Marwa Assaf

Gaelle Martin-Gassin

Benedicte Prelot

Pierre-Marie Gassin
email: pierre-marie.gassin@enscm.fr

The Driving Forces of Cationic Dye Adsorption, Confinement and Long Range Correlation in Zeolitic Materials

Keywords:

Zeolitic materials are commonly used to capture emergent contaminants in water or in complex aqueous effluents. The efficiency of this adsorption depends strongly on the guest-host interactions and on the surrounding environment with possible co-adsorption of the solvent. Only few experimental techniques are available to probe in situ the sequestration processes at the solid / liquid interface. We propose in the present work to combine Second Harmonic Scattering (SHS) technique with Isothermal Titration Calorimetry (ITC) in order to investigate the adsorption and the confinement of an hemicyanine dye adsorbed inside Faujasite materials. The methodology described here permits to quantify the correlations between the confined dyes in the material, and thus give local information about the organization at the nanometer scale. Various impacts, such as the effect of solvent type, the silicon to aluminum ratio of the zeolitic adsorbent, are quantitatively estimated and discussed. This work highlights that the most correlated system matches with the higher adsorption capacity associated with the lower entropic contribution.

Introduction

Zeolites, porous crystalline structures of Aluminum Silicate, have great potential for a wide range in technical [START_REF] Li | Applications of Zeolites in Sustainable Chemistry[END_REF] , industrial [START_REF] Stocker | Characterization and Utilization of Natural Zeolites in Technical Applications[END_REF][START_REF] Colella | Use of Zeolitic Tuff in the Building Industry[END_REF] , agricultural [START_REF] Nakhli | Application of Zeolites for Sustainable Agriculture: a Review on Water and Nutrient Retention[END_REF] , environmental [START_REF] Misaelides | Application of natural zeolites in environmental remediation: A short review[END_REF] and biomedical applications [START_REF] Bacakova | Applications of zeolites in biotechnology and medicine -a review[END_REF] . These applications are based on their outstanding properties for sorption, ion-exchange and catalytic processes. Numerous studies have also emphasized the ability of zeolitic material for the purification and the capture of emergent contaminants from water [START_REF] Shaw | Surface Engineered Zeolite: An Active Interface for Rapid Adsorption and Degradation of Toxic Contaminants in Water[END_REF][START_REF] Jiang | High-silica zeolites for adsorption of organic micro-pollutants in water treatment: A review[END_REF][START_REF] Samanta | Advanced Porous Materials for Sensing, Capture and Detoxification of Organic Pollutants toward Water Remediation[END_REF][START_REF] Wang | Natural zeolites as effective adsorbents in water and wastewater treatment[END_REF][START_REF] Wang | Physical and chemical regeneration of zeolitic adsorbents for dye removal in wastewater treatment[END_REF][START_REF] De Smedt | Removal of Pesticides from Aqueous Solutions by Adsorption on Zeolites as Solid Adsorbents[END_REF][START_REF] Martucci | Influence of water on the retention of methyl tertiary-butyl ether by high silica ZSM-5 and Y zeolites: a multidisciplinary study on the adsorption from liquid and gas phase[END_REF][START_REF] Wang | Application of zeolite MCM-22 for basic dye removal from wastewater[END_REF][START_REF] Sarti | High-Silica Zeolites as Sorbent Media for Adsorption and Pre-Concentration of Pharmaceuticals in Aqueous Solutions[END_REF] or complex beverage like wine [START_REF] Noviello | Synthetic zeolite materials from recycled glass and aluminium food packaging as potential oenological adjuvant[END_REF] . Indeed, in addition to their valuable sorption selectivity, or exchange capacity, these materials otherwise exhibit high chemical stability, good recyclability and above all non-toxicity.

Zeolitic materials have been the object of numerous studies to tune their chemistry, their structure, their textural properties, to optimize their efficiency on the one hand [START_REF] Pagis | Hollow Zeolite Structures: An Overview of Synthesis Methods[END_REF][START_REF] Serrano | Synthesis strategies in the search for hierarchical zeolites[END_REF][START_REF] Shamzhy | New trends in tailoring active sites in zeolite-based catalysts[END_REF] and to understand the driving force of adsorption and confinement processes on the other hand.

The porous network and the presence of confined spaces bring new properties such as chemical reactivity under nanoconfinement [START_REF] Grommet | Chemical reactivity under nanoconfinement[END_REF] in general and for the case of zeolite their use in heterogeneous catalysis. Indeed, catalytic reactions in confined spaces exhibit unique behaviors compared to those observed on bulk materials [START_REF] Dai | Recent Advances in Catalytic Confinement Effect within Micro/Meso-Porous Crystalline Materials[END_REF][START_REF] Sastre | The confinement effect in zeolites[END_REF] . The confinement effects thus imply modification on catalytic reaction, as well as on diffusion and on adsorption/desorption. For application on sorption, there are various properties responsible for the modification of sorption behavior, with the structural and the textural properties together with the surface chemistry. This is particularly the case at the solid / liquid interface with additional physicochemical properties, and where the sorption is indeed a displacement. This involves the influence of the nature of the solvent [START_REF] Varghese | Origins of complex solvent effects on chemical reactivity and computational tools to investigate them: a review[END_REF][START_REF] Gould | Understanding solvent effects on adsorption and protonation in porous catalysts[END_REF] and the specific influence of the hydration of adsorbed and desorbed species [START_REF] Prelot | What are the main contributions to the total enthalpy of displacement accompanying divalent metal adsorption at the silica-electrolyte interface?[END_REF] . For zeolites, the adsorption properties are governed by the ability of adsorbed molecules to penetrate into zeolite voids and by their surface chemistry, with hydrophilic/hydrophobic balances, surfaces acidities (Lewis and Bronsted), and mainly their permanent charge. This charge is designed as the Cation Exchange Capacity (CEC), resulting from the isomorphic substitution of Silicon atoms by Aluminum within the crystallographic framework. Even if interesting theoretical results are reported concerning the understanding of the Bronsted Acid Site (BAS) role [START_REF] Gould | Understanding solvent effects on adsorption and protonation in porous catalysts[END_REF][START_REF] Bordiga | Probing zeolites by vibrational spectroscopies[END_REF][START_REF] Vimont | Analysing and understanding the active site by IR spectroscopy[END_REF] in sorption and catalytic properties [START_REF] Grifoni | Confinement effects and acid strength in zeolites[END_REF] , the various contributions of the mechanisms [START_REF] Grifoni | Confinement effects and acid strength in zeolites[END_REF][START_REF] Tabacchi | Supramolecular Organization in Confined Nanospaces[END_REF][START_REF] Ramamurthy | Controlling photochemical reactions via confinement: zeolites[END_REF] are not commonly investigated from the experimental and local point of view.

The aim of this work is to investigate the guest-host interactions and to rationalize the significance of electrostatic, Van Der Waals and steric interactions in the global adsorption process. To achieve that, the adsorption of a pyridinium iodide salt inside Faujasite materials has been chosen as reference system. The adsorption is studied by modulating the nature of the solvent and the charge inside the material as depicted in Figure 1. To characterize and quantify these interactions, UV-visible absorption method is used to determine the macroscopic adsorption isotherm, Isothermal Titration Calorimetry (ITC) measurement are performed in order to get macroscopic information about the thermodynamic of the adsorption and an original approach based on Second Harmonic Scattering (SHS), is employed to probe the local organization of the dye inside the material, within the micropores. This last technique, based on Second Harmonic Generation (SHG) phenomena, in which two photons at frequency ω are converted into a photon at the double frequency 2ω, has become increasingly popular since recent studies [START_REF] Van Cleuvenbergen | Morphology and structure of ZIF-8 during crystallisation measured by dynamic angle-resolved second harmonic scattering[END_REF][START_REF] Moris | Harmonic light scattering study reveals structured clusters upon the supramolecular aggregation of regioregular poly(3-alkylthiophene)[END_REF][START_REF] Duboisset | Salt-induced Long-to-Short Range Orientational Transition in Water[END_REF][START_REF] Gassin | Second-Harmonic Scattering in Layered Double Hydroxide Colloids: A Microscopic View of Adsorption and Intercalation[END_REF][START_REF] Gassin | Second-Harmonic Scattering Can Probe Hydration and Specific Ion Effects in Clay Particles[END_REF][START_REF] Pardon | Nonlinear optical signature of nanostructural transition in ionic liquids[END_REF][START_REF] Chen | Electrolytes induce long-range orientational order and free energy changes in the H-bond network of bulk water[END_REF][START_REF] Tocci | Second-Harmonic Scattering as a Probe of Structural Correlations in Liquids[END_REF][START_REF] Kim | Aligned Inclusion of Hemicyanine Dyes into Silica Zeolite Films for Second Harmonic Generation[END_REF][START_REF] Van Der Veen | Localization of pnitroaniline chains inside zeolite ZSM-5 with second-harmonic generation microscopy[END_REF] demonstrated that it is suitable to obtained information about organization in long range structured system. Recent works [START_REF] Duboisset | Salt-induced Long-to-Short Range Orientational Transition in Water[END_REF][START_REF] Boudjema | PySHS: Python Open Source Software for Second Harmonic Scattering[END_REF][START_REF] Duboisset | Long-Range Orientational Organization of Dipolar and Steric Liquids[END_REF] have developed a methodology to quantify the correlation between the nonlinear emitter, by analyzing the polarization resolved SHS signal. The novelty of this work is to combine these two approaches, global calorimetric measurement and local SHS measurement used until now to quantify correlation in homogeneous liquid, to probe the order of the guest molecules confinement inside the microporous host material. The hemicyanine dye is chosen because it is an efficient SHG active molecule, and previous works have demonstrated its insertion in zeolitic system [START_REF] Kim | Aligned Inclusion of Hemicyanine Dyes into Silica Zeolite Films for Second Harmonic Generation[END_REF] . The material studied here is a Faujasite type zeolite which consists of cubicoctahedrons called sodalite cages or β-cages. The assembly of these cages linked together by hexagonal prisms leads into a larger cavity of diameter 13 Å called supercage (SC) which is approximately the size of the guest hemicyanine dye molecule chosen. The β-cages are only accessible to solvent, whereas the supercages are accessible for solvent and organic dye [START_REF] Ramamurthy | Controlling photochemical reactions via confinement: zeolites[END_REF] . Faujasite Y with different Si/Al ratio were purchased from Zeolyst International: Faujasite CBV400, CBV720, and CBV780 with Si/Al ratio equal to respectively to 2.5, 15 and 40 and referred as FAU400, FAU720 and FAU780 in the following. The materials have been used as received, without modification, and are in the acidic form. The CEC of these different materials are theoretically deduced by the Si/Al ratio and equal to 4.7 meq.g -1 , 1.0 meq.g -1 and 0.4 meq.g -1 respectively (additional details are given in the SI, including the XRD patterns).

Acetonitrile, Ethanol, and DMSO were obtained from Sigma Aldrich at HPLC grade. Ultrapure water is 18 MOhm.cm resistivity. The hemicyanine dye is 4-(4-Diethylaminostyryl)-1methylpyridinium iodide, M=394.29 g.mol -1 also refereed as sDia in the following, purchased from Sigma Aldrich. All chemicals were used as received without further purification.

Fresh suspensions of Faujasite in the considered solvent were prepared at a solid / liquid concentration of 0.1 g.L -1 and used within 12h. Suspensions were sonicated 10 minutes and kept under constant stirring at 25 °C to prevent solid particles aggregation before using. The characteristic size of all the particles in all solvent is around 800 nm diameter +/-200 nm determined with DLS (additional details in SI).

2°) Adsorption isotherm:

The adsorption isotherms were determined using the depletion method in batch conditions by putting in contact the Faujasite suspension at 0.05 g.L -1 coming from the fresh suspensions and solutions of hemicyanine at various initial concentrations prepared from a 2 mM stock solution.

After overnight stirring at 25 °C, the suspensions are centrifugated at 11000 rpm during 1h and then filtered with polypropylene 0.2µm filter. The dye adsorbed onto the material is determined according to:

Γ 𝑎𝑑𝑠 = (𝐶 0 -𝐶 𝑒 )𝑉 𝑚 (1)
Where Γ 𝑎𝑑𝑠 is the dye amount adsorbed into the material, V is the volume (L) of the solution, and m is the mass of the solid phase (g), C0 is the initial concentration of the dye and Ce is the equilibrium concentration in the supernatant. This Ce was measured by UV-vis absorbance (JASCO V-670) at 460 nm on the supernatant.

3°) ITC measurements

Isothermal Titration Calorimetry (ITC) measurements were performed with a TAM III multichannel calorimetric device with nanocalorimeters and a Micro Reaction System (TA Waters). Experimental conditions (concentrations, solid /liquid ratio) have been optimized to be in correlation with the adsorption isotherm, and taking into account the technical constraints (cell or syringe volume, solubility of the dye) to ensure that the signal is high enough to be properly detected and quantified, and to reach the complete loading at the end of the experiment. ITC experiments were carried out in cells (hastelloy) with 800 µL of zeolithic suspension, at 298 K and consist of 25 injections of 10 seconds with 10 µL of the dye solution at 2 mM. The system is equipped with a gold paddle stirrer used at 90 rpm. In order to allow the system to stabilize between injections, the latter are spaced 45 min apart. The dilution effect is evaluated under similar conditions with only the solvent. The global heat effect were calculated after appropriate subtraction of the dilution effect, and then for each injection is has been related to the real adsorbed amount from the isotherm. [START_REF] Zajac | Calorimetry at the Solid-Liquid Interface[END_REF] The final results are represented as the cumulative displacement enthalpy since the enthalpy represents a global effect combining various contributions involved in the overall sorption process including adsorption / exchange, hydration and displacement. [START_REF] Prelot | What are the main contributions to the total enthalpy of displacement accompanying divalent metal adsorption at the silica-electrolyte interface?[END_REF] 4°) SHS measurement:

The polarization resolved SHS experimental setup is detailed elsewhere [START_REF] Gassin | Adsorbed Dyes onto Nanoparticles: Large Wavelength Dependence in Second Harmonic Scattering[END_REF] and additional details are given in SI. Briefly, the SHS intensity is monitored in the right angle direction as a function of the input polarization angle γ, which was selected with a rotating half-wave plate and the second harmonic light was selected in the vertical state by an analyzer, placed in front of the spectrometer.

All experimental data were recorded for 10 s under stirring conditions, a magnetic stirrer is operated in the cell measurement. For each system, two separate SHS measurements are performed. On the one hand, the SHS measurement of the whole suspension and on the other hand, the SHS measurement of the supernatant solution obtained after centrifugation. Formally, the SHS intensity of the suspension comes from 3 different contributions as depicted in formula ( 2 At the concentrations where the SHS experiment are performed, the re-absorption of the 2𝜔 photon by the solution is negligible [START_REF] Houbrechts | Hyper-Rayleigh scattering investigation of nitrobenzyl pyridine model compounds for optical modulation of the hyperpolarisability[END_REF] as it is discussed in SI. Thus, the SHS signal coming from the dye adsorbed onto and into the material is obtained by the subtraction equation (2) minus equation ( 3):

𝐼 𝑆𝐻𝑆 2𝜔 (𝛾) = 𝐼 𝑠𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛 2𝜔 (𝛾) -𝐼 𝑠𝑢𝑝𝑒𝑟𝑛𝑎𝑡𝑎𝑛𝑡 2𝜔 (𝛾) (4)
All the SHS data presented in the manuscript are treated according to equation ( 4).

Results and discussions

1°) The Si/Al ratio effect on the adsorption Figure 2A shows the adsorption isotherms of sDiA in water onto the different faujasite materials. According to the nature of the initial portion of the curve and its slope, the isotherms exhibit a shape corresponding to a L type (or Langmuir type) following Giles classification. The adsorption isotherms are thus well fitted with the Langmuir like model, equation ( 5), even if liquidphase adsorption of molecule into zeolites is not a situation traditionally characteristic of a Langmuir isotherm. The applicability of this model is discussed here [START_REF] Gould | Understanding solvent effects on adsorption and protonation in porous catalysts[END_REF] and results from a relatively strong interaction of the dyes with the pore walls.

Γ 𝑎𝑑𝑠 = Γ 𝑚𝑎𝑥 𝐶 𝑒 𝐾 𝐿 1+𝐶 𝑒 𝐾 𝐿 (5)
Through this simplified adsorption model used in its linearized form, the maximum amount of adsorbed dye Γ max, as well as the Langmuir constant KL have been evaluated for sDiA sorption on the various zeolites. For the sDiA adsorption, the maximum amount capacity varies from 0.622 mmol.g -1 for the FAU400 material, to 0.338 mmol.g -1 for FAU780 material. This finding is explained by the number of adsorption sites available depending on the Si/Al ratio of the material.

Indeed, these adsorption sites are located on the oxygen atom neighboring an aluminum atom and are thus refereed as Brønsted Acid Site (BAS) [START_REF] Gould | Understanding solvent effects on adsorption and protonation in porous catalysts[END_REF][START_REF] Bordiga | Probing zeolites by vibrational spectroscopies[END_REF][START_REF] Vimont | Analysing and understanding the active site by IR spectroscopy[END_REF][START_REF] Kondo | A comparative IR characterization of acidic sites on HY zeolite by pyridine and CO probes with silica-alumina and γ-alumina references[END_REF] . The different material FAU400, FAU720

and FAU780 exhibits different BAS number, or it is the same thing different CEC, which are respectively equal to 4.7 meq.g -1 (6.7 sites per SC), 1.0 meq.g -1 (1.5 sites per SC) and 0.4 meq.g -1 (0.6 sites per SC). For FAU780 material, the BAS amount limits the adsorption process because the Γmax measured to 0.49 molecule per SC is close to the theoretical CEC of 0.59 site per SC. It confirms that the cationic exchange is the driving force of the adsorption, and that other interactions like Van der Waals interaction are not efficient to yield dye adsorption. At higher charge density in the material, the adsorption is limited by the empty space available in the pore.

When the ratio of 1 molecule per SC is reached, no more adsorption can occur even if there is more BAS. The adsorption capacity Γmax is nearly the same for FAU720 and FAU400 materials, around 1 molecule adsorbed per SC, far lower from the theoretical CEC of 6.7 sites per SC for FAU400. It shows that the amount adsorbed depends on the number of available exchange sites, with a maximum limit due to the steric hindrance because of the size of the adsorbate. Figure 2B shows the displacement enthalpy measured by ITC during the adsorption process on the three different materials. The observed behavior is different for the various considered materials. ITC approach provides the chance to measure directly the heat effect all along the isotherm. The displacement enthalpy is determined for various surface coverage corresponding to the complete loading of the materials. Three 3 distinct areas can be observed depending on the amount adsorbed range. In the first region, at low adsorption, i.e. when Γ 𝑎𝑑𝑠 < 0.2 molecule per SC, the displacement enthalpy, obtained by the slope in the graphic 2B, is the same for the 3 materials with a value around ∆Hads = -10 kJ.mol -1 . In the intermediate region, 0.2 < Γ 𝑎𝑑𝑠 < 0.5 molecule per SC, the displacement enthalpy is nearly stable for FAU400, but it increases for FAU720 and 780 at around ∆Hads = -28 kJ.mol -1 . In the high adsorption region, this quantity continues to increase for FAU720 with high ∆H estimated at -50 kJ.mol -1 , whereas it slightly decreases for FAU400. For FAU780 at sorption higher than 0.32 mmol.g -1 , calculation does make sense since saturation is reached and no adsorption occurs. Those results can be discussed with a microscopic model that supposed independent filling of the SC from the neighboring SC. Indeed, in each Faujasite material, a SC is linked by a dodecagonal window to four other SC. In the low adsorption region, the adsorbed dye is in average involved in a SC surrounded by four empty SC.

Thus, no interaction between the dye occurs in this regime. At higher adsorption, 0.2 < Γ 𝑎𝑑𝑠 < 0.5 molecule per SC, an adsorbed dye begin to interact with another dye located in a neighboring SC.

For FAU720 and FAU780 material, the increase in the displacement enthalpy may be explained by this interaction. On the contrary, for FAU400 material, the presence of unoccupied sites could change this behavior with more spatial configuration available for the adsorption as depicted in Figure 2 right. To go further in the thermodynamics, the enthalpy measurements are compared with other thermodynamics quantities. As the system is a dilute solution of a charged adsorbate, the global free adsorption enthalpy Gads can be deduced from the constant extracted from equation ( 5) applied to the adsorption isotherm [START_REF] Liu | Is the Free Energy Change of Adsorption Correctly Calculated[END_REF] :

Δ𝐺 𝑎𝑑𝑠 = -𝑅𝑇𝑙𝑛(𝐾 𝐿 ) (6).
An estimation of the entropic term TSads can be done with the combination of the measured Hads from ITC and the calculated Gads. The results are given in table 1 and Figure 2D. Two ranges on the surface coverages have been distinguished, with ads lower or higher than 0.2 molecule per SC.

For low surface coverage, the entropic terms TSads is in the range of 30-20 kJ.mol -1 for the 3 materials, with a slight decrease from CBV-400 to CBV-780 in line with the small decrease of Gads. For higher loading, the difference in the entropic contribution between materials is more pronounced, with a high decrease of TSads for CBV-400 to CBV-780. FAU400 and FAU720 exhibit nearly the same adsorption capacity, but the adsorption thermodynamics quantities are strongly different with a higher entropic contribution for the FAU400. As mentioned above, this difference should be attributed to the presence of unoccupied sites. To probe locally the order/disorder of the dye organization inside the material, polarization resolved SHS measurement are presented in figure 2C. These data were analyzed as a Fourier series [START_REF] Duboisset | Salt-induced Long-to-Short Range Orientational Transition in Water[END_REF] :

𝐼 𝑆𝐻𝑆 (𝛾) = 𝑖 0 + 𝑖 2 cos(2𝛾) + 𝑖 4 cos (4𝛾) (7)
where i0, i2 and i4 are the amplitudes of the constant, the harmonic 2γ and the harmonic 4γ terms.

In Eq. ( 7), the parameters i0 and i2 are related to the local microscopic structure, i.e. the first hyperpolarizability of the dye and i4 to the long-range correlations. In the case of uncorrelated species, SHS is a purely incoherent phenomenon and the amplitude i4 vanishes [START_REF] Duboisset | Salt-induced Long-to-Short Range Orientational Transition in Water[END_REF] . On the contrary, when molecular orientations are correlated, the scattered photons have well-defined phase relationship and i4 differs from 0. To quantify this term, the normalized parameter I4 is introduced as :

I4= 𝑖 4 𝑖 0 (8)
I4 is equal to 0 when no correlation occurs between adsorbed dyes and is different from 0 when correlations between dyes exist. The I4 parameter deduced from the fit are presented in Table 1 and its absolute value is clearly higher for FAU720 compared to FAU400 and FAU780. As discussed above, FAU400 and FAU720 exhibit nearly the same filling of the material, and the higher I4 value for the FAU720 is puzzling. This trend is in line with the higher FAU400 entropic contribution and the most convincing explanation involves the unoccupied sites. For FAU 400 material, the dye molecule may probably change from one configuration to other one and for FAU720 the dye molecule is most confined as depicted in Figure 2 right. Thus, at high loading, unoccupied sites result in a loss of long range order and in a lower value for I4. 2°) The solvent effect on the adsorption Figure 3A shows the sDiA adsorption isotherms onto the FAU720 material with different solvents.

The maximum amount of adsorption varies with the solvent in the order Water>acetonitrile>Ethanol>DMSO. Those results follow the order recently published [START_REF] Gould | Understanding solvent effects on adsorption and protonation in porous catalysts[END_REF] obtained with pyridine onto various zeolites. These findings are explained by the efficiency of the exchange which depends on the ionic properties of the solvent. Indeed, if the proton is solvated, as depicted in Figure 3C, the cationic exchange occurs, and not on the otherwise. For the water solvent, Γ max reaches nearly the CEC, which means that BAS can be considered as "active" for the exchange.

For the other solvents, Γ max is lower than the CEC, and the BAS are partially "inactive" in respect with the lower relative permittivity value of the solvent. As depicted in figures 3C and 3D, the solvent effect could be explained by this difference in proton solvation. The impact of the material for the case of ethanol solvent is presented in Figure 4A. The results greatly differ from the water solvent. The adsorption capacity follows monotonically the Si/Al ratio. This shows that the adsorption is limited by the number of active BAS and not by the space available in the framework. Those results confirm that only a small part of the BAS are "active" with a proton solvated in ethanol. Figures 4B presents the SHS polarization plot for these systems.

Those results also confirm that when the material is few filled, no correlation is detected and that I4 increases with the filling of the material. A summary taking into account all the SHS results, is presented in Figure 5. It shows that I4

quantification draws a distinction between three kind of dye intercalations. For the case of low adsorption with Γmax lower than 0.4 molecule per SC, the I4 value is around 0, and thus the system exhibits no correlation between the dye. The molecules don't really interact with the others. For the case with moderate or high filling, i.e. Γmax higher than 0.5 dye per SC, I4 increase up to -0.25.

This evolution shows that long range correlation appears in the system when molecules interact with other located in the immediate surrounding SC. This could induce a percolation phase transition of the intercalated dye and explains the long rang correlation emergence. The last point in Figure 5B represent the case of the FAU400 materials in water. This system exhibits a specific behavior not in line with the other. This is probably explained by the unoccupied sites, which decrease the long range order in the system.

Conclusion

As a conclusion, this work demonstrates that the adsorption process involved in this system is a cation exchange mechanism. The amount adsorbed depends on the number of available exchange sites, with a maximum limit due to the steric hindrance because of the size of the adsorbate. The efficiency of the exchange depends on the ionic properties of the solvent. At high loading, the porous structure of the zeolite leads to long range order and to non-zero values for I4 parameter.

When the number of exchange sites is higher than the maximum achieved loading, unoccupied sites result in a loss of long range order and in a lower value for I4. The calorimetric measurements also show an increase of the entropic contribution with the unoccupied sites. Finally, this work demonstrates the ability of SHS measurements to probe the organization in buried system. In particular, it opens new perspectives to probe the local order and confinement of molecular inclusion in other class of material, thanks to the SHS measurement of correlations.

ASSOCIATED CONTENT Supporting Information. Additional information are given about the SHS experimental setup, the 2𝜔 photons reabsorption, the suspension characterization by DLS measurement, and the calculus of the theoretical cationic exchange capacity. The following files are available free of charge.

Funding Sources

Financial support for this work by the ANR project CAMOMILS (ANR-15-CE21-0002) is greatly acknowledged.

ACKNOWLEDGMENT

Figure 1 .

 1 Figure 1. Schematic representation of the three various elements (adsorbate, adsorbent and

  𝐼 𝐻𝑅𝑆,𝑑𝑦𝑒 2𝜔 is the uncoherent Second Harmonic light coming from the free dye in solution also referred in the literature as the Hyper Rayleigh Scattering (HRS) contribution of the dye; 𝐼 𝐻𝑅𝑆,𝑠𝑜𝑙𝑣𝑒𝑛𝑡 2𝜔 is the Hyper Rayleigh Scattering contribution of the solvent, and 𝐼 𝑆𝐻𝑆,𝑑𝑦𝑒+𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠 2𝜔 is the Second Harmonic light contribution of the dye adsorbed onto and into the material. The contribution of the bare Faujasite can be neglected as it is shown in the additional data presented in SI. The SHS intensity of the supernatant can be written as:

Figure 2 .

 2 Figure 2. A) sDiA in water adsorption isotherm onto the three Faujasite, B) Corresponding ITC

  Figures 3 B presents the SHS polarization plots for these systems. The case of water solvent exhibits a significative I4 contribution, as discussed above. All the other SHS polarization plots show a nearly zero I4 contribution as summarized in Figure 5.

Figure 3 .

 3 Figure 3. A) Adsorption isotherm for sDiA onto FAU720 with different solvents: water

Figure 4 .

 4 Figure 4. A) Adsorption isotherm for sDiA onto FAU400, FAU720 and FAU780 in ethanol, B)

Figure 5 .

 5 Figure 5. A), I4 parameters deduced from SHS experiments for sDiA adsorption onto the different

Table 1 .

 1 Results of the fitting parameters extracted from macroscopic experiment adsorption of sDiA, with the Γmax, the maximum amount adsorbed, KL the Langmuir like constant, Gads the corresponding free enthalpy obtained from equation (6), Hads measured by ITC, deduced TSads and I4 the normalized SHS correlation parameter. For Hads and TSads, the values are calculated for the two various regimes, with low surface coverage (in italic) and high surface coverage (in

	bold).
	Material	Γ max	Γ max	KL	Δ𝐺 𝑎𝑑𝑠	Δ𝐻 𝑎𝑑𝑠	TΔ𝑆 𝑎𝑑𝑠	SHS-I4 parameter
		[𝜇mol.g -1 ]	[molecule/SC]	[L.𝜇mol -1 ]	[kJ.mol -1 ]	[kJ.mol -1 ]	[kJ.mol -1 ]
						-9.3	30.0
	FAU400	622	0.9	10	-39.3			-0.10
						-8.7	30.6
						-11.0	23.9
	FAU720	588	0.85	1.7	-34.9			-0.26
						-26.7	8.2
						-12.1	19.8
	FAU780	338	0.49	0.5	-31.9			-0.01
						-29.2	2.7

We thank Lotfi Boudjema and Anne Galarneau for regular discussions about the zeolite structure. ABBREVIATIONS SHS, Second Harmonic Scattering; BAS, Bronsted Acid site; CEC, Cationic Exchange Capacity, SC, SuperCage.


Alberto Rodríguez Cuevas

Ricardo D Brancher
email: ricardo.d.brancher@gmail.com

Frédéric Topin

Stéphane Le Calvé
email: slecalve@unistra.fr

Irina Graur

Alberto Rodriguez-Cuevas
email: alberto.rodriguezc@alumnos.unican.es

Fréderic Topin
email: frederic.topin@univ-amu.fr

Fréderic Topin

Numerical simulation of the sorption phenomena during the transport of VOCs inside a capillary GC column

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Introduction

When two different solutes α and β flow through a Gas Chromatography (GC) column carried by a carrier gas, they can interact differently with the stationary phase. If species α is more strongly adsorbed on the stationary phase than the species β, this means its displacement along the column is slower, consequently, this species α will spend more time to reach the outlet of the column than the species β [START_REF] Harris | Quantitative Chemical Analysis[END_REF].

There are two main factors which indicate how well the compounds are separated inside the column: the retention time (RT) and the width of the peaks (WOP). The RT is a measure of the time taken for a solute to pass through a chromatography column. It is calculated as the time from injection to detection, which corresponds, in the absence of significant dead volumes between the injector, the column and the detector, to the time spent in the column. The difference in the RT of the species is crucial in the chromatographic process. If the values of retention time of two species are too similar, the separation will not be clear and therefore the identification of each analyte will not be possible.

The quality of analysis is also related to the WOP the narrower and taller the peaks the better. If the peaks of two species are too wide, they can overlap each other making it difficult to obtain accurate measurement of the area under the peaks, usually called peak area integration. Thus, it is indispensable that the peaks are separated from each other. Nonetheless, the RT and the width of the peak for a compound are not fixed as many factors can influence them even if the same GC system and column are used. These factors include carrier gas flow rate, temperature differences, varying from one analysis to another, or column degradation.

In the ideal case, the columns should be very long and narrow, while the retention times of the different analytes should be different enough to have fully independent peaks. However, in the reality the concentration of a solute moving through a chromatography column tends to spread into a Gaussian shape [START_REF] Harris | Quantitative Chemical Analysis[END_REF]. The longer the solute resides in the column, the wider the Gaussian peak becomes. Therefore, once satisfied the condition that the peaks should not overlap, it is desirable that the retention times are as short as possible, in order to decrease the time consumed by the separation. Aiming to obtain a chromatogram close to the ideal one, it is critical to select the optimal operating conditions, that is, the best parameters such as stationary phase, carrier gas, pressure difference, and temperature, among others.

In actual practice, when there is a necessity to acquire a new gas chromatography column, to separate one or more families of VOCs, the choice is based on some general guidelines and previous experience from the experts working in the field [START_REF] Barry | Columns for gas chromatography: performance and selection[END_REF]. The selection of the stationary phase polarity, its thickness, inner diameter and length of the column are often done upon some generic approximations and concepts. For example, in many occasion, the choice of the stationary phase composition is done using the concept "equal dissolve equal". Consequently, often the best column option is not the one eventually selected. Generally, the variables influencing the performance of the chromatographic process are so abundant that selecting a good chromatography column is indeed a very difficult task. The best choice of the parameters like temperature, use of isothermal condition, temperature ramps, inlet and outlet pressures, volume and duration of the injection, optimal velocity of the gas through the column or carrier gas type require a huge amount of systematic experiments. This means a lot of time, resources and efforts.

The objective of the work presented here is to develop a mathematical model and implement it in a numerical code to predict the behavior of the BTEX molecules separation inside a chromatographic column. It should be clarified that the objective here is not to create a complete program taking into account all the above-mentioned parameters, but developing a numerical code considering some of these parameters, which could be used as the first step in the process of choosing a column, namely, temperature, pressure, concentration of the species, properties of the carrier gas, length and radius of the GC column. Other parameters such as flow distortions due to the injections will not be considered. Moreover, the model will be evaluated in 1D following the column length, to reduce the computational cost, but considering the radial fluxes through integration over the column cross-section.

Several authors have developed alternative models to estimate retention times of aromatic molecules inside GC-columns [START_REF] Dose | Simulation of gas chromatographic retention and peak width using thermodynamic retention indexes[END_REF], [START_REF] Aldaeus | Prediction of retention times of polycyclic aromatic hydrocarbons and n-alkanes in temperature-programmed gas chromatography[END_REF], [START_REF] Aldaeus | Prediction of retention times and peak widths in temperature-programmed gas chromatography using the finite element method[END_REF], [START_REF] Bryan | Prediction of gas chromatographic retention time via an additive thermodynamic model[END_REF], [START_REF] Shukui | Predictions of comprehensive two-dimensional gas chromatography separations from isothermal data[END_REF]. This approach could be applied to the transport of various molecules flowing through a GC column. Among all the pollutants detected by using gas chromatography, the BTEX compounds (benzene, toluene, ethylbenzene, o-xylene, p-xylene and m-xylene) are the ones used as a reference in the present study to create and validate our numerical model. The harmful effects of this family of volatile organic compounds (VOCs) are well documented [START_REF] Hazrati | Preliminary assessment of BTEX concentrations in indoor air of residential buildings and atmospheric ambient air in Ardabil, Iran[END_REF], [START_REF] Rinsky | Benzene and leukemia: An epidemiologic risk assessment[END_REF] and several groups and companies are developing gas analyzers to detect them more accurately in the indoor environment [START_REF] Nasreddine | Portable novel micro-device for BTEX real-time monitoring : Assessment during a field campaign in a low consumption energy junior high school classroom[END_REF], [START_REF] Lara-Ibeas | Sub-ppb level detection of BTEX gaseous mixtures with a compact prototype gc equipped with a preconcentration unit[END_REF]. In light of the efforts being done to develop more and more accurate sensing systems [START_REF] Lyu | Tio 2 hollow heterophase junction with enhanced polluant adsorption, light harvesting, and charge separation for photocatalytic degradation of volatile organic compounds[END_REF], [START_REF] Yavarinasab | Graphene/poly (methyl mathacrylate) elecrochemical impedancetransdued chemiresistor for detection of volatile organic compounds in aqueous medium[END_REF], proper VOCs separation is highly needed to quantify even the smallest concentrations of each individual species. The chemical family studied in these articles is a good example of this necessity. The BTEX compounds have very similar chemical properties however the impact on human health varies a lot from one specie to the other. Benzene is a well-known carcinogenic when chronicle exposure at ppb levels while xylene only generate irritation in sharp exposures over dozens of ppm. Thus, a correct separation and identification is critical.

Table 1 presents some important physical and chemical characteristics of the BTEX compounds, while Figure 1 shows the chemical structure of the BTEX molecules. We propose a new methodology for the modeling of the properties of gas transport in the chromatography column, including advection and diffusion mechanisms, and also considering the adsorption and desorption on the stationary phase. This approach could be applied to the transport of various molecules flowing through a GC column even if we implement it to a particular column for the BTEX molecules analysis. In addition, the adsorption-desorption mech- anism could be also used for the pollutant detection, not only by CG column, but also by the microfluidics biosensors, see for example Ref. [START_REF] Hansen | Transient convection, diffusion, and adsorption in surface-based biosensors[END_REF]. This article is organized as following. First, the mathematical model is proposed to describe the interaction (in terms of adsorption-desorption) between the stationary phase and each of the analytes in the GC column in combination with the effects of the transport of the molecules by the carrier gas and diffusion. Afterwards, the proposed model is applied to simulate the behavior of the BTEX molecules in the chromatography column with PDMS as the stationary phase. Finally, to check the accuracy of the proposed model, the comparison between experimental data and the numerical results is presented.

Mathematical model

We start by a short description of the adsorption-desorption model used for the simulation of the interaction between the pollution species and the stationary phase. Then, we present the model of single gas transport through a long tube generated by the pressure difference applied at the tube ends. Finally, the diffusion and advection effects are quantified and the interaction between the mobile phase and stationary phase, in terms of the adsorption and desorption processes, are discussed.

Model of adsorption and desorption

Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface [START_REF] Langmuir | The construction and fundamental properties of solid and liquids[END_REF], [START_REF] Butt | Physics and Chemistry of Interfaces[END_REF]. This process creates a film of the adsorbate on the surface of the adsorbent. Adsorption is often brought on by ion exchange, that causes the mobile tracer to adhere to the surface and thus it becomes immobile.

In the simplest case, the adsorption process can be envisioned as a reversible chemical reaction where one adsorption site on the solid reacts with a molecule to produce an absorbed molecule. To describe the adsorption and desorption processes we denote N (t, x) the amount of adsorbed solute. Since the molecules become attached to solid particles, this amount N of adsorbed molecules is usually measured in number of adsorbed molecules per unit of solid surface area, [m -2 ].

The law of the evaluation of the number of the molecules on the solid surface, N , can be written as [START_REF] Wolkenstein | Electron Theory of catalysis on Semiconductors[END_REF]

∂N ∂t = R ads -R des , (1)
where R ads and R des are the adsorption and desorption rates, respectively. If we assume that the adsorption-desorption process equilibrates on a faster time scale compared to that of the gas transport, then the adsorption and desorption are always in equilibrium, i.e. R ads = R des .

To estimate the number of gas molecules hitting the surface per second, we recall the kinetic theory of ideal gases [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF]. In textbooks of physical chemistry [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF], [START_REF] Butt | Physics and Chemistry of Interfaces[END_REF] the rate of effusion of an ideal gas through a small hole, which can be associated to the adsorption rate, is given by

R ads = σAp √ 2πmk B T N a , (2)
where A is the cross-sectional area of one binding site. It can be estimated as a molecular cross-section area, which is of the order of Å2 , σ is the adsorption probability. Additionally, p is the gas pressure (the partial pressure is used instead of pressure, in the case of gas mixture), k B is the Boltzmann constant, m is the molecular mass, T is the temperature, N a is the number of the site available for adsorption per unit surface area. When the maximal adsorption capacity of a solid surface, N max , i.e. the total number of available adsorption sites per unit of area, is non infinite number, then the number of the sites available for adsorption could be written as:

N a = N max -N. (3)
If we assume that each molecule that hits the surface sticks to it with a probability σ, expression (2), then we can define the adsorption coefficient as:

K ads = σA √ 2πmk B T . (4)
On the other hand, the desorption rate is proportional to the number of the molecules on the solid surface, N , and it is expressed as:

R des = K des N, (5)
where the desorption constant, K des is calculated from:

K des = ν exp - Q R g T , ( 6
)
here ν is the oscillation frequency, ν = 1/τ 0 , τ 0 is the adsorption time, R g =8.314

[J•K -1 mol -1
] is universal gas constant and Q is the heat of adsorption [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF], [START_REF] Butt | Physics and Chemistry of Interfaces[END_REF].

If the reaction proceeds so slowly that it does not have time to come to local chemical equilibrium, then the kinetics of adsorption-desorption is not instantaneous and requires a dynamic rate law for its description, as Eq. [START_REF] Harris | Quantitative Chemical Analysis[END_REF]. Therefore, the rate of adsorption depends on both quantities, n and N , being n the gas number density, which is related to the pressure through equation of state, n = p/(k B T ). We could reason, for example, that the adsorption rate should increase as the number density n of solute increases but, as more and more chemical is adsorbed, the ability of the solid to adsorb additional chemical decreases. The simplest model with these characteristics is:

∂N ∂t = K ads (N max -N )nk B T -K des N. (7)
When the simultaneous adsorption and desorption of different species is considered, the changes in the surface number density for species i, N i , is expressed as

∂N i ∂t = K ads,i n i k B T   N max - l j=1 N j   -K des,i N i , i = 1, l, (8)
l is the total number of pollution species, excluding the carrier gas. Since the flow of carrier gas is constant, we assume that its adsorption and desorption rates are in equilibrium and do not change the total number of available adsorption site per unit of surface area. In the following we assume that the surface covered by the carrier gas molecules is already accounted in the evaluation of N max and it does not change during the chromatographic analysis. Some authors have studied adsorption and desorption phenomenon of the BTEX molecules on different adsorbents [START_REF] Zhao | Adsorption and Diffusion of Benzene on Chromium-Based Metal Organic Framework MIL-101 Synthesized by Microwave Irradiation[END_REF], [START_REF] Lara-Ibeas | Adsorbent screening for airbone btex análisis and removal[END_REF]. However, there is no available data relative to coefficients of adsorption and desorption of BTEX molecules on the stationary phase of this particular column.

Model of carrier gas flow in a column

The flow of a carrier gas through the chromatography column transports a very small quantity (order of tens of ppb) of molecules of BTEX. Since the amount of BTEX is indeed very small, these molecules cannot perturb the dynamics of the carrier gas flow. Therefore, this flow is modeled from Navier-Stokes equations applied to compressible gas, flowing through a circular tube driven by a constant static pressure difference. The gas properties are supposed to be those of carrier gas. Besides, the gas flow inside the GC column can be considered as isothermal, since the temperature is set by the operator and maintained at a constant value during an analysis. In addition, a typical GC column have very small radius to length ratio and the gas velocity along the column is much higher than the radial velocity. These properties allow us to reduce the system of compressible Navier Stokes equations to the Stokes equation, see Refs. [START_REF] Van Den Berg | Compressible laminar flow[END_REF], [START_REF] Graur | Analytical and numerical description for isothermal gas flows in microchannels[END_REF].

The Stokes equation completed by the non-slip boundary condition on the column surface and zero velocity gradient on the column axe of symmetry was solved analytically in Ref. [START_REF] Graur | Analytical and numerical description for isothermal gas flows in microchannels[END_REF]. The complete expression of the gas velocity u parallel to the column axis, denoted in following as x, reads

u(r, x) = - R 2 4µ dp dx 1 - r 2 R 2 , ( 9
)
where r is the radial axis and R is the column radius. The viscosity coefficient µ is calculated taking into account its temperature dependence in the form [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]:

µ = µ ref T T ref ω . (10)
In the previous expression µ ref is the viscosity at the reference temperature

T ref = 273.15 [K],
and ω is the viscosity index [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF].

The dependency of the velocity from r variable can be eliminated by averaging over the column cross-section:

u(x) = 1 πR 2 R 0 2π 0 u(r, x)rdrdϕ = 2 R 2 R 0 u(r, x)rdr = - R 2 8µ dp dx . (11)
The mass flow rate through a column cross section, A c , is calculated from the following expression:

Ṁ = πR 2 ρu(x) = A c ρu(x) = Ac ρu(r, x)dA c = - πR 4 p 8µRT dp dx . (12)
In the previous equation, ρ is the gas density, R is the specific gas constant, R = R g /M, and M is the gas molar mass. By integrating Eq. ( 12) from 0 to x, the pressure distribution along the column can be calculated from:

p(x) = p in 1 - x L (1 -P 2 ), P = p out p in (13)
and the pressure gradient along the column by:

dp(x) dx = - p in 1 -P 2 2L 1 - x L 1 -P 2 -1/2 . ( 14
)
As previously mentioned, the flow is isothermal. Thus, if the pressure distribution, Eq. ( 13), is known, the number density, n, or density, ρ, distributions can also be calculated from the equation of state. Finally, to calculate the carrier gas velocity distribution along the GC column, ( 9), ( 14), the geometrical parameters of the column, the pressures at the column ends and the gas nature must be specified. One example of the velocity and pressure profiles along the column is provided in Appendix A.

Model of multi-species flow in a GC column

To describe the transport of a gas species inside the CG column we use the continuity equation of a species i, Ref. [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF], called also the convection-diffusion equation. This equation is expressed in cylindrical coordinates as:

∂n i ∂t + ∂(n i u) ∂x = 1 r ∂ ∂r rD iN2 ∂n i ∂r + ∂ ∂x D iN2 ∂n i ∂x , (15)
where n i = n i (t, r, x) is number density of species i, which is supposed here to be also the column cross-section coordinate, r, dependent. The carrier gas velocity, u(r, x), depends also from two r and x coordinates and is given by Eqs. ( 9), [START_REF] Hansen | Transient convection, diffusion, and adsorption in surface-based biosensors[END_REF]. As it was done for the carrier gas transport modeling, in Section 2.2, we neglected the radial component of macroscopic velocity compared to the longitudinal one. In addition, the terms of the thermo-diffusion are also neglected and only mass diffusion is considered. As the concentration of the analyzed species in the mixture is very small, of the order of several ppb, compared to that of the carrier gas, only a binary diffusion coefficient is considered. Therefore the effect of the collisions among all the other species is neglected. Therefore, here D iN2 is the binary diffusion coefficient of a molecule of BTEX (i index) with the carrier gas, N 2 (N 2 index), see Appendix B for the details of binary diffusion coefficient calculation and its typical values. Equation ( 15) can be averaged over the column cross-section, introducing the averaged over cross-section number density as

n i (t, x) = 2 R 2 R 0 n i (t, r, x)rdr. (16)
In addition, we use the average over the cross-section velocity u according to Eq. [START_REF] Lara-Ibeas | Sub-ppb level detection of BTEX gaseous mixtures with a compact prototype gc equipped with a preconcentration unit[END_REF]. Besides, the integral over the cross-section for the first term of the right-hand-side of Eq. ( 15) can be calculated explicitly, therefore we obtain:

∂n i ∂t + (∂n i u) ∂x = - 2 R D iN2 ∂n i ∂r r=R + ∂ ∂x D iN2 ∂n i ∂x . ( 17
)
It is important to note that several terms were neglected during this averaging procedure, but they are considered not essential for the presented analysis. The first term in the right-hand side of previous equation represents the mass flux from (or to) the column surface. This flux can be evaluated using the adsorptiondesorption fluxes balance as

- 2 R D iN2 ∂n i ∂r r=R = 2 R (R adsi -R desi ) . ( 18
)
This condition is a balance between diffusive flux in the normal direction to the surface and net adsorption-desorption rate, so condition [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF] allows to evaluate the molecular flux on the column surface, r = R, in Eq. ( 17). In addition, this adsorption-desorption flux balance can be calculated via Eq. ( 7). Thereby, the concentration of a species i can be obtained from:

∂n i ∂t + (∂n i u) ∂x = - 2 R ∂N i ∂t + ∂ ∂x D iN2 ∂n i ∂x . ( 19
)
For simplicity we omitted the "bar" over the velocity and concentration in Eq. [START_REF] Zhao | Adsorption and Diffusion of Benzene on Chromium-Based Metal Organic Framework MIL-101 Synthesized by Microwave Irradiation[END_REF]. The term ∂N i /∂t quantifies the evolution in time of the number of adsorbed molecules and represents the balance of the adsorption-desorption fluxes, R adsi -R desi . The explicit expressions for the adsorption and desorption fluxes depend on the adsorption-desorption model used in the analysis. The evaluation of the arial concentration of the adsorbed molecules in time is calculated here from Eq. ( 8). Therefore, in the following we solve numerically the system of two equations for each species i (i = 1, l):

∂n i ∂t + ∂(n i u) ∂x = - 2 R ∂N i ∂t + ∂ ∂x D iN2 ∂n i ∂x , ∂N i ∂t = K adsi   N max - l j=1 N j   n i k B T -K desi N i . (20)
It should be mentioned that the second equation of system [START_REF] Lara-Ibeas | Adsorbent screening for airbone btex análisis and removal[END_REF] requires the information about the part of surface covered by other pollutant molecules, so this system has to be solved simultaneously with the analogous systems for other species.

The initial and boundary conditions have to be specified for the system [START_REF] Lara-Ibeas | Adsorbent screening for airbone btex análisis and removal[END_REF]. We can assume that initially there is not any species in the CG column

n i (0, x) = 0, 0 < x < L, (21)
and that the column surface is bare (non pollutant molecule adsorbed):

N i (0, x) = 0, 0 < x < L. (22)
The determination of the maximum of adsorption capacity, N max value, is provided in Section 5.2. The composition of gas at inlet is known (see Section 5.1) and the pollutant is injected during a controlled time, thus the associated inlet boundary condition is simply:

n i (t, 0) = n inj , 0 ≤ t ≤ t inj , (23)

Experimental Approach

In order to validate the mathematical model and its numerical implementation we acquired experimental data using the facilities and devices provided by In'Air Solutions and ICPEES. The experimental setup consists of one gas cylinder of pure nitrogen, one gas cylinder of 100 ppb of each species of the BTEX mixture, 600 ppb in total. A flow regulator that controls the flow of BTEX mixture and a pressure regulator that controls the pressure at the inlet of the column and therefore the velocity of the carrier gas, N 2 , through the chromatography column. A 6-port valve in combination with a sampling loop, sends the right volume of mixture towards the column in a two steps process. That way, 200 µL of 100 ppb of each species in the BTEX mixture is accurately injected into the GC column. When the injection is started the software starts recording the signal from the PhotoIonization Detector (PID).

The characteristics of all the components previously introduced are presented in Table 2, while a schematic representation of experimental configuration is displayed in Fig. 2. The two components that affect more directly the BTEX molecules separation and detection are the GC column and the PhotoIonization detector. The GC column is a 20-m long capillary column with 0.18 mm of internal diameter, RXi-624 stationary phase, 1 µm film thickness (Restek, Bellefonte, PA, USA). Its coating is a combination of 95% PDMS + 5% cyanopropyl phenyl. This kind of coating is slightly polar and was selected among others based on experimental data. The detection unit is an eVx Blue mini PhotoIonization Detector (PID) (Baseline MOCON, Lyons, CO, USA) equipped with a high performance 10.6eV ultra-violet lamp which has different sensitivities for each compound. Different sensitivities means also different response factors. The BTEX analyzer do not incorporate any sort of purge, trap, headspace of precconcentrator that can influence the flow. More details about the GC system and BTEX analyzer can be obtained in Refs. [START_REF] Nasreddine | Portable novel micro-device for BTEX real-time monitoring : Assessment during a field campaign in a low consumption energy junior high school classroom[END_REF], [START_REF] Lara-Ibeas | Sub-ppb level detection of BTEX gaseous mixtures with a compact prototype gc equipped with a preconcentration unit[END_REF], [START_REF] Rodríguez-Cuevas | Easy-to-manufacture micro gas preconcentrator integrated in a portable gc for enhanced trace detection of btex[END_REF].

Experimental conditions

A series of the measurements are carried out, varying two parameters: the pressure imposed in the inlet of the column, p in , (p out fixed at atmospheric pressure) and temperature of the column. These nine operating conditions are presented in Table 3.

Below we provide the details on the "standard case", case E, see Table 4. For the case E the pressures at the inlet and outlet sections of the GC column are fixed to 4bar and 1bar, respectively, with the constant temperature equal to 65 • C. The volumetric flow rate is measured on both inlet and outlet sections of the column, see Table 4. From these flow rates, the carrier gas velocities in the inlet and outlet cross-sections of the column are calculated. Obviously, as ∆p/p > 1, the variations of the gas density with position are significant and the volumetric flow rate (velocity) varies along the column. For case E, inlet velocity is 0.4 m/s while outlet one is 1.64 m/s, see also

Numerical implementation

As the BTEX concentration is very low, dilute approximation is chosen for modeling the advection, and only binary diffusion (pollutant in the carrier gas) is considered. On the other hand, as the species adsorb simultaneously on the solid, the available adsorption sites for a given species at a time moment depend on the already adsorbed amount of all species. Finally, we introduce the specific surface (m 2 /m 3 ) of the solid, 2/R, to express all molecule density per unit of fluid volume, i.e. N V i = 2N i /R, so N V i is now the number of molecules on the column surface per unit of volume. Thus, the system [START_REF] Lara-Ibeas | Adsorbent screening for airbone btex análisis and removal[END_REF] one (all equations are solved simultaneously):

∂n i ∂t + ∂(n i u) ∂x = - ∂N V i ∂t + ∂ ∂x D iN2 ∂n i ∂x , ∂N V i ∂t = K adsi   N V max - l j=1 N V j   n i k B T -K desi N V i . (24)
To numerically solve the system (24) we used a simple Finite Volume approach, with a spacial discretization step, ∆x, of 1mm and a temporal one, ∆t, of 0.5 ms, so the volume of each numerical cell, V cell = πR 2 ∆x, is equal to 2.49 • 10 -11 m 3 . For convenience we choose to express both densities, n i and N V i in number of molecules per computational cell, by multiplying both equations of system (24) by the volume of the computational cell, V cell . Now n is the number of molecules of species i in one millimeter of the column in gas phase, while N i is the number of molecules of species i adsorbed in one millimeter of the stationary phase of the column, so the system (24) becomes

∂n i ∂t + ∂(n i u) ∂x = - ∂N i ∂t + ∂ ∂x D iN2 ∂n i ∂x , ∂N i ∂t = K * adsi   N max - l j=1 N j   n i k B T -K desi N i . (25)
In the second equation K * adsi is equal to K adsi /V cell . The system (25) represents simultaneously the variation of number of molecules of species i in each numer-ical cell, or in other words, the variation in the number of adsorbed, desorbed and transported molecules of species i per millimeter of the column.

The initial conditions (injection of the BTEX species) and the boundary conditions (the maximum site available for adsorption on the column surface) need to be estimated.

Calculation sampling loop

The sampling loop has a length of 0.25 m and an internal volume of 200 µL. It is filled with a homogeneous mixture of BTEX at atmospheric pressure. Nevertheless, when the six-port valve is switched to the second position the pressure suddenly increases up to 4 bars. This increase is generated by the stream of N 2 coming from the second tubing line. The nitrogen compresses the initial content of the sampling loop in a space 4 times smaller, 50 µL, and therefore increases the concentration by a factor of 4. Thus, even though the injection time needed to transport the content of the sampling loop into the column is 16 seconds, the first 4 seconds contains the large majority of the BTEX molecules. The simulation of the injection in the GC-column were performed taking this fact into consideration.

The experimental chromatograms were obtained when injecting an initial concentration of 100 ppb of each species. Considering that the temperature of the column is T = 338 K, and the pressure at the inlet is 4 bars we can transform the concentration from ppb to molecules per cubic meter using the well-known ideal gas law

pV = N R g T, ( 26
)
where N is amount of substance, [mol], V is the gas volume [m 3 ]. Under our injection condition, from Eq. ( 26) we calculate N /V equal to 142.33 mol•m -3 .

To obtain the initial concentration in molecules per cubic meter we use the formula

n inj = N V c inj N A , (27)
where N A =6.022•10 23 [mol -1 ] is the Avogadro number. From Eq. ( 27) we calculate n inj =8.57•10 18 [molecules•m -3 ]. Thus, the inlet concentration is fixed at n i = 2.134 • 10 8 molecule/mm during the first 4 second.

Calculation of the number of free sites

The RXt-624 column is coated with a product which consists in 95% PDMS and 5% cyanopropyl phenyl. The total available for adsorption specific surface area of this coated layer is unknown, however we can get an approximation using published data from the literature. The authors of Ref. [START_REF] Noh | Fabrication and characterization of Aerogel-Polydimethyl Siloxane (PDMS) insulation film[END_REF] studied a similar PDMS sample and they obtained a specific area of 832 m 2 •g -1 and a density of 1.095 g•cm -3 . Our column is 20 m long, has an internal radius of 90 µm and a coating thickness of 1 µm resulting in a total volume of coating of 1.13 • 10 -2 mL or an equivalent mass of 1.24 • 10 -2 g. If we multiply this mass by the specific area, we obtain the available surface of 10.3 m 2 . We can estimate the effective surface of each molecules using their diameter, see Table 1. However, six BTEX molecules can adsorb to the surface, so we use the average over all six molecules radius, equal to 6.23•10 -10 m. Assuming a monolayer of adsorption in accordance to the Langmuir theory [START_REF] Butt | Physics and Chemistry of Interfaces[END_REF], [START_REF] Langmuir | The construction and fundamental properties of solid and liquids[END_REF], and identifying a surface needed to adsorb one molecule as a square of 6.23 • 10 -10 m of side, we obtain a total number of free places for "an average BTEX molecule", of N V max = 5.23 • 10 25 in the whole column and N max = 1.302 • 10 15 in a computational cell. This last value is several orders of magnitude higher than the number of molecules that we are injecting in the column. Therefore, even if the error of this assumption is of one or two orders of magnitude it would make no difference in the overall results.

Numerical scheme

The system [START_REF] Rodríguez-Cuevas | Easy-to-manufacture micro gas preconcentrator integrated in a portable gc for enhanced trace detection of btex[END_REF] was splited in two parts: the advection-diffusion and the adsorption-desorption one. At each time step we first solve the advectiondiffusion part using a simple explicit upwind scheme [START_REF] Regmi | Micro gas chromatography: An overview of critical components and their integration[END_REF] then add the contribution of the adsorption-desorption (30) -(32).

The numerical scheme for the advection-diffusion part is displayed here

(n i ) * k = (n i ) j k -∆t/∆x (n i u) j k -(n i u) j k-1 + ∆t/(∆x 2 ) D j k+1/2 (n i ) j k+1 -(n i ) j k -D j k-1/2 (n i ) j k -(n i ) j k-1 , k = 1, N L -1. ( 28
)
In previous equation, k index denotes the point k of the grid in physical space (along the column), j index denotes the time space, * superscript indicates intermediate results during a time step, D j k±1/2 = 0.5 (D iN2 ) j k + (D iN2 ) j k±1 . To approximate the reaction equation the number of free sites available for adsorption is introduced as (N free ) j k = (N max ) k -l q=1 (N q ) j k , so the adsorption and desorption terms are approximated as following

(R adsi ) j k =K adsi (N free ) j k (n i ) * k k B T, (R desi ) j k = K desi (N i ) j k . ( 29
)
Then the update of all parameters at the new time step is carried out, for the number of molecules in the gas phase

(n i ) j+1 k = (n i ) * k -∆t (R adsi ) j k -(R desi ) j k , (30)
for the number of adsorbed molecules

(N i ) j+1 k = (N i ) j k + ∆t (R adsi ) j k -(R desi ) j k (31)
and finally for the free places available for adsorption considering the balance of all six species:

(N free ) j+1 k = (N free ) j k -∆t l q=1 (R adsq ) j k -(R desq ) j k . ( 32
)
The last relation ensure the conservation of number of particles during one time step. The described numerical procedure was reproduced up to the injected molecules left the column.

Adsorption and desorption constants

When it comes to the numerical implementation, model input parameters and equations of state are not easily available. This happens not only for this particular combination of stationary phase and mobile phase, but also for most of the analysis in pharmaceutical, biochemical, environmental and general gas chromatography. Therefore, the constants of adsorption and desorption, K ads and K des , were obtained following an iterative process using all the measured retention time values for all compounds.

Results

In this Section, first the chromatograms of BTEX compounds obtained from nine experiments for different operating conditions, Table 3, are presented and analyzed. After that, based on the chromatogram obtained from the experiment under standard operating condition, case E, the adsorption and desorption coefficients, K ads , (4), and K des , ( 6), of all the BTEX compounds are calibrated. Then, the numerical simulations are carried out for five from nine available operating conditions and the numerical results reproducing the variation of the concentration of all BTEX species in time along the column are compared to the experimental results.

Experimental results

The series of the experimental measurements are carried out, by varying two parameters: the pressure imposed in the inlet of the column, p in , and temperature of the column, T , see Table 3. The injection condition remains the same, i.e. the volume of 200 µL (at 4 atm and room temperature) with the injected concentration of BTEX of 100 ppb.

The experimental chromatograms are presented in Figure 3. The tendencies of the graphs are clearly mark. The separation process is longer when lower inlet pressures are used while it takes less time for higher temperatures. In other words, the retention time of each compound tend to increase by using lower pressures and temperatures. The lower the inlet pressure, the slower the flow inside the column and therefore the longer the average time of the molecules to leave the column. On the other hand, the lower is the temperature, the lower is the kinetic energy of the molecules and the longer the average interactions between the two phases through of adsorption and desorption processes.

When it comes to a commercial application, a balance is needed between the chromatogram resolution and the time used for the process. If the separation process of every chromatogram last too long, it does not match many of the requirements needed in the monitoring process. Many of which require continuous monitoring and a good time resolution to correlate the sources of pollution 3.

with the concentrations present in the air. However, if the separation process is too quick, the retention times of some of the compounds are too similar to each other and they may leave the column too close. Then, the compounds become coleluted and their peaks become overlapped inducing larger quantification errors of those species concentrations. Taking these facts into consideration, the chromatogram at 4 bar of inlet pressure and 65 • C of temperature was taken as the standard one, (case E), being these conditions recommended by In'Air Solutions for the optimal separation of BTEX. Some of the experimental chromatograms exhibited baseline drift, which happened due to the lack of stability in the PID. In practice, the lamp becomes stable after 24 hours being on and some time for stabilization is also require when the temperature or flow rate is modified. Nevertheless, the baseline instability does not affect to the retention time in any way.

Comparison between experimental and numerical results

In Figure 4 the temporal evolution of the pollutants inside the column simulated using the previously described method is displayed at p in = 4 bar and T = 65 • C (case E). At time of 5s all injected species are seen in form of a Guassian pick, while at t=20s the first separation effects are clearly present. At time 120s benzene left the column and it is quantified by the PID detector, while at t=510s last species, i.e. o-xylene, also went out from the column.

The number of molecules at the very end of the column for every moment of time is then multiplied, so by multiplying this value by the PID response factor, varying between 0.50 and 0.53, depending on the aromatic compound, we obtain the simulated chromatogram. This later is compared to the experimental one in Fig. 5 for the standard conditions (case E). There are some obvious differences between the two chromatograms displayed in Fig. 5. The numerical results show ideal chromatography peaks with a nice sharp symmetrical shape (Gaussian peak) and a flat baseline. In real conditions, the chromatographic peaks can deviate from this ideal. In Figures 3 and5, the experimental peaks become asymmetric, broader, and sometimes the baseline slightly rises or decreases. This common shift away from a Gaussian peak is when the back half of the peak falls away. If the peak were split into two, vertically, the second half would be wider than the first half of the peak. This effect is known as peak tailing and is clearly observed in Figures 3 and5 for all the experimental chromatograms. This effect can be originated due to several factors, namely contamination, column overloading, secondary silanol interactions, reversible chemical reactions or flow path disruption [START_REF] De Zeeuw | Ghost peaks in gas chromatography part i: The carrier gas and carrier gas lines[END_REF]. Indeed, determining the source, it is complex and often there are more than one. In our case we can neglect the column overloading, since concentrations 100 times higher have been successfully tested. Contamination can be neglected too, since no impurities are present in the commercial gas. Secondary, silanol interactions could contribute partially to the distortion of the experimental peaks even if this chromatography column is known to be very adapted to the BTEX separation [START_REF] Nasreddine | Portable novel micro-device for BTEX real-time monitoring : Assessment during a field campaign in a low consumption energy junior high school classroom[END_REF], [START_REF] Lara-Ibeas | Sub-ppb level detection of BTEX gaseous mixtures with a compact prototype gc equipped with a preconcentration unit[END_REF]. Ideally, this phenomenon could be also simulated by adding a secondary adsorption desorption reaction between the BTEX molecules and free silanol groups. However, activity coefficients and surface density of the silanol groups must be known.

Another reason for the peak tailing is a flow path disruption. In Section 5.1 we discussed the filling of the sampling loop and the effect of the compression of the sample. We suspect this is a source for peak tailing. When the injection is started the 6-port valve switches sending the 4-bars-pressure flow of nitrogen through the sampling loop. Even though, the core of the BTEX molecules is compressed in the front part of the sampling loop (first four seconds of injection) there is a part of the molecules that tend to spread back again to the initial part of the sampling loop. Therefore, and since the total time of injection is 20 seconds it might happened that the injection is already tailed before entering the GC-column. After all a troubleshooting process should be done experimentally to identify the source of the peak tailing. If the peak tailing comes effectively from a flow path disruption in the sampling loop or from a secondary silanol interactions it could be described mathematically and simulated either way.

The second and most important disagreement is the width of the peaks. The mini PID detector was chosen regarding its compact size for the GC portability issue. It is connected to the GC column outlet using a T-shape connector, so that the gas flow leaving the chromatography column does not cross directly the PID detection cell. In fact, the gas mixture diffuses from the GC column outlet to the detection cell through a permeation membrane, as illustrated in Fig. C.10. More precisely, pollutant molecules and carrier gas pass through the permeation membrane and fill the cavity of the PID detection cell by passive diffusion due to concentration gradient. Therefore, the gas molecules require longer time to enter and leave the PID detection cell, increasing the width of the peaks. In addition, the filling process is faster than the emptying process because a rapid increase in the concentration gradient is expected when the pollutant enters into the PID detection cell, but the emptying process lasts longer since the more molecules leave the PID chamber the smaller the gradient becomes and the slower they abandon the PID detection cell. The geometry of the PID connection with the column is thus probably the main reason for the tailing effect, observed and discussed previously.

Predictions for other operating conditions

The prediction capability of the model can only be tested if the retention times are a priori unknown. Therefore, first two cases with the same operating temperature, 65 • C, but different inlet pressure, equal to 3 and 5 bars, namely the cases D and F, see Table 3, are simulated, without using any information about measured retention time. Nonetheless, to do so, the K ads and K des constants were required. For standard case, case E, both constants are obtained by a fitting procedure using the measured retention time for each species. To simulate the cases D and F the same values of adsorption and desorption coefficients are kept, since these coefficients are pressure independent according to Eqs. ( 4) and [START_REF] Bryan | Prediction of gas chromatographic retention time via an additive thermodynamic model[END_REF].

The comparison between experimental and numerical results are presented in Fig. 6 (a) and (b), for the case D and F, performed at 3 and 5 bars, respectively, see Table 3. The comparison between the experimental and numerical curves shows that the general prediction is relatively good and the model allows to mimic the inlet column pressure changes. However, the intensity of the simulated peaks does not change with the inlet pressure changing, therefore the discrepancy with the simulations is larger for the lower pressures, where intensity of the measured signal is lower, especially for the heavier species. The second, more complex comparison has been done between experimental and numerical results for two different working temperature, T = 55 • C and 75 • C, but using the same intet pressure of 4 bars. By analyzing the available analytical expressions for both adsorption and desorption coefficients, Eqs. ( 4) and ( 6), it is clear that they depend only on the temperature for the same species. By using the known temperature dependency of each coefficient from the temperature, i.e. K ads ∝ 1/ √ T and K des ∝ exp(1/T ) we adjusted the values of these coefficients for different temperature.

The comparisons between experimental and numerical results, for cases B and H, are shown on Fig. 7 (a) and (b), respectively. Here the influence of the temperature inside the column is evaluated. The same comments as for the previous results can be done: the numerical results do not allow to predict the intensity of peak when the conditions, here the temperature, change. The results of experimental and numerical retention times for the cases B, D, F and H, are presented in Table 5. The overall agreement between the experimental and numerical results is good. However, it is much better, of the order of 1.9-6.4%, for the cases D and F, where only inlet pressure changes. This agreement becomes less good, of the order of 8-24.4%, in the cases B and H, where the working temperature varies. This is due to the difficulties to model the temperature dependence of coefficients K ads and K des .

Conclusion

The mathematical model developed in this work has been proved valid for simulating gas-chromatographic interactions inside a gas-chromatographic column. The numerical evaluation based on this mathematical model resulted in accurate results that are in correlation to the experimental results. When either the column temperature is increased and, therefore, the intensity of the adsorption process is reduced, or the pressure at the inlet of the column is increased, the flow through the column is speeded up. Both conditions result in a reduction of the retention time as it was derived from the experimental and numerical results. The major differences between the numerical and experimental results come from the shape of the chromatogram and the retention time. The peak tailing occurs mainly due to the experimental connection way between the GC column and the PID detector, inducing long diffusion time for gas to enter and to leave the detection cell. Besides, retention times variation can be associated with the approximations done in the calculation of K ads and K des . Although the model presents some limitations, it can be a useful tool to perform the first analysis in the development and optimization of a chromatographic microcolumn, such as those fabricated by MEMS-techniques [START_REF] Regmi | Micro gas chromatography: An overview of critical components and their integration[END_REF]. Since the activity coefficient models or values are not published for most of the potential gas chromatography simulations, there is a need of having at least one known result. must be done must be cordata for alternative analytes in order to be able to benefit from rigorous dynamic simulations using this model. Nonetheless, this approach has the potential to allow engineers and scientist to model and understand the chromatographic separation process for a wide range of applications, resulting in higher chromatographic separation. Additionally, it can be very useful at identifying optimal designs and improving profitability.

Figure 1 :

 1 Figure 1: Molecular structure of all the BTEX compounds.

Figure 2 :

 2 Figure 2: Scheme of the BTEX analyzing system setup in the analysis mode.

  Fig. A.8 in Appendix A. By using the outlet velocity the Reynolds number is estimated to be less then 10, so the flow inside column is essentially laminar. Operating Conditions p in [bar] T [

Figure 3 :

 3 Figure 3: Chromatograms of BTEX separated inside the column for nine different operating conditions, see Table3.

Figure 4 :

 4 Figure 4: Temporal evolution of the pollutants inside the GC-Column, results of the numerical simulations in the standard conditions, case E.

Figure 5 :

 5 Figure 5: Chromatograms of BTEX separation for the standard conditions, case E, obtained from experiment (dashed line) and from numerical simulations (solid line).

Figure 6 :

 6 Figure 6: Chromatograms of BTEX species separation obtained from experiments (dashed line) and numerical simulation (solid line) for T = 65 • C: (a) p in = 3bar, case D, and (b) p in = 5bar, case F.

Figure 7 :

 7 Figure 7: Chromatograms of BTEX species separation obtained from experiments and numerical simulations for inlet pressure 4 bar: (a) B case, T = 55 • C, and (b) H case, T = 75 • C.

Figure B. 9 :

 9 Figure B.9: Diffusion coefficient for some BTEX gases.

Figure C. 10 :

 10 Figure C.10: Scheme of the PID detection system and its location in parallel to the flow at the outlet of the GC column. Red hexagons represent the BTEX molecules while the blue dashed arrows represent the flow direction of the carrier gas.

Table 1 :

 1 Physical and chemical parameters of BTEX and N 2 .

	Molecule	Benzene Toluene Ethylbenz. p-Xylene m-Xylene o-Xylene	Nitrogen
	molecular weight (M)[g/mol]	78.11	92.14	106.17	106.16	106.16	106.16	20.01
	Molecular mass (m) ×10 27 [kg]	129.71	153.00	176.30	176.29	176.29	176.29	46.52
	Molecular diameter (d) ×10 10 [m]	5.26	5.68	6.02	6.80	6.80	6.80	4.17

Table 2 :

 2 Models and brands of the components used in the experimental apparatus

Table 3 :

 3 Experimental operating conditions

Table 4 :

 4 Characteristics of the flow of carrier gas for the standard condition, case E.

Table 5 :

 5 Retention time experimentally and numerically obtained for 4 operating conditions.

	Condition	Species	t R,exp [s]	t R,num [s]	error [%]
		Benzene	144.3	126.9	12.1
		Toluene	290.9	245.1	15.7
	B p in = 4bar, T=55 • C	Ethylbenzene	581.6	469.8	19.2
		m and p-Xylene	624.2	502.4	19.5
		o-Xylene	751.0	597.6	24.4
		Benzene	177.9	168.9	5.1
		Toluene	328.9	317.0	3.6
	D p in = 3bar, T=65 • C	Ethylbenzene	612.8	592.2	3.4
		m and p-Xylene	675.0	631.7	6.4
		o-Xylene	780.8	746.6	4.4
		Benzene	91.6	93.2	1.7
		Toluene	169.9	173.7	2.2
	F p in = 5bar, T=65 • C	Ethylbenzene	315.7	323.6	2.4
		m and p-Xylene	337.3	345.1	2.3
		o-Xylene	399.8	407.7	1.9
		Benzene	104.0	113.1	8.0
		Toluene	177.8	205.0	13.2
	H p in = 4bar, T=75 • C	Ethylbenzene	313.0	372.1	15.9
		m and p-Xylene	328.7	395.9	17.0
		o-Xylene	382.2	464.8	17.8

Acknowledgements

The authors would like to acknowledge the financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No.643095.

Appendix A. Example of macroscopic parameters profiles. Case E Figure A.8 shows the profiles of averaged over the column cross-section carrier gas velocity [START_REF] Lara-Ibeas | Sub-ppb level detection of BTEX gaseous mixtures with a compact prototype gc equipped with a preconcentration unit[END_REF] and pressure distribution [START_REF] Yavarinasab | Graphene/poly (methyl mathacrylate) elecrochemical impedancetransdued chemiresistor for detection of volatile organic compounds in aqueous medium[END_REF] along the GC column for the standard case (case E) conditions, see Table 4. As it can be seen, the velocity increases along the column, according to expected. This happens because the pressure, and consequently the density, is higher at the inlet in comparison to the outlet of the column. Then, to satisfy the mass conservation along all the cross-sections of the tube, the velocity is lower for a higher density. Moreover, both velocity and pressure distributions are not linear.

Appendix B. Binary Diffusion Coefficient

In the column the gas mixture is flowing, but the concentration of the pollution is very low, so the binary diffusion coefficient of the carrier gas and each type of the pollution molecule has to be calculated. The binary diffusion coefficient of two and D ij , is [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF]:

where d m is the mean collision cross-section diameter of species i and j:

It is to underline that the Hard Sphere (HS) [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF] model is used when calculated the binary diffusion coefficient. The values of the molecular diameter and molecular mass, needed for the calculation of binary mixture coefficients, are given in Table 1. The diffusion coefficient variation for all six species along the column is shown on Fig. B.9, where its strong variation can be observed.