The Effect of Temperature on London Dispersive Properties of H-β-Zeolite / Rhodium Catalysts Using New 2D-Chromatographic Models

1. Introduction

Two-dimensional (2D) inverse gas chromatography (IGC) technique at infinite dilution [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] was widely used for the surface characterization of solid materials such as oxides, metals, polymers, fibers, biomaterials and nanomaterials, ceramics, catalysts, and pharmaceutical and food products. The chromatographic measurements led to the net retention volume $V_n$ of the organic solvents adsorbed on the solid surfaces. This thermodynamic parameter allowed obtaining the dispersive free energy $\left(-\Delta G_a^0\right)$, the London dispersive surface energy ${\gamma }_s^d$, the polar enthalpy $\left(-\Delta H_a^p\right)$ and entropy $\left(-\Delta S_a^p\right)$ of adsorption, and the Lewis acid-base properties of materials.

The surface physicochemical properties of solid materials or nanomaterials were studied by several scientists using the inverse gas chromatography at infinite dilution [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. The determination of these surface properties is of vital interest in many industrial processes such as catalysis, adhesion, chemical engineering, colloidal dispersions, and other industrial applications. Different chromatographic methods [21,22,23,24,25,26] and molecular models [27,30,31,32] were applied in the literature to determine the surface energetic properties of solid particles. The experimental determination of the net retention volume was utilized for the calculation of the dispersive $\left(-\Delta G_a^d\right)$ and polar $\left(-\Delta G_a^p\right)$ free energies of organic solvents adsorbed on solid materials using the following thermodynamic equation:

$$\left(-\Delta G_a^0\right)=RTln V_n+C\left(T\right)=\left(-\Delta G_a^d\right)+\left(-\Delta G_a^p\right)$$

(1)

where R is the perfect gas constant, $T$ the absolute temperature, and $C\left(T\right)$ a constant parameter depending on $T$ and the interaction between the solid materials and the organic molecules.

The first method used to calculate the London dispersive surface energy of materials was based on Dorris and Gray [33] and the well-known relationship of Fowkes [34] relating the dispersive work of adhesion $W_a^d$ to the free dispersive energy of adsorption $\Delta G_a^d$ by Equation 2:

$$\left(-\Delta G_a^d\right)=\mathcal{N}a W_a^d=2\mathcal{N}a \sqrt{{\gamma }_l^d{\gamma }_s^d}$$

(2)

Where $\mathcal{N}$ is Avogadro’s number, $a$ the surface area of adsorbed solvent, and ${\gamma }_l^d$ and ${\gamma }_s^d$ are respectively the dispersive components of the liquid solvent and the solid.

Dorris and Gray were the first who determined the dispersive component of the surface energy of a solid by defining the increment $\Delta G_{-CH2-}^0$ of methylene group:

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

(3)

Where $C_n{H}_{2\left(n+1\right)}$ and $C_n{H}_{2\left(n+1\right)}$ represents the general formula of two consecutive n-alkanes.

The dispersive surface energy ${\gamma }_s^d$ of a the solid can be therefore determined by the equation (4):

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

(4)

Where $a_{-CH2-}$ is the surface area of methylene group taken equal to 6 Ų with a surface energy given by Equation 5:

$${\gamma }_{-CH2-}\left(\mathrm{mJ}/{\mathrm{m}}^2\right)=52.603–0.058T \left(K\right)$$

(5)

$$RTln V_n=2\mathcal{N}a \sqrt{{\gamma }_l^d{\gamma }_s^d}+A\left(T\right)$$

(6)

Where $A\left(T\right)$ is a constant depending on the temperature. A straight line is obtained and its slope is equal to $\sqrt{{\gamma }_s^d}$ .

In a previous paper [35], we criticized the above methods and proposed a new thermal model proving the temperature effect of the surface area $a\left(T\right)$ and the London dispersive surface energy of the organic solvents ${\gamma }_l^d\left(T\right)$. The use of the Hamieh thermal model [35,36] led to more accurate values of the London dispersive surface energy of solid surfaces.

Furthermore, different other methods were applied in the literature to determine the polar variables of adsorption and the Lewis acid-base parameters of solids. All these methods used the linear properties of the thermodynamic parameters of the organic solvents. A general linear equation relative to the adsorption of n-alkanes on solid surfaces was previously proposed:

$$RTln{V}_n\left(T\right)={\alpha }_j{x}_j+{\beta }_j$$

(7)

where $x_j$ represents a surface chromatographic parameter, and ${\alpha }_j$ and ${\beta }_j$ are two constants relative to the adsorption of n-alkanes on the solid surfaces experimentally determined at each temperature from the slope and the ordinate at the origin of the n-alkanes straight line thus leading to the polar free energy, enthalpy, and entropy of adsorption. The reference parameter $x_j$ can be taken from the following variables: the boiling point $T_{B.P.}$ of the solvent proposed by Sawyer and Brookman [20], the vapor pressure $P_0$ of the solvent, $ln{P}_0$ used by Saint-Flour and Papirer [21,22], the London dispersive surface energy ${\gamma }_{l }^L$ of the solvent considered by Schultz et al. [37], the deformation polarizability ${\alpha }_0$ of the organic molecule suggested by Donnet et al. [23], the topological index ${\chi }_T$ determined by Brendlé and Papirer [24,25], and the Hamieh thermal model $a\left(T\right){\gamma }_{l }^L\left(T\right)$ using the temperature effect on the surface area and the London dispersive surface energy of organic molecules [35,36].

The dispersive adhesion work $W_a^d\left(T\right)$ of n-alkanes on H-β-zeolite / rhodium catalysts as a function of temperature given by Equation 8:

$$W_a^d\left(T\right)=2 \sqrt{{\gamma }_l^d{\gamma }_s^d}$$

(8)

was never studied in the literature.

In this paper, we applied our new thermal model to determine the effect of temperature on the London dispersive surface energy and dispersive adhesion work of H-β-zeolite and the rhodium impregnated in H-β-zeolite at different rhodium percentages.

2. Materials and Methods

2.1. Materials and Solvents

The rhodium with different percentages supported by H-β-zeolites were previously synthetized in a previous paper [32] using the method developed by Navio et al. [38] and Zhang et al. [39]. Whereas, the organic solvents utilized for chromatographic measurements obtained from Aldrich (Paris, France) were of highly purity grade (i.e., 99%). The chosen solvents were n-alkanes such as pentane, hexane, heptane, octane, and nonane, amphoteric solvents such as methanol, acetone, trichloroethylene, tetrachloroethylene, basic solvents such as diethyl ether and benzene (weak base), acidic solvents such as chloroform and cyclohexane (weak acid). Corrected donor $D{N}^{\prime }$ and acceptor $A{N}^{\prime }$ electron numbers of the used polar probes [26,40] were previously normalized [26,41].

2.2. Experimental

The chromatographic measurements were carried out on a commercial Focus GC gas chromatograph (from Sigma-Aldrich, St. Quentin Fallavier, France) equipped with a flame ionization detector. Dried nitrogen was the carrier gas. The gas flow rate was set at 20 mL/min. The injector and detector temperatures were maintained at 450 K during the experiments [42]. To achieve infinite dilution, 0.1 μL of each probe vapor was injected with 1 μL Hamilton syringes. All chromatographic columns were prepared using a stainless-steel column with a 2 mm inner diameter and with an approximate length of 20 cm. Each column was packed with 1 g of solid particles with a size not exceeding 250 μm. The temperature of columns varied from 300 K to 430 K. 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.

The surface specific area of the various catalysts was determined in a previous study by using Brunauer-Emmett-Teller (BET) [32]. The obtained specific surface area S_BET and microporous volume V_m of the various catalysts at different rhodium percentages were given in Table 1.

2.3. Retention Volume

The most important experimental parameter derived from chromatographic measurements was the net retention volume $V_n$ was obtained from Equation 9:

$$V_n=j{D}_c\left(t_R-t_0\right)$$

(9)

where $t_R$ is the retention time of the adsorbed solvent, $t_0$ the zero-retention reference time measured with a non-adsorbing probe such as methane, $D_c$ the corrected flow rate and $j$ a correction factor taking into account the gas compression [42]. The factors $D_c$ and $j$ were respectively given by the following relations

$$D_c=D_m{\displaystyle \frac{T}{T_a}} {\displaystyle \frac{\eta \left(T\right)}{\eta \left(T_a\right)}}$$

(10)

and

$$j={\displaystyle \frac{3}{2}} {\displaystyle \frac{{\left(\frac{\mathsf{\Delta }P+P_0}{P_0}\right)}^2-1}{{\left(\frac{\mathsf{\Delta }P+P_0}{P_0}\right)}^3-1}}$$

(11)

where $D_m$ is the measured flow rate, $T$ the column temperature, $T_a$ the room temperature, $\eta$ the gas viscosity, $P_0$ the atmospheric pressure, and $\mathsf{\Delta }P$ the pressure variation.

2.4. London Dispersive Surface Energy of Catalysts Using the Hamieh Thermal Model

Various molecular models were previously used for the determination of the London dispersive surface energy ${\gamma }_s^d$ and polar free energy of sold particles at different temperatures. It was showed that the values of ${\gamma }_s^d$ strongly depend on the chosen molecular model and the difference between the values varied and sometimes reached four times from a molecular model to another. In fact, the surface area of organic solvents depends on the temperature, while the molecular models give constant values of the surface area. This will lead to wrong values of the surface thermodynamic properties of materials. The correction of the above properties can be made by applying the Hamieh thermal model [35}. Indeed, in a recent study, we proved the temperature effect on the surface areas of molecules and proposed the following relation of the surface area $a\left(n,T\right)$ of n-alkanes as a function of the temperature:

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

(12)

We also showed the surface area $a_{-CH2-}$ of methylene group depends on the temperature and showed the following Equation:

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

(13)

Another expression of the surface area $a_X\left(T\right)$ of a polar molecule X was given as a function of temperature:

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

(14)

Where $T_{Max.1}$, $T_{Max.\left(X\right)}$, and $a_{Xmin.}$ are constant characteristics of the polar molecules [35].

The London dispersive surface energy of the different catalysts was therefore obtained using Relation 12 given as a function of temperature:

$$RTln V_n\left(T\right)=\sqrt{{\gamma }_s^d\left(T\right)} \left[2\mathcal{N}a\left(T\right) \sqrt{{\gamma }_l^d\left(T\right)}\right]+A\left(T\right)$$

(15)

Plotting $RTln V_n\left(T\right)$ as a function of $\left[2\mathcal{N}a\left(T\right) \sqrt{{\gamma }_l^d\left(T\right)}\right]$ of n-alkanes adsorbed on zeolite material for example, we obtained a straight line giving a slope equal to $\sqrt{{\gamma }_s^d\left(T\right)}$ and therefore the value of ${\gamma }_s^d\left(T\right)$ for the different temperatures (Figure 1).

3. Results

3.1. Determination of $RTln{V}_n$ of n-Alkanes Adsorbed on H−β−Zeolite/Rhodium Catalysts

The chromatographic measurements led to the values of $RTln{V}_n$ of n-alkanes adsorbed on H-β-zeolite/rhodium at different percentages of rhodium (from 0 to 2%) at different temperatures (varying 300 K to 430 K). The experimental results were given in Table S1. The variations of $RTln{V}_n$ of n-alkanes adsorbed on H-β-zeolite/rhodium were represented in Figure 2 as a function of temperature for various rhodium percentages. The results showed linear variations for the different n-alkanes. It was observed in Figure 2 an important effect of the rhodium percentage on the free energy of adsorption. The largest values of $RTln{V}_n$ was obtained with H-β-zeolite. The free energy of adsorption decreased when the temperature increased.

The representation of $RTln{V}_n$ of adsorbed n-alkanes as a function of the rhodium percentage plotted in Figure 3 showed a brutal decrease from H-β-zeolite to H-β-zeolite/rhodium highlighting a maximum of $RTln{V}_n$ at a rhodium percentage equal to 0.75% and for different temperatures.

3.2. London Dispersive Surface Energy of H−β-Zeolite/Rhodium Catalysts

The London dispersive surface energy ${\gamma }_s^d$ of H-β-zeolite/rhodium catalysts was determined for different percentages of rhodium and various temperatures using Equation 14 and applying the Hamieh thermal model taking into account the temperature effect on the surface area of n-alkanes. The values of ${\gamma }_s^d$ of H-β-zeolite/rhodium catalysts were given in Table S2 for different temperatures and rhodium percentages using the straight-line method and Hamieh thermal model. The variations of ${\gamma }_s^d$ were plotted in Figure 4 as a function of temperature at different rhodium percentages. A decrease in the values of ${\gamma }_s^d\left(T\right)$ was observed (Figure 4) versus the temperature and for various rhodium percentages. The variations of ${\gamma }_s^d\left(T\right)$ were represented by a second-degree equation with an excellent regression coefficient (R² = 0.9993) as follows:

$${\gamma }_s^d\left(T\right)=a\times T^2+b\times T+c$$

(16)

where the coefficients $a$, $b$, and $c$ are constants depending on the solid materials.

Table S2 and Figure 4 allowed giving in Table 2 the different equations of ${\gamma }_s^d\left(T\right)$ of the various catalysts. An interesting result was deduced from Table 2. Indeed, the same value of the coefficient $a$ was obtained for all H-β-zeolite/rhodium catalysts whatever the rhodium percentage, $a=-4.8\times {10}^{-3}\mathrm{mJ}\times {\mathrm{m}}^{-2}\times {\mathrm{K}}^{-2}$.

It can be concluded from equations in Table 2 that the coefficients $a$, $b$, and $c$ are function of the different derivatives of ${\gamma }_s^d\left(T\right)$ as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d^2{\gamma }_s^d}{d{T}^2}}=2a=-9.6\times {10}^{-3}mJ\times {\mathrm{m}}^{-2}\times {\mathrm{K}}^{-2}\\ {\displaystyle \frac{d{\gamma }_s^d}{dT}}=b \left(in \mathrm{mJ}\times {\mathrm{m}}^{-2}\times {\mathrm{K}}^{-1}\right) \\ {\gamma }_s^d\left(OK\right)=c \left(in \mathrm{mJ}\times {\mathrm{m}}^{-2}\right) \end{array}\right.$$

(17)

The variations of the coefficients $b$ and $c$ were represented in Figure 5 as a function of the rhodium percentage in the zeolite catalyst. Let us consider $\theta$ the rhodium percentage. The curves of coefficients $b\left(\theta \right)$ and $c\left(\theta \right)$ shown in Figure 5 had parabolic variations for $0\le \theta \le 1.25\%$ with a minimum of $b\left(\theta \right)$ for $\theta =0.50\%$ and a maximum of $c\left(\theta \right)$ for the same rhodium percentage both followed by a pallier.

The results in Figure 5 led to the following expressions of $b\left(\theta \right)$ and $c\left(\theta \right)$ in the interval $0\le \theta \le 1.25\%$:

$$\mathrm{b}\left(\theta \right)=0.712\times {\theta }^2-0.639\times \theta +2.040$$

(18)

$$\mathrm{c}\left(\theta \right)=-397.48\times {\theta }^2+383.53\times \theta +163.73$$

(19)

These equations led to conclude that the London dispersive surface energy of H-β-zeolite/rhodium catalysts can be written as a function of two variables $\left(T, \theta \right)$ as follows:

$${\gamma }_s^d\left(T,\theta \right)=-4.8\times {10}^{-3}\times T^2+b\left(\theta \right)\times T+c\left(\theta \right)$$

(20)

Where $b\left(\theta \right)$ and $c\left(\theta \right)$ are given by Equations 18 and 19 in the interval $0\le \theta \le 1.25\%$. We observed that ${\gamma }_s^d\left(T,\theta \right)$ is constant independent from the rhodium percentage $\theta$ for $\theta >1.25\%$. These results were perfectly confirmed by the variations of ${\gamma }_s^d\left(T,\theta \right)$ plotted in Figure 6 as a function of $\theta$ for different values of temperature. A maximum of ${\gamma }_s^d$ was also confirmed at $\theta =0.50\%$.

We plotted the variations ${\gamma }_s^d\left(T\right)$ of H-β-zeolite/rhodium catalysts in Figure 7 as a function of the specific surface area S using the values in Table 1. The maximum of ${\gamma }_s^d$ was obtained for $S=603{\mathrm{m}}^2/\mathrm{g}$ corresponding to a rhodium percentage equal to $\theta =0.50\%$. A minimum of ${\gamma }_s^d$ was obtained at $S=622{\mathrm{m}}^2/\mathrm{g}$ corresponding to $\theta =1.0\%$. Whereas, the London dispersive surface energy became constant for $\theta \ge 1.25\%$ certainly due to the effect of the rhodium percentage of the surface energy of the catalyst.

3.2. Dispersive Adhesion Work of n-Alkanes on H−β-Zeolite/Rhodium Catalysts

By applying Equation 8, the various dispersive adhesion works $W_a^d$ of the different n-alkanes on H−β-zeolite/rhodium catalysts were determined as a function of temperature. The results were given in Table S3. The corresponding curves of the variations of $W_a^d\left(T\right)$ of the different n-alkanes were plotted in Figure 8. The results showed that the dispersive adhesion work increases when the carbon atom number of n-alkanes increases, while a decrease of $W_a^d$ was observed when the temperature increases (Figure 8). This result is a direct consequence of the decrease of the London dispersive surface energies of the solid surface and the organic solvents when the temperature decreases.

Furthermore, the curves plotted in Figure 8 showed linear variations of $W_a^d\left(T\right)$ versus the temperature. The equations of ${\mathit{W}}_{\mathit{a}}^{\mathit{d}}\left(\mathbf{T}\right)$ of the different n-alkanes adsorbed on H-β-zeolite/rhodium catalysts were given in Table 3 as a function of temperature at different rhodium percentages. The excellent linearity of ${\mathit{W}}_{\mathit{a}}^{\mathit{d}}\left(\mathit{T}\right)$ showed that the dispersive adhesion work can be represented by the general following:

$$W_a^d\left(T\right)=H_S^d-T{S}_S^d$$

(21)

where $H_S^d$ and $S_S^d$ are respectively the dispersive surface enthalpy and entropy of adhesion given in Table 3 for the different n-alkanes adsorbed on H-β-zeolite/rhodium catalysts.

The results in Table 3 led to the variations of the dispersive surface enthalpy ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}$ (mJ m^-2) and entropy ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}$ (mJ m^-2 K^-1) of n-alkanes adsorbed on H-β-zeolite/rhodium catalysts as a function of the rhodium percentage, plotted in Figure 9.

The dispersive surface parameters ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}$ and ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}$ given in Figure 9 strongly depended on the rhodium percentage $\theta$. Two zones were distinguished for the above surface variables. The first one was characterized by parabolic variations in the interval [$\theta =0\%$; $\theta =1.25\%$] with a maximum obtained at $\theta =0.50\%$. While a constant pallier for the curves of ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}$ and ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}$ was observed after $\theta =1.25\%$. The results in Table 3 and Figure 9 showing parabolic variations allowed giving the different second-degree equations of ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)$ and ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)$.

From Table 4, we can deduce the general equation ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)$ and ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)$ of n-alkanes adsorbed on H-β-zeolite/rhodium catalysts:

$${\mathit{H}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)=\mathit{a} {\theta }^2+\mathit{b} \theta +\mathit{c}$$

(22)

$${\mathit{S}}_{\mathit{S}}^{\mathit{d}}\left(\theta \right)=0.235 {\theta }^2-0.232 \theta +\mathit{d}$$

(23)

where $\mathit{a}$, $\mathit{b}$, $\mathit{c}$, and $d$ are coefficients depending on the carbon atom number $n$ present in the general formula $C_n{H}_{2n+2}$ of n-alkanes.

The coefficients $\mathit{a}$, $\mathit{b}$, $\mathit{c}$, and $\mathit{d}$ obtained using Table 4 and the equations 22 and 23 were given in Table 5 as a function of the carbon atom number $n$.

The results in Table 5 led to conclude that dispersive surface enthalpy and entropy of n-alkanes are function of the rhodium percentage and the carbon atom number of n-alkanes and can be therefore written as ${\mathit{H}}_{\mathit{S}}^{\mathit{d}}\left(\theta ,n\right)$ and ${\mathit{S}}_{\mathit{S}}^{\mathit{d}}\left(\theta ,n\right)$. The dispersive adhesion work of n-alkanes adsorbed on H-β-zeolite/rhodium catalysts can be given by Equation 24:

$${\mathit{W}}_{\mathit{a}}^{\mathit{d}}\left(\mathit{T}, \mathit{\theta },\mathit{n}\right)={\mathit{H}}_{\mathit{S}}^{\mathit{d}}\left(\mathit{\theta },\mathit{n}\right)-\mathit{T}{\mathit{S}}_{\mathit{S}}^{\mathit{d}}\left(\theta ,\mathit{n}\right)$$

(24)

The dispersive adhesion work is then function of the temperature, rhodium percentage, and the carbon atom number. This new and original result proved for the first time that the adhesion work is function of three surface variables $\mathit{T}, \theta ,\mathrm{and} n$.

5. Conclusions

The 2D- inverse gas chromatography technique at infinite dilution was used to determine the dispersive surface properties of H-β-zeolite/rhodium catalysts at different rhodium percentages by varying the temperature. The chromatographic measurements led to the values of the net retention volume of adsorption of n-alkanes on the solid surfaces and therefore to the dispersive free energy of adsorption. The use of our new thermal model taking into account the temperature effect on the surface area of n-alkanes allowed to determine the accurate values of the London dispersive surface energy ${\gamma }_s^d\left(T\right)$ as a function of temperature. The results showed parabolic variations of ${\gamma }_s^d\left(T,\theta \right)$ as a function of the temperature $T$ and the rhodium coefficient $\theta$. A similar variation of ${\gamma }_s^d$ as a function of the specific surface area of solid catalysts was also observed. The dispersive adhesion work, surface enthalpy, and surface entropy of n-alkanes adsorbed on H-β-zeolite/rhodium catalysts were determined. The results showed linear variations of the adhesion work $W_a^d\left(T\right)$ as a function of temperature for fixed rhodium percentage and n-alkane, and parabolic variations as a function of the rhodium percentage and the carbon atom number of n-alkanes. When the rhodium percentage and the carbon atom number varied, it was showed that the dispersive adhesion work $W_a^d\left(T, \mathit{\theta },\mathit{n}\right)$ was function of temperature, rhodium percentage, and carbon atom number.