the dispersive (or London) and polar (or specific) interactions between a solid substrate and a polar molecule was proposed by Saint-Flour and Papirer [12,13]. These authors used the representation of $RTlnVn$ versus the logarithm of the vapor pressure $P_0$ of probes:

$$RTlnVn=\alpha P_0+\beta$$

(1)

where R is the ideal gas constant, T the absolute temperature and $\alpha$ and $\beta$ constants depending on the interface solid-solvent. The distance relating the representative point of $RTlnVn$ of a polar molecule to its hypothetic point located on the n-alkane straight-line determined the specific free energy of adsorption $\Delta G_a^{sp}$. The variation of $\Delta G_a^{sp}$ versus the temperature led to the specific enthalpy $\Delta H_a^{sp}$ and entropy $\Delta S_a^{sp}$ of polar molecule adsorbed and therefore to the Lewis acid-base parameters. Five other IGC methods were proposed, to characterize the solid surfaces, a similar linearity to separate the two dispersive and polar components of the specific interactions. Two similar methods were used to determine the dispersive component ${\gamma }_s^d$ of the surface energy of the solid. These methods are given below:

1． Dorris and Gray [71] first determined ${\gamma }_s^d$ of solid materials by using Fowkes relation [72] and correlating the work of adhesion $W_a$ to the free energy of adsorption $\Delta G_a^0$ by the following relation:

$$\Delta G_a^0=\mathcal{N}a W_a=2\mathcal{N}a \sqrt{{\gamma }_l^d{\gamma }_s^d}$$

(2)

where a is the surface area of adsorbed molecule, ${\gamma }_l^d$ the dispersive component of the liquid solvent ${\gamma }_l^d$ and $\mathcal{N}$ the Avogadro’s number.

Dorris and Gray introduced the increment $\Delta G_{-CH2-}^0$ of two consecutive n-alkanes $C_n{H}_{2\left(n+1\right)}$ and $C_n{H}_{2\left(n+1\right)}$:

$$\Delta G_{-CH2-}^0=\Delta G^0\left(C_{n+1}{H}_{2\left(n+2\right)}\right)-\Delta G^0\left(C_n{H}_{2\left(n+1\right)}\right)$$

(3)

By supposing the surface area of methylene group, $a_{-CH2-}=6\text{Å}$, independent from the temperature and the surface energy ${\gamma }_{-CH2-}\left(\mathrm{in}\mathrm{mJ}/{\mathrm{m}}^2\right)$ of –CH2- equal to:

$${\gamma }_{-CH2-}=52.603–0.058T \left(T in K\right)$$

Dorris and Gray [71] then deduced the value of ${\gamma }_s^d$ by the equation (3):

$${\gamma }_s^d=\frac{{\left[RTln\left[\frac{V_n\left(C_{n+1}{H}_{2\left(n+2\right)}\right)}{V_n\left(C_n{H}_{2\left(n+1\right)}\right)}\right]\right]}^2}{4{\mathcal{N}}^{2 }{a}_{-CH2-}^2{\gamma }_{-CH2-}}$$

(4)

2. The method proposed by Schultz et al. [73] using Fowkes relation [72] similarly gave the free energy of adsorption $\Delta G_a^0$ as a function of the geometric mean of the respective dispersive components of the surface energy of the liquid solvent ${\gamma }_l^d$and the solid ${\gamma }_s^d$ :

$$\Delta G_a^0=RTlnVn+C=2\mathcal{N}a {\left({\gamma }_l^d{\gamma }_s^d\right)}^{1/2}+D$$

(5)

where a is the surface area of probes supposed constant for all temperatures and C and D two constants depending on the used materials and the temperature. The variations of $RTlnVn$ versus $2\mathcal{N}a {\left({\gamma }_l^d\right)}^{1/2}$ of n-alkanes and polar molecules gave both the ${\gamma }_s^d$ and $\Delta G_a^{sp}\left(T\right)$ of the solid.

In previous studies, one determined the dispersive component of many solid materials by using the various molecular areas of Kiselev, Van der Waals (VDW), Redlich-Kwong (R-K), Kiselev, geometric, cylindrical or spherical models [74,75,76,77,78,79].

3. The method deduced from the works of Sawyer and Brookman [70] used:

$$RTlnVn=A{T}_{B.P.}+B$$

(6)

where A and B are two constants. This method gave the specific free energy and the acid base properties.

4. The method of the deformation polarizability ${\alpha }_0$ proposed by Donnet et al. [80]. They proposed the following relation:

$$RTlnVn=E{\left(h{\nu }_L\right)}^{1/2} {\alpha }_{0, L}+F$$

(7)

where ${\nu }_L$ is the electronic frequency of the probe, h the Planck’s constant and E and F constants of interaction.

5. Chehimi et al. [59] used the standard enthalpy of vaporization $\Delta H_{vap.}^0$ (supposed constant) of n-alkanes and polar molecules:

$$RTlnVn=I\Delta H_{vap.}^0+J$$

(8)

where I and J are two constants. This method is similar to Saint-Flour and Papirer method using $ln{P}_0$ and that of Sawyer and Brookman using $T_{B.P.}$.

6. The method of Brendlé and Papirer [2] used the concept of the topological index ${\chi }_T$ that is a parameter considering the topology and the local electronic density in the polar probe structure. They gave the following relation:

$$RTlnVn=Mf\left({\chi }_T\right)+N$$

(9)

where M and N are two adsorption constants.

In all previous cases, the determination of $\Delta G_a^{sp}\left(T\right)$ of polar solvents versus the temperature will allow to deduce the specific enthalpy $\left(-\Delta H_a^{sp}\right)$ and entropy $\left(\Delta S_a^{sp}\right)$ of polar probes adsorbed on the solid surfaces by using equation (1):

$$\Delta G_a^{sp}\left(T\right)=\Delta H_a^{sp}\Delta S_a^{sp}$$

(10)

Knowing of $\Delta H_a^{sp}$ polar solvents, the two respective acid base constants K_A and K_D of solids can be determined by Papirer following relation [12,13]:

$$-\Delta H^{Sp}= K_A\times DN+K_D\times AN$$

(11)

That can be also written as:

$$\frac{-\Delta H^{Sp}}{AN}=K_A \frac{DN}{AN}+K_D$$

(12)

where AN and DN respectively represent the electron donor and acceptor numbers of the polar molecule given by Gutmann [62] and corrected by Fowkes, $K_A$ and $K_D$ are the respective acid and base constant.

2.2. New methodology

2.2.1. Molecular models

The surface areas of solvents used in literature are those proposed by Kiselev and supposed constant for all temperatures. Hamieh et al [79] proposed six new molecular models that allowed the calculation of the surface areas of organic molecules: spherical (Sph.), geometric (Geom.), cylindric (Cyl.), Redlich-Kwong (R-K) and van der Waals (VDW) models compared to Kiselev results given in Table 1.

All these molecular models were used for the determination of the dispersive surface energy and Lewis’s acid base parameters of sold particles.

2.2.2. Hamieh’s thermal model

In a recent study, Hamieh [69] proved the dependency of the surface areas of molecules on the temperature. He gave the following relation of the surface area $a\left(n,T\right)$ of n-alkanes as a function of the temperature:

$$a\left(n,T\right)=\frac{69.939n+313.228}{{\left(563.02-T\right)}^{1/2}}$$

(13)

By showing the failure of Dorris-Gray method that was largely used to determine ${\gamma }_s^d$ of solids. This method considered the surface area $a_{-CH2-}$ of methylene group equal to 6 Ų and constant for any used temperature. Hamieh [69] proved the non-validity of Dorris-Gray method and gave the following expression of $a_{-CH2-}$ (in Å2) as a function of the temperature T (in K):

$$a_{-CH2-}=\frac{69.939}{{\left(563.02-T\right)}^{1/2}}$$

(14)

Hamieh also gave the surface areas of polar molecules against the temperature by defining three new surface parameters: a first maximum temperature $T_{Max.1}$ characteristics of the interaction between the polar solvents and the PTFE fibers, a second maximum temperature $T_{Max.\left(X\right)}$ , an intrinsic characteristics of the dispersive surface tension of the polar molecules and a third geometric parameter $a_{Xmin.}$ proper to the molecule X itself. The general expression of the surface area $a_X\left(T\right)$ of polar molecules was given below:

$$a_X\left(T\right)=a_{Xmin.}\times \frac{\left(T_{Max.1}-T\right) }{{\left(563.02-T\right)}^{1/2}{\left(T_{Max.\left(X\right)}-T\right)}^{1/2}}$$

(15)

The large effect of the temperature on surface areas of molecules was here highlighted.

On the other hand, we highlighted an important effect of the temperature on the standard enthalpy of vaporization $\Delta H_{vap.}^0$ of organic molecules supposed constant by Chehimi et al. [59] and therefore proposed the following relation that can be used to determine the specific contribution of the Gibbs free energy of adsrption of poalr molecules on alumina particles:

$$\Delta G_a^0\left(T\right)=\delta \Delta H_{vap.}^0\left(T\right)+\epsilon$$

(16)

where $\delta$ and $\epsilon$ are two constants of interaction.

2.2.3. The new Lewis’s acid base parameters

In several cases, the Gutmann method cannot be applied because the linearity of equation 11 is not satisfied for many solid substrates. This classical relationship was corrected and a new equation was proposed by Hamieh et al. [77,81] by adding a third parameter K reflecting the amphoteric character of the oxide or polymer according to:

$$-\Delta H^{Sp}= K_A\times DN+K_D\times AN-K\times DN\times AN$$

(17)

By dividing by AN, we obtain:

$$\frac{-\Delta H^{Sp}}{AN}=K_A \frac{DN}{AN}+K_D-K\times DN$$

(18)

or:

X1 = K_D + K_A X2 - K X3

(19)

With:

$$X_1=-\frac{\Delta H^{sp}}{AN},X_2=\frac{DN}{AN},X_3=DN$$

(20)

and K = K(K_A,K_D)

X1, X2 and X3 are known for every polar molecule, whereas K_D, K_A and K are the unknown parameters. The problem given by equation (16) is represented by a linear system for N solvents and admits a unique solution for N ≥ 3 giving the three unkown constants numbers: K_D, K_A and K.

On the other hand, in previous study Hamieh [83] proposed new entropic acidic ${\omega }_A$ and basic ${\omega }_D$ parameters to determine the entropic acid base character of the solid surfaces by writing:

$$\left(-\Delta S_a^{sp}\right)={\omega }_A D{N}^{’}+{\omega }_D A{N}^{’}$$

(21)

or

$$\left(\frac{-\Delta S_a^{sp}}{A{N}^{’}}\right)={\omega }_A \left(\frac{D{N}^{’}}{A{N}^{’}}\right)+{\omega }_D$$

(22)

3. Materials and solvents

The different solid particles used in this study are furnished by Aldrich. Classical organic probes, characterized by their donor and acceptor numbers, were used in this study. Corrected acceptor number AN′ = AN-AN^d, given by Riddle and Fowkes [82], who subtracted the contribution of Van der Waals interactions (or dispersion forces). This acceptor number was normalized by Hamieh et al. [77,81] proposed to use dimensionless donor number DN′ and dimensionless acceptor number AN′. All probes (Aldrich) were highly pure grade (i.e., 99%). The probes used were n-alkanes (pentane, hexane, heptane, octane, and nonane); amphoteric solvents: acetonitrile, acetone; basic solvents: ethyl acetate, tetrahydrofuran (THF) and acidic solvent: chloroform and nitromethane.

On Table 2 gives, we gave the donor and acceptor numbers of polar probes used in this study.Chromatographic injections were made using Hamilton microsyringes. Measurements were carried out with a Focus GC Chromatograph equipped with a flame ionization detector of high sensitivity. The IGC measurements were performed on a commercial Focus GC gas chromatograph equipped with a flame ionization detector. Dried nitrogen was the carrier gas. The gas flow rate was set at 25 mL/min. The injector and detector temperatures were maintained at 400 K during the experiments.

Chromatographic injections were made using Hamilton microsyringes. Measurements were carried out with a Focus GC Chromatograph equipped with a flame ionization detector of high sensitivity. The IGC measurements were performed on a commercial Focus GC gas chromatograph equipped with a flame ionization detector. Dried nitrogen was the carrier gas. The gas flow rate was set at 25 mL/min. The injector and detector temperatures were maintained at 400 K during the experiments.

To achieve infinite dilution, 0.1 L of each probe vapor was injected with 1 L Hamilton syringes, in order to approach linear condition gas chromatography. The column temperatures were 300 K to 460 K, varied in 20 ºC steps. Each probe injection was repeated three times, and the average retention time, t_R, was used for the calculation. The standard deviation was less than 1% in all measurements. All columns used in this study were prepared using a stainless-steel column with a 2 mm inner diameter and with an approximate length of 20 cm.

3. Results

3.1. Determination of the Gibbs free energy of adsorption

The results of the IGC technique at infinite dilution gave the net retention volume Vn of n-alkanes and polar molecules adosrbed on alumina at various temperatures in the interval [323.15K, 463.15K]. This allowed to obtain the Gibbs free energy $\Delta G_a^0$ of adsorption by using the following fundamental thermodynamic relation of IGC:

$$\Delta G_a^0=-RT ln V_n+\beta \left(T\right)$$

(23)

and $\beta \left(T\right)$ is given by:

$$\beta \left(T\right)=RT ln \left(\frac{sm{\pi }_0}{P_0}\right)$$

(24)

where s is the specific surface area of alumina and m the mass of the solid particles introduced in the coumn. Whereas, $P_0$ and ${\pi }_0$ are respectively given by the Kemball and Rideal reference state [21] at $T_0=0$ºC by:

$$P_0=1.013\times {10}^{5}Pa\mathrm{and}{\pi }_0=6.08\times {10}^{-5} N m^{-1}$$

(25)

The obtained results are presented in Table 3.

Table 3 clearly showed that the standard free energy of adsorption $\left(-\Delta G_a^0\right)$ decreases for every probe when the tempertaure increases (Figure 1) and decreasing at the same time the adsorption of molecules. One also observed that the values of the free energy of adsorption of polar solvents are greater than that of n-alkanes showing the strong polar interaction between alumina and polar molecules and proving the important Lewis’s acid base character of the solid surfaces.

From Table 3 and Figure 1, one deduced the values of the standard enthalpy and entropy of adsorption for all orgnaic molecules as shown in Table 4.

The polar molecules on Table 4 can be classified in decreasing order of standard entalpy of adsorption:

Diethyl ether > CH2Cl2 > THF > CHCl3 > Toluene > CCl4

(26)

Inequalities (21) showed that the alumina material exhibits an amphoteric character because the stronger adsorption is obtained with the amphoteric molecule such as diethyl ether.

In order to quantify the Lewis acid base properties, one used relation (10) by using the values of $\left(-\Delta H_a^0\right)$ of polar molecules. A linear relation was obtained giving

$$\frac{\left(-\Delta H_a^0\right) }{AN}=1.112 \frac{DN}{AN}+3.045$$

(27)

and the values of the Lewis enthalpic acid base constants are respectively $K_A=1.112 kJ/mol$ and $K_D=3.045 kJ/mol$, whereas, those relative to the entropic constants are ${\omega }_A=1.44 J K^{-1}mo{l}^{-1}$ and ${\omega }_D=4.56 J K^{-1}mo{l}^{-1}$. These can be written in unitless as:

$K_A=0.664$, $K_D=1.820$, ${\omega }_A=8.6\times {10}^{-4}$ and ${\omega }_D=2.7\times {10}^{-3}$. The respective ratios of enthalpic and entropic base on acid constants are therefore given by:

$$\frac{K_D}{K_A}=2.74\mathrm{and}\frac{{\omega }_D}{{\omega }_A}=3.2$$

(28)

These results confirmed that the alumina surfaces have an amphoteric behavior with stronger Lewis’s base character 2.74 times greater than the acidic character. In this case, we proved that the Lewis and Bronsted concepts are very close showing the amphoteric behavior in both aqueous medium and solid surfaces.

3.2. London dispersive surface energy of alumina particles

In this section, we calculated the London dispersive component of alumina particles by using relations (3), (4), (12) and (13) and applying the various models of Van der Waals (VDW), Redlich-Kwong (R-K), Kiselev, geometric, cylindrical or spherical models [75,76,77,78,79], Dorris-Gray [69] and thermal model [71]. On Table 5, we gave the values of the London dispersive surface energy ${\gamma }_s^d \left(T\right)$ of alumina particles as a function of the temperature for the various methods and models.

The results on Table 5 and Figure 2 showed the linear decrease of ${\gamma }_s^d$ when the temperature increases for all applied models and methods.

Three groups can be distinguished between the above models and methods:

1. The group constituted by Kiselev, cylindrical, VDW, geometric and Doris-Gray models taking into account the geometric form of n-alkanes and they presented very close values of ${\gamma }_s^d$ and the surface of methylene group (Figure 3).

2. The second group concerns the models relative to thermal model, Redlich-Kwong equation and the global average results that concluded to the more accurate values of ${\gamma }_s^d$ of alumina surfaces (Figure 3).

3. The third group is relative to spherical model and Gray-Hamieh model [69]. The obtained values of ${\gamma }_s^d$ are higher than the true value, certainly because of the non-accurate values of the surface area of n-alkanes for these models (Figure 3).

The different equations giving ${\gamma }_s^d\left(T\right)$ of alumina particles as a function of the temperature for the various molecular models of n-alkanes were presented on Table 6 with the values of the dispersive surface entropy ${\epsilon }_s^d$, the extrapolated values ${\gamma }_s^d\left(T=0K\right)$ and the maximum of temperature $T_{Max}$ allowed by by the chosen molecular model. $T_{Max}$ Was defined by the following relation:

$$T_{Max}=-\frac{{\gamma }_s^d\left(T=0K\right)}{{\epsilon }_s^d}$$

(29)

Table 6 showed the large difference between the values of dispersive surface entropy of the different molecular models compared to that given by Hamieh thermal model $({\epsilon }_s^d=-0.480 mJ m^{-2}{K}^{-1}$) and the highest value obtained by the spherical model. The important result obtained here is the average value of $T_{Max}=523.42 K$ with comparable values to the various models excepted for the case of Dorris -Gray.

3.2. Surface thermodynamic of alumina particles

3.2.1. The Gibbs specific free energy of adsorption

The variations of the Gibbs specific free energy ($\Delta G_a^{sp}\left(T\right)$) of the various polar solvents adsorbed on silica particle surface as a function of the temperature T were given on Table S1 (Supporting Information) for the various models such as Kiselev, Van der Waals, Redlich-Kwong, geometric, spherical and thermal models and IGC methods such as Boiling point, vapor pressure, deformation polarizability, enthalpy of vaporization, $\Delta H_{vap.}^0$, $\Delta H_{vap.}^0 \left(T\right)$ and Topological index methods.

The values $\Delta G_a^{sp}\left(T\right)$ presented on Table S1 proved that there is no universal method that can give accurate results excepted the thermal model that took into account the temperature effect on the surface area of molecules. We observed on Table S1 the large difference in results obtained with the different models and methods that can reach 100% of the results given by the thermal model. We showed on Figure 4 the disparity in obtained results with different polar molecules adsorbed on alumina surfaces. The linear variations of $\Delta G_a^{sp}\left(T\right)$ plotted on Figure 4 and Table 7 giving the corresponding equations clearly proved that the slope and the ordinate at origin strongly depend on the chosen model or used method and for every polar molecule.

3.2.2. Lewis’s acid base parameters

From the equations $\Delta G_a^{sp}\left(T\right)$ on Tables 7, we deduced the values of ($-\Delta H_a^{sp})$ and ($-\Delta S_a^{sp})$ relative to the adsorption of CCl4, CH2Cl2, CHCl3, diethyl ether, THF and toluene on alumina particles by using the various molecular models and methods. The results were presented on Table 8 and Table 9.

The calculations of the average, standard deviation and error percentage committed on the values of the specific enthalpy of adsorption on alumina reflect the dispersion of results obtained by the models and methods compared to the thermal model. One conluded here that the error with CCl4 is 51.86% followed by toluene (21.96%), CH2Cl2 (20.47%), CHCl3 (13.91%), THF (8.18%) and diethyl ether (5.01%). The results on Table 8 showed that the IGC methods that match better the thermal models are the following: boiling point, vapor pressure and enthalpy of vaporization; followed by the other molecular models such as cylindrical and Kiselev models.

For the specific entrop of adsorption (Table 9), the error percentage reaches 77.02 with CCl4, 31.97 with CH2Cl2 followed by THF (12.98%), CHCl3 (12.46%), THF (8.18%) and diethyl ether (5.01%). The closer methods to the thermal models are identical to those obtained with the specific enthalpy of adsorption proving the effect of the temperature on the surface area of organic molecules.

The acid base parameters of alumina were obtained from Table 8 and Table 9 and allowed to plot Figure 5 and Figure 6 representing the respective variations of $\left(\frac{-\Delta H_a^{sp}}{A{N}^{’}}\right)$ and $\left(\frac{-\Delta S_a^{sp}}{A{N}^{’}}\right)$ as a function of $\left(\frac{D{N}^{’}}{A{N}^{’}}\right)$. The excellent linearity obtained with the different models gave on Table 10 the values of Lewis enthalpic acid base constants $K_A$ and $K_D$and Lewis entropic acid base constants ${\omega }_A$ and ${\omega }_D$of alumina surface. The values of the acid base parameters obtained the thermal model are the following:

$$\begin{gathered} K_A=0.624,K_D=1.831\mathrm{and}{K}_D/K_A=2.93 \\ {\omega }_A=0.72,{\omega }_D=2.79\mathrm{and}{\omega }_D / {\omega }_A=3.9 \end{gathered}$$

These results proved the amphoteric character of alumina surfaces with stronger Lewis’s basicity about three times more important than the Lewis’s acidity of alumina. The same tendency was observed with the entropic acid base parameters.

Table 10 showed that the best models or methods that gave results comparable to those obtained by the thermal model are the following: boiling point, vapor pressure and enthalpy of vaporization methods proving the important effect of the temperature on the surface areas adsorbed on alumina particles.

4. Discussion

Since 1982, scientists were interested to the determination of physicochemical properties of materials by using inverse gas chromatography at infinite dilution. Saint-Flour and Papirer [12,13] have first tried to separate the two dispersive and polar contributions of the Gibbs free energy of adsorption of polar solvents on solid substrates by using the notion of vapor pressure of organic molecules. Schultz et al [73], Donnet et al. [80], Brendlé and Papirer [2] and other scientists [10,14,20,23,33,36,59] were also interested to the determination of specific and dispersive properties of materials. There was no universal method or model to be used in IGC at infinite dilution for an accurate characterization of solid particles. Even Dorris-Gray relation [71] cannot give a precise value of the dispersive component of the solid surface of solid materials. Hamieh et al. [35,77,78,81 have nevertheless succeeded in carrying out a very precise determination of the second-order temperatures (such as the glass transition) of polymers such as poly (methyl methacrylate) or pol (a-n-alkyl) methacrylate in bulk phase or in adsorbed state. The serious difficulty faced by researchers during 40 years was the problem of the diversity of methods and models used to determine the surface properties of a solid material such as the dispersive free surface energy, the specific free energy and the Lewis’s acid-base constants without obtaining the same results. Indeed, we proved in several previous studies that the various models and methods of IGC technique did not give the same results. On the contrary, the methods used gave results completely different from each other. The error percentage sometimes exceeds 100% from one method to another. Our previous works highlighted the important effect of the temperature on the surface area of organic solvents. We failed the methods proposed by Dorris-Gray [71] and Schultz et al. [73] that supposed a constant value of the surface area of organic molecules and we proposed various relations of the surface area of probes depending on the temperature and corrected the values of the dispersive surface energy and polar properties of materials. Our new methodology was applied to the alumina particles and determined with an excellent accuracy the surface physicochemical properties of alumina. The thermal model [69] has to be applied in the future for a good characterization of solid surfaces.

On the other hand, we observed similar results between the thermal model [69] and those given by equations (27) and (28) by using the values of the standard enthalpy ($-\Delta H_a^0$) and entropy ($-\Delta S_a^0 )$ of adsorption of polar molecules on alumina surfaces. The last way can resolve some difficulties related to the choose of the best model for more accurate results of the surface thermodynamic properties of materials.

5. Conclusions

The surface thermodynamic properties such as the London dispersive component of the surface energy, the Gibbs free energy of adsorption, the specific interactions and the Lewis’s acid base parameters of alumina particles were determined by using the inverse gas chromatography technique at infinite dilution an applying 15 molecular models and chromatographic methods.

The determination of the London dispersive surface energy of alumina by using the various molecular models showed that Dorris-Gray formula [71] and Schultz et al. method [73] cannot be used for an accurate determination of ${\gamma }_s^d$of solid alumina. We corrected the calculation of ${\gamma }_s^d \left(T\right)$ by using the new thermal model that took into account the change of the surface areas $a\left(T\right)$ of organic molecules as a function of the temperature. The following relation was therefore obtained:

$${\gamma }_s^d\left(T\right)=0.480T+233.9$$

with a dispersive surface entropy equal to ${\epsilon }_s^d=-0.480 mJ m^{-2}{K}^{-1}$ and the maximum of temperature, characteristic of alumina, equal to $T_{Max}=487.21K$.

The specific Gibbs free energy was obtained by the different molecular models and methods. All methods were proved to be inaccurate except that of the thermal model. However, the methods based on the effect of the temperature on the thermodynamic parameters such as the vapor pressure, the boiling point temperature and the standard enthalpy of vaporization gave results closer to those obtained by Hamieh thermal model [69]. We gave below the variations of the Specific Gibbs free energy $\Delta G_a^{sp}\left(T\right)$ of polar molecules adsorbed on alumina surfaces as a function of the temperature with the values of the specific enthalpy and entropy of adsorption.

Polar molecule	$\Delta G_a^{sp}\left(T\right)$(kJ/mol)	$\Delta H_a^{sp} \left(kJ/mol\right)$	$\Delta S_a^{sp}\left(J{K}^{-1}mo{l}^{-1}\right)$
CCl4	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.021 T + 12.489	12.489	21
CH2Cl2	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.135 T + 83.700	83.700	135
CHCl3	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.136 T + 65.871	65.871	136
Diethyl ether	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.088 T + 68.367	68.367	88
THF	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.071 T + 53.71	53.71	71
Toluene	$-\Delta {\mathrm{G}}_{\mathrm{a}}^{\mathrm{sp}}\left(\mathrm{T}\right)$= -0.041 T + 21.91	21.91	41

These results clearly showed the strong Lewis basicity of alumina proved by the strong interactions between the more acidic solvent (CH2Cl2) with $\Delta H_a^{sp}$= 83.700 kJ/mol and also an important amphoteric character.

The use of the thermal model led to the following Lewis acid base parameters:

$$\begin{gathered} K_A=0.624,K_D=1.831\mathrm{and}{K}_D/K_A=2.93 \\ {\omega }_A=0.72,{\omega }_D=2.79\mathrm{and}{\omega }_D / {\omega }_A=3.9 \end{gathered}$$

Thus, showing that the alumina surface is about three times more basic than acidic. The same tendency was observed with the entropic acid base parameters. Comparable values of the acid base parameters of alumina were obtained by using the methods of vapor pressure, boiling point or the standard enthalpy of vaporization.

The new methodology proposed in this study will therefore allow to obtain an accurate determination of the London dispersive surface energy, the Gibbs free energy, the specific enthalpy and entropy of adsorption of polar molecules on the solid surfaces as well as the Lewis acid base parameters of solid substrates.