Statistical Physics Modeling of Sorption Isotherms of Aluminum, Iron, and Indium on Tetraphenylporphyrin (H2TPP) and Tetrakis(4-tolylphenyl)porphyrin (H2TTPP): Phenomenological Investigation of Metalloporphyrins at the Molecular Level

A quartz crystal adsorbent functionalized with two promising porphyrins (the 5,10,15,20-tetrakis(4-tolylphenyl)porphyrin and the 5,10,15,20-tetraphenylporphyrin) was applied for the investigation of the adsorption phenomenon of aluminum chloride, iron chloride, and indium chloride. The aim is to prove new insights about the appropriate adsorption materials for metalloporphyrin fabrication. The equilibrium isotherms were measured at five adsorption temperatures (from 290 to 330 K) through the microbalance (QCM) method. The discussion of the experimental observations indicated that the adsorption of the aluminum chloride and the iron chloride was performed via a monolayer process. On contrary, the participation of the chloride ions in the double-layer adsorption of the indium chloride was explained by the layer-by-layer process. Overall, the statistical physics modeling of the experimental curves indicated that the number of ions per adsorbent site n was found inferior to 1 for all the adsorption systems (multi-interaction process for the three ions). Interestingly, the physicochemical investigation of the three adopted models showed that the complexation mechanism of the tested porphyrins was an endothermic process since the two steric parameters ( n and P M ) increased with the rise of the temperature. The FeCl3 curves were discussed via a monolayer adsorption model which includes the parameters a and b (lateral interaction description), indicating the lowest stability of the formed iron-porphyrin complex. The energetic study showed that the adsorption energies ∣ − Δ E 1 / 2 ∣ of AlCl3 on H2TTPP and H2TPP are superior to 40 kJ/mol (chemical adsorption mechanism), whereas the adsorption mechanisms of FeCl3 and InCl3 took place via a physical process since they presented adsorption energy values lower than 40 kJ/mol.


Introduction
In recent years, porphyrins have been studied as chemical sensor materials of various metals leading to the formation of metalloporphyrin complexes [1,2]. The metal-porphyrin complexes have received increased interest in several biological and environmental processes such as stereochemistry study and molecular recognition [3][4][5][6]. Note here that the nature of the central ion has influenced the photophysical properties of porphyrins [7,8]. For example, these tetrapyrrolic macrocycles are involved in the chlorophyll structure (magnesium-porphyrin complex) and the hemoglobin structure (iron-porphyrin complex), so they are known as pigments of life [8]. In addition, metalloporphyrin complexes were also used as ionophores in the development of potentiometric recognition sensors supported with their structural variety and coordination chemistry [7,8]. In particular, the photosensitizing properties of aluminum(III)-porphyrin have promoted its use as potential ionophore of fluoride [9,10]. Moreover, the aluminum(III)-porphyrin and the indium(III)-porphyrin have been potentially used as supramolecular building blocks [11] and numerous other areas (sensing, molecular recognition, polymerisation reactions, etc.) [12,13]. Surprisingly, it was remarked that while several papers have been devoted to metalloporphyrin complexes including the metal centers zinc, rhodium, and cobalt [14,15], little attention has been paid to the use of porphyrins as complexing compounds of aluminum(III), iron(III), and indium(III) (Figure 1) despite their contribution in the interesting fields indicated above. In fact, the adsorption of these three metals on two tested porphyrins (H 2 TPP and H 2 TTPP) was examined in the present investigation.
In the literature, many approaches have been applied for the investigation of the adsorption mechanism of metals such as the sorption/reduction of heavy metal ions in environmental pollution management [16,17]. Other experimental methods have been applied for the investigation of the adsorption mechanism [18][19][20], but in this work, the adsorption mechanisms of aluminum chloride, iron chloride, and indium chloride have been investigated by the QCM technique for many reasons. Firstly, this experimental method is a simple mass detector technique which requires the immobilization of porphyrins on the solid support of the quartz crystal. Then, the complexed mass of the metal into the macromolecule cavities can be controlled through the microbalance apparatus. Secondly, the metal-porphyrin complexes do not require an inert environment, and they are stable in the presence of water. Thirdly, the use of the QCM technique enables to plot the adsorption curves which describe the complexed amount of metallic ions on porphyrins at different temperatures. Lastly, the microscopic characteristics of the resulting film can be followed by easily fitting the experimental isotherms with analytical models [21].
According to the literature, the physical models were adopted by numerous authors to examine the adsorption problem. For example, the empirical expressions of the Langmuir and Freundlich models [22,23] have been applied for the description of the isotherm curves, but they did not give a deep physicochemical analysis of the adsorption process. Importantly, the analytical adsorption models of the present investigation were established through the innovative statistical physics theory [24,25] which provides the new vision about the metalloporphyrin description. Based on this idea, the fundamental aim of the modeling work of adsorption isotherms is to find the sufficient systematic model that can anticipate physical insights of porphyrin adsorption based on the physicochemical parameters of the statistical physics models [26].
The tetrakis(4-tolylphenyl)porphyrin (H 2 TTPP) and the tetraphenylporphyrin (H 2 TPP) are the porphyrins used in these experiments. They were synthesized based on a published method (the Adler-Longo strategy) [27]. Figure 2 shows the synthesis methods of porphyrins H 2 TPP and H 2 TTPP. H 2 TPP (Figure 2(a)) was synthesized through the reaction between propanoic acid (2 L) and benzaldehyde (40 mL). The synthesis of H 2 TTPP (Figure 2(b)) was performed by introducing 4-tolualdehyde (12.5 g) with the propanoic acid (110 mL). Note that the synthesis of the two tested adsorbents has been performed by following the same procedure (reflux for 30 min, allowing for cooling, filtration, and drying under vacuum for 3 hours). The dried solid mass of H 2 TPP was 9.76 g while the obtained mass of H 2 TTPP was 3 g. The two solid compounds of porphyrins were dissolved in chloroform giving two solutions with concentration of 2.9 10 -2 mol·L -1 .

Experimental
Measurements. The experimental QCM setup is presented in Figure 3.
The QCM measurements were performed based on the piezoelectric quartz crystal [28,29]. The crystal is a thin disc which is cut from the polished quartz (AT-cut). The fundamental resonant frequency of the quartz is 5 MHz, and the diameter of the disc is 2.54 cm [29]. A cleaning treatment was applied on the crystals by means of a Piranha solution at room temperature followed with rinsing with ethanol and drying with high purity nitrogen.
For the adsorption measurement, 60 μL of the adsorbents (porphyrins H 2 TPP and H 2 TTPP) was doped onto the clean crystal surface by the spin coating technique at 3500 rpm for 30 seconds. The functionalized crystals were dried at 393 K for 2 hours.
The functionalized crystal (adsorption cell) was placed in a Teflon probe, which was covered by a protective ring, and it was immersed in V s = 100 mL of pure water contained in the bain-marie. The stabilization of the resonant frequency in the reactor took about 1 hour, and we noted the crystal frequency F 0 after the stabilization. Then, we proceed with the injection of volumes of stock solutions of the adsorbates (AlCl 3 /Fe-Cl 3 /InCl 3 ) in the reactor. The added volumes of adsorbates V ad should be calculated according to the next equation: where V s is the initial volume in the reactor, c 0 is the concentration of the prepared adsorbate solution, and c f is the final concentration in the reactor.
At least 15 injections into the reactor were performed using a micropipette, then the variation of the resonant frequency (corresponding to the new adsorbate concentration in the reactor) was determined as follows: with F i as the resonant frequency after each adsorbate injection and F 0 characterizing the effect of the hydrostatic Adsorption Science & Technology     3 Adsorption Science & Technology pressure on the quartz crystal. It can be understood from this equation that the frequency variation Δf is only the result of the mass variation Δm of the metal on the porphyrin surface and does not consider any effect from the quartz crystal on the adsorption process since the hydrostatic pressure effect on the quartz (frequency F 0 ) is not considered in the measured frequency variation Δf .

Sauerbrey's Equation.
The deposited mass of the metallic ions on the surface after each injection was determined according to the Sauerbrey hypothesis which links the mass variation to the frequency change [30][31][32]: where C is the crystal sensitivity factor (Hz·cm 2 ·μg -1 ). Here, keep in mind that this relation should be applied when the resonant frequency variation is mainly due to the change of the deposited mass on the adsorption surface.
In order to justify the application of the Sauerbrey equation in a liquid medium, the total frequency should be quantified according to the crystal properties. This allows writing the total frequency as follows [19,33,34]: with, Δf m being the effect of the mass variation, Δf T being the effect of the temperature variation, Δf p being the effect of the pressure variation, Δf r being the effect of the crystal roughness, and Δf η,ρ being the influences of the liquid properties (viscosity and density of the solution) described by Kanazawa and Gordon [33].
Indeed, the influences of the temperature and the roughness on the total frequency should be neglected since the present investigation was carried out using polished crystals at a fixed temperature. The hydrostatic pressure influence was not taken into account according to Equation (2). Furthermore, low effects of the solution properties were noted, so they were not considered. Thus, the impact of the variation of the deposited mass is the dominant factor for the frequency change (Δf = Δf m ). This means that the experimental measurements of the frequency variations characterize only the interactions between the metals and the adsorbent sites of porphyrins. Therefore, the physicochemical characterization of the adsorption process should be only focus on the analysis of the interactions between the ions and the porphyrins without taking account of the quartz crystal sensor because the adsorption process is performed without the intervention of the crystal properties.
In fact, the adsorbed masses Δm (also noted Q A ) of the three adsorbates on the surface after each injection was determined from the resonant frequency variation (Equation (3)).

Results of Experimental
Measurements. The experimental adsorbed quantities of Al 3+ , Fe 3+ , and In 3+ on H 2 TPP and H 2 TTPP are plotted in Figure 4 at 290-330 K.
Observing carefully the experimental data of the six adsorption systems, it is clear that the behavior of the iso-therm curve depends on the type of adsorbate. The isotherms of aluminum chloride on H 2 TTPP and H 2 TPP (systems (a) and (a ′ )) show the same behavior (a unique saturation level for all the temperatures). This indicates that the two porphyrin supports adsorbed one layer of aluminum ions without contribution of the chloride ions in the adsorption process. The adsorbed quantities of FeCl 3 (systems (b) and (b ′ )) decrease after the saturation level. In this case, the presence of an irreversible phenomenon is possible at a high iron chloride concentration which reflects weak iron-porphyrin binding. The indium chloride isotherms (systems (c) and (c ′ )) present two stability states (multilayer ionic adsorption). In this situation, it can be suggested that the two tested porphyrins H 2 TPP and H 2 TTPP adsorb many layers based on charge neutralization between particles having opposite charge signs (layer-by-layer (LBL) process) [24,25,35,36].
In addition, comparing the performance of the six adsorption systems in terms of quantity, we can note the following order of adsorption performance: ). The adsorbed quantities are the highest for AlCl 3 -H 2 TTPP. Then, the adsorption compounds aluminum chloride and porphyrin H 2 TTPP can be recommended for a real application of the metalloporphyrin complex. In the following section, the microscopic investigation of this experimental result is carried out through the physical modeling of the experimental isotherms.

Statistical Physics Theory.
The estimated values of the physical model parameters represent the key to understand the adsorption mechanism. The progress of this physical treatment is firstly seen against the well-known model of Langmuir (empirical model) [22]. The Langmuir model is developed by considering that one particle can interact with only one adsorbent site without taking account of the external factors. In general, this assumption led to wrong scientific conclusions. On the contrary, our statistical physics models correct this assumption by introducing a parameter defined by the number of adsorbed particles per adsorption site; it is noted in general by "n." This correction can simply provide useful interpretations regarding the adsorption mechanism.
Based on the adsorption isotherm profiles, our statistical physics models can suggest various energies which are linked to different functional groups of the adsorbent surface contrary to the traditional equations of Langmuir [22] or Freundlich [23]. In the same direction, the statistical physics models are able to estimate the total number of the formed layer at all the reaction temperatures [24][25][26]. Note also that the multilayer adsorption of charged ions can be only carried out via a layer-by-layer (LBL) process [25,36].
A general analysis of the adsorption data led to test two models: the monolayer model for AlCl 3 and FeCl 3 and the LBL multilayer model for InCl 3 . The modeling work is then arranged as follows: (a) it is worth mentioning the general methodology which allows developing the physical models expressing the adsorbed amount corresponding to the adsorbate concentration, (b) the statistical physics model can be applied and discussed on the experimental data, and (c) the choice of the adopted model led to describing the adsorption reaction via its parameters.

Advanced Adsorption Models.
To establish the physical model expressions, the equilibrium of the adsorption reaction can be expressed via the next equation [25,37]: where A represents the tested adsorbates (AlCl 3 /Fe-Cl 3 /InCl 3 ), S is the adsorbent receptor site of porphyrin (H 2 TPP or H 2 TTPP), and A n − S is the resulting complex. The variable n is a stoichiometric parameter which describes the number of bonded ions per adsorbent site. In general, this parameter can identify the nature of the adsorption process (n ≤ 1: multi-interaction process, n ≥ 1: multi-ionic process) [38,39].
This system is characterized by the chemical potential (μ) and the temperature (T) imposing from the outside towards the considered system (grand canonical situation). In this case, the general expression of the partition function is written as a function of the occupation number (N i ), the adsorption energy (−E i ), and the Boltzmann factor (β) [37]: Concerning the adsorption via one energy level (−E) (one adsorbed layer), it can be expressed by [39] For the LBL adsorption process (double-layer or multilayer), two energies can be responsible for this process. Note that the adsorption of the first layer is characterized by the energy (−E 1 ), and the second energy (−E 2 ) is in relationship with the formation of the additional adsorbed layers. The partition function is written as [38] follows: For the double-layer model: For the multilayer model: with L as the number of adsorbed layers. Then, the average occupation number of P M porphyrin sites for the three models should be calculated using the next equation [37,40]: In this modeling work, the analytical development can be carried out by introducing the chemical potential of the perfect gas (μ p ). μ p is introduced in the mathematical development of the models through its expression which is written as a function of the partition-function of translation (z Tr ) and the number of adsorbates (N) [40,41]: where z tr can be written as a function of Planck's constant, the volume V, and the mass m of the adsorbed particle [37]: The investigation of the adsorption problem can be also performed using the chemical potential (μ r ) of the real gas. In this case, the lateral interactions involving the adsorbates at free state are taken into account by means of the parameters a (the pressure of cohesion) and b (the covolume) [24,25]: Then, the adsorbed quantity Q A (μg/cm 2 ) can be determined by the following equation [37,40]: Finally, we get the analytical expressions of the six adsorption models: The monolayer model of the ideal gas approach: where c 1/2 involves the molar adsorption energy (−ΔE 1/2 ) (kJ/mol), the adsorption temperature T (K), and the adsorbate solubility S (mol/L). It has the following formula: The monolayer model of the real gas approach: where the energetic coefficient w 1/2 is The double-layer and multilayer models include two energetic coefficients involving two molar adsorption energies (−ΔE 1 ) and (−ΔE 2 ). 6 Adsorption Science & Technology  The double-layer model of the ideal gas approach: where the energetic variables c 1 and c 2 are The double-layer model of the real gas approach: where w 1 and w 2 are The multilayer model of the ideal gas approach: where c 1 and c 2 are given by Equation (20). The multilayer model of the real gas approach: where c 1 and c 2 are written as with w 1 and w 2 are previously mentioned (Equation (22)).

Adjustment of Adsorption Models with Experimental
Isotherms. The six physical adsorption models were applied on the experimental isotherms by means of a numerical fitting program [42]. The mathematical fitting method is based on the Levenberg-Marquardt iterating algorithm using a multivariable nonlinear regression program [40,43]. The criteria to adopt a descriptive model are the determination coefficient R 2 , the RMSE coefficient (residual root mean square error), and the well-known AIC (Akaike Information Criterion) coefficient. The first error coefficient (R 2 ) is a standardized measure of the fit goodness [43,44]. If the coefficient R 2 value is near the unit, we are in the case of the best fitting model. The second error coefficient (RMSE) is a nonstandardized measure of the fit goodness [44][45][46][47]. The best fitting model is obtained when the RMSE value is inferior to 2. The third adjustment coefficient (AIC) shows the best fitting with the model having the lowest AIC values [36]. These three adjustment coefficients are interpreted to adopt a physical model. Tables 1 and 2 show the numerical values of the error adjustment coefficients.
According to Tables 1 and 2, the aluminum chloride isotherms (systems (a) and (a′)) should be analyzed by the monolayer model of ideal gas which shows R 2 values near the unit and low values of RMSE and AIC for the five reaction temperatures. However, the adsorption isotherms of FeCl 3 (systems (b) and (b′)) show the best coefficients of adjustment with the monolayer model of real gas. This explains that the decline of the FeCl 3 isotherms at high equilibrium concentration is fundamentally due to the lateral interaction impacts and confirms that the  Figure 5 illustrates the monolayer adsorptions of aluminum chloride and iron chloride and the double-layer adsorption of indium chloride on H 2 TTP and H 2 TPP.

Analysis of Physicochemical Parameters.
The mon-layer model (ideal gas), devoted for the description of the AlCl 3 isotherms, was governed by three parameters (the number of aluminum ions per receptor site n, the number of porphyrin sites P M , and the energetic parameter c 1/2 (Equation (16))). For FeCl 3 , the monolayer model (real gas) presents the parameter a (cohesion pressure) and the parameter b (covolume) in addition to the steric variables n and P m and the energetic parameter w 1/2 . The LBL double-layer adsorption of InCl 3 can be interpreted via four physicochemical variables (n and P M (steric variables) and c 1 and c 2 (Equation (20))).
All fitting parameters values are given in Tables 3 and 4. The study of the adsorption model parameters attributes interesting microscopic interpretations of the porphyrin complexation.
3.4.1. Steric Interpretation. Basically, the parameters n and P M are typified by a steric aspect. The product of these parameters is the result of the maximum adsorption capacity [48].
The evolution of the two steric parameters versus temperature is reported in Figure 6.    Figure 6(a), the n values are found inferior to 1 for the six adsorption systems. For example, regarding the values of this parameter that are inferior to 0.5, it is clear that the complexation mechanism is a multi-interaction process (the ions can be interacted with two adsorption sites) [39].
The fitted values of P M describe the number of porphyrin sites accessible to the adsorbate ions at each temperature. Furthermore, it is noted from Figure 4 that the temperature exerts exactly the same influence on the six 10 Adsorption Science & Technology complexation systems: once the temperature increases, the adsorption capacities increase. This is often explained from Figure 6 which shows that the values of the coefficients n ( Figure 6(a)) and P M (Figure 6(b)) rise with the temperature. It can be concluded that the thermal agitation effect favors the adsorption dynamics: the rise of temperature activates other receptor sites to contribute in the complexation process.  [49]. It can be concluded that the decrease of FeCl 3 isotherms (Figures 4(b) and 4(b ′ )) can be the result of the high adsorbate-adsorbate interaction which reflects a weak binding Fe 3+ -porphyrin compared to the Al 3+ -porphyrin binding.

Van der
Overall, it can be concluded that the use of aluminum chloride guarantees more stability during the metalloporphyrin formation.
The values of these two parameters for H 2 TPP and H 2 TTPP are summarized in Figure 7.

Adsorption Science & Technology
Generally, the interactions of the particles (lateral interactions) are characterized by the cohesion pressure a. One can see from Figure 7 that aðFeCl 3 -H 2 TTPPÞ < aðFeCl 3 -H 2 TPPÞ. Therefore, the H 2 TTPP is the best complexing porphyrin of the iron ions because it presents the lowest impact of the lateral interactions on the adsorption process. On the other hand, it should be noted, from Figure 7, that the H 2 TTPP presents the highest values of the covolume b: bð FeCl 3 -H 2 TTPPÞ > bðFeCl 3 -H 2 TPPÞ. In this case, the distance between the adsorbates is high which indicates that the iron ions can be easily adsorbed by the surface and causes an expansion in the adsorbed amount. This result confirms the suggestion of the porphyrin H 2 TTPP as a suitable material for the metalloporphyrin adsorption.
Furthermore, it is seen from Figure 7 that the parameter a declines with the expansion of the temperature for the FeCl3 adsorption while the covolume b increases. The decrease of the cohesion pressure indicates that the lateral interaction effect is low at high temperature. The increase of the parameter b reflects a strong distance between the adsorbates. The lateral interaction description explains the highest performance of the iron adsorption at 330 K and shows the endothermic nature of the studied process. (16), (18), and (20), the molar energies of the six adsorption mechanisms can be calculated by means of the energetic coefficients, c 1/2 for AlCl 3 , w 1/2 for FeCl 3 , and c 1 and c 2 for InCl 3 , which are calculated from the numerical simulation of the three adopted models [36,48].

Energetic Calculation. According to Equations
For aluminum chloride adsorption, For iron chloride adsorption, For indium chloride adsorption, The values of the molar energies and their variations via the temperature are reported in Table 5 and Figure 8.
According to Table 5, for the indium chloride adsorption, it is obviously remarked that the calculated values of |ð−Δ E 1 Þ | which characterizes the indium-porphyrin interaction are greater than those of |ð−ΔE 2 Þ | (interaction between the two adsorbed layers [36]). Therefore, we can conclude that the interaction |ð−ΔE 1 Þ | should be compared to the other adsorption system energies (|ð−ΔE 1/2 Þ | ) to evaluate the stability of the formed metalloporphyrin complexes.
It can be also observed that the molar energies rise with the expansion of the temperature (Figure 8). This can be interpreted by the endothermic behavior of the six studied processes.
Furthermore, the energies |ð−ΔE 1/2 Þ | of AlCl 3 are superior to 40 kJ/mol for all the temperatures. For this system, the adsorption is carried out via a chemical process involving covalent bonds (irreversible phenomenon) [39,[50][51][52]. In this case, the adsorption is a dissociative mechanism (there is no possibility of a desorption process) because there is a change in the electron density between the adsorbent and the adsorbed particle. Moreover, a multilayer adsorption is impossible due to the high interaction between the adsorbed layer and the adsorbent surface (e.g., AlCl 3 adsorption).
In contrast, all the other adsorption mechanisms (FeCl 3 and InCl 3 ) take place via a physical process since they present adsorption energy values lower than 40 kJ/mol [42,50,51]. During physisorption, the adsorbed particles retain their individual properties and there is no change in the structure of the components of the adsorption system. The system is weakly energetic (low adsorbate-adsorbent interaction), so the presence of a desorption phenomenon is possible (e.g., FeCl 3 adsorption). Moreover, the adsorbed layer can interact with other adsorbed layers and can act as a new template for the adsorption of the following layer (multilayer adsorption mechanism) such as the case of InCl 3 in the present investigation.

Conclusion
In this work, the experimental QCM tool is used to control the adsorption of aluminum chloride, iron chloride, and indium chloride on porphyrins H 2 TTPP and H 2 TPP. By analyzing the experimental isotherms, the AlCl 3 -H 2 TTPP complex was found as the best adsorption system in terms of reproducibility since it showed the highest adsorbed quantities. Theoretically, statistical physical models were suggested for the analysis of the experimental results. The numerical investigation confirmed that the chloride ions contributed only in the adsorption of InCl 3 (double-layer adsorption). The iron chloride adsorption was described by a monolayer model that considers the lateral interactions of the ions indicating the lowest stability of the formed iron-porphyrin complex. The physicochemical parameters of these models gave interesting microscopic insights about the complexation of the two porphyrins. Indeed, it was found that the six adsorption processes were governed by a multi-interaction process (n is inferior to 1). The thermal agitation caused an increase of the number of occupied sites P M which indicated that some porphyrin sites contributed to the adsorption only at high temperature (endothermic process). The energetic study, through adsorption energy calculation, demonstrated that the adsorption of AlCl 3 on H 2 TTPP is a chemisorption process involving covalent bonds.

Data Availability
The data that supports the findings of this study are available within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.