Physical meaning of the parameters used in fractal kinetic and generalized adsorption models of Brouers-Sotolongo Taher Selmi1

Physical meaning of the parameters used in fractal kinetic and generalized adsorption models of Brouers-Sotolongo
Taher Selmi1, Mongi Seffen1, Habib Sammouda1, Sandrine Mathieu2, Jacek Jagiello3, Alain Celzard4 and Vanessa Fierro4*
1 Laboratory of Energy and Materials (LabEM). High School of Sciences and Technology of Hammam Sousse – Sousse University, BP 4011, Hammam Sousse, Tunisia.

2 Institut Jean Lamour, UMR Université de Lorraine – CNRS 7198, Parc de Saurupt, CS 50840, 54011 Nancy Cedex, France.

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3 Micromeritics Instrument Corporation, 4356 Communications Drive, Norcross, GA 30093, USA.

4 Institut Jean Lamour, UMR Université de Lorraine – CNRS 7198, BP 21042, 88051 Epinal Cedex 9, France.

*Corresponding author (Vanessa Fierro)
Tel: + 33 372 74 96 77 Fax: + 33 372 74 96 38
E-mail address: [email protected]

Abstract
The aim of the present study was clarifying the physical meaning of the parameters used in fractal kinetic and generalized isotherm models of Brouers-Sotolongo. For this purpose, adsorption of methylene blue (MB) and methyl orange (MO) onto four activated carbons (ACs) was carried out. These ACs were characterised in terms of composition, surface area, pore volumes and pore size distributions, carbon nanotexture and surface chemistry. Adsorption isotherms were carried out at 25°C, and at pH 2.5 and 8 for MO and MB, respectively, and fitted with Langmuir, Freundlich, Jovanovich, Hill-Sips (HS), Brouers-Sotolongo (BS), Brouers-Gaspard (BG) and General Brouers-Sotolongo (GBS) models. Adsorption kinetics were fitted by traditional pseudo-first and pseudo-second order models and compared to the Brouers-Sotolongo (BSf) fractal kinetic model. GBS and BSf were found to be the best models describing adsorption isotherms and kinetics, respectively. This finding suggests that MB and MO adsorption is probabilistic and closely correlated to the heterogeneous character of the adsorbent surface. Moreover, BSf and GBS parameters were correlated with surface area and amount of surface functional groups. In particular, higher surface area and amount of functional groups respectively decreased and increased the constants ?c and ? of the BSf stochastic model.

Keywords: Dyes adsorption; activated carbon; fractal kinetics; stochastic isotherm; surface heterogeneity; adsorption isotherms.

Introduction
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In order to understand the adsorption phenomena, different models can be applied for describing adsorption kinetics and isotherms data. For that purpose, stochastic isotherms and fractal kinetics have become increasingly employed PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5TYW5kcm88L0F1dGhvcj48WWVhcj4yMDA5PC9ZZWFyPjxS
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ADDIN EN.CITE.DATA (Sandro et al. 2009; Gaspard et al. 2006). Indeed, ACs can be considered as fractal materials due to their intricate porous network, developed during the activation process ADDIN EN.CITE <EndNote><Cite><Author>Neimark</Author><Year>1992</Year><RecNum>292</RecNum><DisplayText>(Neimark 1992)</DisplayText><record><rec-number>292</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493131635″>292</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Neimark, A.</author></authors></contributors><titles><title>A new approach to the determination of the surface fractal dimension of porous solids</title><secondary-title>Physica A: Statistical Mechanics and its Applications</secondary-title></titles><periodical><full-title>Physica A: Statistical Mechanics and its Applications</full-title><abbr-1>Phys A: Stat Mech Appl</abbr-1><abbr-2>Phys A: Stat. Mech. Appl</abbr-2><abbr-3>Phys A: Stat Mech Appl</abbr-3></periodical><pages>258-262</pages><volume>191</volume><number>1</number><dates><year>1992</year><pub-dates><date>1992/12/15</date></pub-dates></dates><isbn>0378-4371</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/037843719290536Y</url></related-urls></urls><electronic-resource-num>http://dx.doi.org/10.1016/0378-4371(92)90536-Y</electronic-resource-num></record></Cite></EndNote>(Neimark 1992), and this has an influence on their adsorption properties. The adsorption process of a molecule dissolved in a solvent indeed takes place at the liquid-solid inter-phase with dimensional or topological constraints ADDIN EN.CITE <EndNote><Cite><Author>Kopelman</Author><Year>1988</Year><RecNum>221</RecNum><DisplayText>(Kopelman 1988)</DisplayText><record><rec-number>221</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1476511658″>221</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kopelman, Raoul</author></authors></contributors><titles><title>Fractal Reaction Kinetics</title><secondary-title>Science</secondary-title></titles><periodical><full-title>Science</full-title><abbr-1>Science</abbr-1><abbr-2>Science</abbr-2><abbr-3>Science</abbr-3></periodical><pages>1620-1626</pages><volume>241</volume><dates><year>1988</year><pub-dates><date>23 SEPTEMBER I988</date></pub-dates></dates><urls><related-urls><url>http://dx.doi.org/10.1126/science.241.4873.1620</url></related-urls></urls><electronic-resource-num>doi.org/10.1126/science.241.4873.1620</electronic-resource-num></record></Cite></EndNote>(Kopelman 1988). Thus, some physical properties of the adsorbate/adsorbent systems not only depend on the random behaviour of the mass distribution of adsorbent, but also on the fractal and stochastic character of its surface ADDIN EN.CITE <EndNote><Cite><Author>Sokolowska</Author><Year>2001</Year><RecNum>219</RecNum><DisplayText>(Sokolowska et al. 2001)</DisplayText><record><rec-number>219</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1476475253″>219</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Sokolowska, Z</author><author>M. Hajnos,</author><author>C. Hoffmann,</author><author>M.S. Manfred,</author><author>S. Sokolowski</author></authors></contributors><titles><title> Comparison of fractal dimensions of soils estimated from adsorption isotherms,mercury intrusion, and particle size distribution. </title><secondary-title>Journal of Plant Nutrition and Soil Science</secondary-title></titles><periodical><full-title>Journal of Plant Nutrition and Soil Science</full-title><abbr-1>J Plant Nutr Soil Sci</abbr-1><abbr-2>J. Plant. Nutr. Soil. Sci</abbr-2><abbr-3>J Plant Nutr Soil Sci</abbr-3></periodical><pages>591–599</pages><volume>164</volume><number>5</number><section>591</section><dates><year>2001</year><pub-dates><date>16 July 2001</date></pub-dates></dates><isbn>1522-2624</isbn><urls><related-urls><url>http://dx.doi.org/10.1002/1522-2624(200110)164:5&lt;591::AIDLPLN591&gt;3.0.CO;2-Y</url></related-urls></urls><electronic-resource-num>10.1002/1522-2624(200110)164:5&lt;591::AIDLPLN591&gt;3.0.CO;2-Y</electronic-resource-num></record></Cite></EndNote>(Sokolowska et al. 2001).

Meilanov et al. ADDIN EN.CITE <EndNote><Cite><Author>Meilanov</Author><Year>2002</Year><RecNum>222</RecNum><DisplayText>(Meilanov et al. 2002)</DisplayText><record><rec-number>222</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1476512116″>222</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Meilanov, R.P</author><author>D.A Sveshnikova,</author><author>O.M Shabanov</author></authors></contributors><titles><title>Fractal nature of sorption kinetics</title><secondary-title>Journal of Physics and Chemestry A</secondary-title></titles><periodical><full-title>Journal of Physics and Chemestry A</full-title><abbr-1>J Phys Chem A</abbr-1><abbr-2>J. Phys. Chem A</abbr-2><abbr-3>J Phys Chem A</abbr-3></periodical><pages>11771–11774</pages><volume>106</volume><number>48</number><dates><year>2002</year><pub-dates><date>18 July 2002</date></pub-dates></dates><urls><related-urls><url>http:/dx.doi.org/10.1021/jp0216575</url></related-urls></urls><electronic-resource-num>10.1021/jp0216575</electronic-resource-num></record></Cite></EndNote>(Meilanov et al. 2002) expressed the need of developing new studies of sorption kinetics based on fractals that would take the heterogeneity of adsorbents into account. This new approach was developed by Brouers and coworkers PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAwNjwvWWVhcj48
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ADDIN EN.CITE.DATA (Brouers,Sotolongo-Costa 2006; Brouers et al. 2005), who provided a new kinetic model called BSf including former models already applied to water treatment ADDIN EN.CITE <EndNote><Cite><Author>Hamissa</Author><Year>2013</Year><RecNum>295</RecNum><DisplayText>(Ben Hamissa et al. 2013)</DisplayText><record><rec-number>295</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493197243″>295</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Ben Hamissa, Aïcha Menyar,</author><author>Brouers, François,</author><author>Ncibi, Mohamed Chaker,</author><author>Seffen, Mongi,</author></authors></contributors><titles><title>Kinetic Modeling Study on Methylene Blue Sorption onto Agave americana fibers: Fractal Kinetics and Regeneration Studies</title><secondary-title>Separation Science and Technology</secondary-title></titles><periodical><full-title>Separation Science and Technology</full-title><abbr-1>Sep Sci Technol</abbr-1><abbr-2>Sep. Sci. Technol</abbr-2><abbr-3>Sep Sci Technol</abbr-3></periodical><pages>2834-2842</pages><volume>48</volume><number>18</number><dates><year>2013</year><pub-dates><date>2013/12/12</date></pub-dates></dates><publisher>Taylor &amp; Francis</publisher><isbn>0149-6395</isbn><urls><related-urls><url>http://dx.doi.org/10.1080/01496395.2013.809104</url></related-urls></urls><electronic-resource-num>10.1080/01496395.2013.809104</electronic-resource-num></record></Cite></EndNote>(Ben Hamissa et al. 2013) and to pharmacokinetics ADDIN EN.CITE <EndNote><Cite><Author>Pereira</Author><Year>2010</Year><RecNum>296</RecNum><DisplayText>(Pereira 2010)</DisplayText><record><rec-number>296</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493197587″>296</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Pereira, Luis M,</author></authors></contributors><titles><title>Fractal Pharmacokinetics</title><secondary-title>Computational and Mathematical Methods in Medicine</secondary-title></titles><periodical><full-title>Computational and Mathematical Methods in Medicine</full-title><abbr-1>Comput Math Methods in Med</abbr-1><abbr-2>Comput. Math. Methods in Med</abbr-2><abbr-3>Comput Math Methods in Med</abbr-3></periodical><pages>161-184</pages><volume>11</volume><number>2</number><dates><year>2010</year></dates><urls><related-urls><url>http://dx.doi.org/10.1080/17486700903029280</url></related-urls></urls><electronic-resource-num>10.1080/17486700903029280</electronic-resource-num></record></Cite></EndNote>(Pereira 2010).

On the other hand, understanding adsorption isotherms, i.e., at equilibrium, remains a major way to predict the efficiency of some adsorbents for removing a given pollutant from water ADDIN EN.CITE <EndNote><Cite><Author>Ncibi</Author><Year>2008</Year><RecNum>297</RecNum><DisplayText>(Ncibi et al. 2008)</DisplayText><record><rec-number>297</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493197769″>297</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Ncibi, MC</author><author>Altenor, S</author><author>Seffen, M</author><author>Brouers, F</author><author>Gaspard, S</author></authors></contributors><titles><title>Modelling single compound adsorption onto porous and non-porous sorbents using a deformed Weibull exponential isotherm</title><secondary-title>Chemical Engineering Journal</secondary-title></titles><periodical><full-title>Chemical Engineering Journal</full-title><abbr-1>Chem Eng J</abbr-1><abbr-2>Chem. Eng. J</abbr-2><abbr-3>Chem Eng J</abbr-3></periodical><pages>196-202</pages><volume>145</volume><number>2</number><dates><year>2008</year></dates><isbn>1385-8947</isbn><urls><related-urls><url>https://dx.doi:10.1016/j.cej.2008.04.001</url><url>https://www.researchgate.net/publication/244362197</url></related-urls></urls></record></Cite></EndNote>(Ncibi et al. 2008). Consequently, an abundant literature exists on the development of mathematical models and their suitability for describing adsorption phenomena. However, most of these models are empirical, are sometimes based on unrealistic assumptions and, finally, give little information on the physicochemical processes involved. For this reason, Brouers extended the empirical model of Langmuir in a more general one, called General Brouers-Sotolongo (GBS) model, based on a Burr distribution ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2014</Year><RecNum>302</RecNum><DisplayText>(Brouers 2014b)</DisplayText><record><rec-number>302</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493212527″>302</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author></authors></contributors><titles><title>Statistical foundation of empirical isotherms</title><secondary-title>Open Journal of Statistics</secondary-title></titles><periodical><full-title>Open Journal of Statistics</full-title><abbr-1>Open J Stat</abbr-1><abbr-2>Open. J. Stat</abbr-2><abbr-3>Open J Stat</abbr-3></periodical><pages>687-701</pages><volume>4</volume><number>09</number><keywords><keyword>Adsorption Isotherms, Burr Functions, Adsorption Energy Distribution, Maximum Entropy</keyword></keywords><dates><year>2014</year><pub-dates><date>12 September 2014</date></pub-dates></dates><urls><related-urls><url>http://www.scirp.org/journal/ojs</url></related-urls></urls><electronic-resource-num>http://dx.doi.org/10.4236/ojs.2014.49064</electronic-resource-num></record></Cite></EndNote>(Brouers 2014b).

Applying the aforementioned modern models presents many advantages compared to traditional ones. From the kinetics point of view, BSf allows determining valuable information with a very good accuracy such as adsorption capacity, fractal time exponent (see below for details), half-reaction time, and order of reaction. From the isotherms point of view, GBS allows determining the initial reaction kinetics at various concentrations of adsorbate, assessing the heterogeneity of ACs surface in terms of agglomeration and clustering of AC particles, or of fractal distribution of mesopores.

Herein, BSf and GBS models were used to describe adsorption kinetics and isotherms, respectively, of two dyes: methylene blue (MB) and methyl orange (MO), onto four ACs: F200, F300, Acticarbone® and Cecalite® (see below for description). These materials were also thoroughly characterised by elemental analysis, scanning electron microscopy (SEM), X-ray diffraction (XRD), Raman spectroscopy, thermogravimetry, mercury porosimetry, and adsorption-desorption of N2 and CO2 at -196°C and 0°C, respectively. Assessment of the surface functional groups of these ACs was obtained by a potentiometric titration technique. The initial pH and the pH of zero charge, pHPZC, were also determined. We present correlations between the parameters of BSf and GBS models and the ACs’ physicochemical characteristics, namely porous texture and related parameters such as DFT surface area, and their chemical characteristics such as nature and amount of surface functional groups.

Materials and methods
2.1 Raw materials
Four commercial granular activated carbons (ACs) were used in this study. Filtrasorb 200 (F200) and Filtrasorb 300 (F300) from Calgon Corporation were obtained from a local textile industry (Chimitex, Tunisia). Acticarbone® and Cecalite® were purchased from CECA Company. All ACs were thoroughly washed with distilled water to remove surface impurities, followed by drying at 80°C for 48h.

The basic dye, methylene blue (MB), and the acidic one, methyl orange (MO), both 85% pure, were purchased from Sigma-Aldrich. Stock solutions were prepared by dissolving accurately weighed amounts of MB and MO in distilled water to give a concentration of 1 g.L-1. The various solutions used in this study were then prepared by diluting the stock solution of either MB or MO with distilled water. REF _Ref461543183 h Table 1 shows the main characteristics of those dyes, such as their molecular structure and weight, their acidic or basic nature, and the optimum wavelength (?) used in UV-Vis experiments for their detection.

Table SEQ Table * ARABIC 1: Main characteristics of dyes used in the present work.Dye MB (Cationic dye) MO (Anionic dye)
Molecular structure CarbonSulphurHydrogenNitrogen
C16H18ClN3S SulphurCarbonHydrogenNitrogenOxygen

C14H15N3O3S
Molecular weight (g.mol-1) 319.85 305.35
? (nm) 663 464
2.2 Activated carbon characterisation
The pore texture characterisation of ACs was carried out by adsorption-desorption studies of N2 and CO2 at -196°C and 0°C, respectively, in an ASAP 2020 manometric equipment (Micromeritics, USA). For each material, the BET model was applied to determine the apparent surface area, ABET (m2.g-1), whereas the pore size distribution (PSD) was obtained by using two-dimensional (2D) version of the non-local density functional theory (NLDFT) with the Solution of Adsorption Integral Equation Using Splines (SAIEUS®) routine. SAIEUS® has the advantage of combining both CO2 and N2 adsorption data to get more accurate PSDs PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5KYWdpZWxsbzwvQXV0aG9yPjxZZWFyPjIwMTM8L1llYXI+
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ADDIN EN.CITE.DATA (Jagiello,Olivier 2013; Jagiello et al. 2015). The NLDFT method was also used to determine the surface area, SDFT (m2.g-1), by integrating the PSD over the whole range of pore sizes ADDIN EN.CITE ;EndNote;;Cite;;Author;Centeno;/Author;;Year;2010;/Year;;RecNum;346;/RecNum;;DisplayText;(Centeno,Stoeckli 2010);/DisplayText;;record;;rec-number;346;/rec-number;;foreign-keys;;key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496582540″;346;/key;;/foreign-keys;;ref-type name=”Journal Article”;17;/ref-type;;contributors;;authors;;author;Centeno, T.A.;/author;;author;Stoeckli, F;/author;;/authors;;/contributors;;titles;;title;The assessment of surface areas in porous carbons by two model-independent techniques, the DR equation and DFT;/title;;secondary-title;Carbon;/secondary-title;;/titles;;periodical;;full-title;Carbon;/full-title;;abbr-1;Carbon;/abbr-1;;abbr-2;Carbon;/abbr-2;;abbr-3;Carbon;/abbr-3;;/periodical;;pages;2478-2486;/pages;;volume;9;/volume;;number;48;/number;;dates;;year;2010;/year;;/dates;;urls;;/urls;;/record;;/Cite;;/EndNote;(Centeno,Stoeckli 2010).

The total pore volume measurable by adsorption, or Gurvitch volume, was taken at the relative pressure of 0.97, V0.97 (cm3.g-1). The Dubinin-Raduskevic (DR) model ADDIN EN.CITE ;EndNote;;Cite;;Author;Dubinin;/Author;;Year;1981;/Year;;RecNum;347;/RecNum;;DisplayText;(Dubinin 1981);/DisplayText;;record;;rec-number;347;/rec-number;;foreign-keys;;key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496582690″;347;/key;;/foreign-keys;;ref-type name=”Journal Article”;17;/ref-type;;contributors;;authors;;author;Dubinin, M.M,;/author;;/authors;;/contributors;;titles;;title;In homogeneous microporous structures of carbonaceous adsorbents;/title;;secondary-title;Carbon;/secondary-title;;/titles;;periodical;;full-title;Carbon;/full-title;;abbr-1;Carbon;/abbr-1;;abbr-2;Carbon;/abbr-2;;abbr-3;Carbon;/abbr-3;;/periodical;;pages;321-324;/pages;;volume;19;/volume;;dates;;year;1981;/year;;/dates;;urls;;/urls;;/record;;/Cite;;/EndNote;(Dubinin 1981) was applied to obtain the microporous volume from the N2 isotherm, VDR,N2 (cm3.g-1) on one hand, and from the CO2 isotherm, VDR,CO2 (cm3.g-1) on the other hand. The average micropore volume, L0 (nm), was calculated from Stoeckli’s equation ADDIN EN.CITE <EndNote><Cite><Author>Stoeckli</Author><Year>1995</Year><RecNum>348</RecNum><DisplayText>(Stoeckli 1995)</DisplayText><record><rec-number>348</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496659221″>348</key></foreign-keys><ref-type name=”Book”>6</ref-type><contributors><authors><author>Stoeckli, F</author></authors><tertiary-authors><author>Edward Arnold,</author></tertiary-authors></contributors><titles><title>in Porosity in carbon. Characterization and applications</title></titles><dates><year>1995</year></dates><pub-location>London</pub-location><publisher>J. W Patrick</publisher><urls></urls></record></Cite></EndNote>(Stoeckli 1995):
L0= 10.8E0-11.4 ( SEQ _ * ARABIC 1)
where E0 (kJ.mol-1) is the adsorption energy calculated by the DR model. The micropore volume was also calculated by application of the 2D-NLDFT model, V?,NLDFT (cm3.g-1). The mesopore volume, Vmes (cm3.g-1), was obtained from the difference V0.97 – V?,NLDFT.

Meso- and macropore size distributions were also determined by mercury porosimetry using an Autopore IV apparatus (Micromeritics, USA). Mercury intrusion was performed in two steps at low (0.001-0.24 MPa) and at high (0.24-414 MPa) pressure. Application of Washburn’s equation ADDIN EN.CITE ;EndNote;;Cite;;Author;Washburn;/Author;;Year;1921;/Year;;RecNum;178;/RecNum;;DisplayText;(Washburn 1921);/DisplayText;;record;;rec-number;178;/rec-number;;foreign-keys;;key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1471174845″;178;/key;;/foreign-keys;;ref-type name=”Journal Article”;17;/ref-type;;contributors;;authors;;author;Washburn, Edward W,;/author;;/authors;;/contributors;;titles;;title;The dynamics of capillary flow;/title;;secondary-title;Physical review;/secondary-title;;alt-title;Phys. Rev.;/alt-title;;/titles;;periodical;;full-title;Physical review;/full-title;;abbr-1;Phys Rev;/abbr-1;;abbr-2;Phys. Rev;/abbr-2;;abbr-3;Phys Rev;/abbr-3;;/periodical;;pages;273-283;/pages;;volume;17;/volume;;number;3;/number;;dates;;year;1921;/year;;pub-dates;;date;March 1921;/date;;/pub-dates;;/dates;;urls;;related-urls;;url;https://link.aps.org/doi/10.1103/PhysRev.17.273;/url;;/related-urls;;/urls;;electronic-resource-num;10.1103/PhysRev.17.273;/electronic-resource-num;;/record;;/Cite;;/EndNote;(Washburn 1921), equation 2, allowed calculating the pore size as a function of the mercury pressure. Assuming cylindrical pores of diameter d, it reads:
d=-4 ?Hg cos?P( SEQ _ * ARABIC 2)
where P is the pressure (MPa), ?Hg is the surface tension of mercury (0.485 J.m-2 at 20°C), and ? is the contact angle (140°). Pores as narrow as 3.7 nm could be probed at the highest available pressure, 400 MPa.XRD measurements were performed by using the Cu K? radiation generated by an X’Pert Pro diffractometer (Phillips, The Netherlands). The angle range used was 8°-100° with a scan step size of 0.0334°. Raman spectra were recorded in the range 0-3000 cm-1 using a Labram HR800UV confocal Raman microscope (Horiba Jobin Yvon, Japan) equipped with a CCD detector cooled by Peltier effect. The morphology of the activated carbon was analysed by SEM with a FETQuanta 400 scanning electron microscope using an accelerating voltage of 3 kV.

Carbon, hydrogen, nitrogen and sulphur contents were determined with a Vario EL Cube elemental analyser (Elementar, Germany). Oxygen was also directly determined with the same equipment in a second step. Thermogravimetric analysis (TGA) was performed with an SDT Q600 V8.3 Build 101 thermal analyser (TA Instruments, USA) at a heating rate of 10°C min-1 in an air flow of 60 mL.min-1.

The pH at the point of zero charge, known as pHPZC, is the pH at which the net charge of the surface is zero. The value of pHPZC depends both on the nature and on the amount of functional groups at the surface. For determining it, 0.1 g of AC powder was put into contact with 20 mL of 0.1 mol.L-1 NaCl solution and stirred for 48 h. Then, the suspension was filtered and the equilibrium pH was measured. To determine the initial pH of ACs, pHInitial, 0.1 g of AC powder was placed in 20 mL of distilled water (initial pH 5.7 due to dissolved atmospheric carbon dioxide) and equilibrated during the night. Then the pH of the suspension was measured at room temperature.

Potentiometric titration was used to identify and quantify the functional groups on the AC surface. For that purpose, 0.1 g of AC was placed in 50 mL of NaNO3 solution (0.01 mol.L-1), used both as electrolyte and as suspension medium, to which 1 mL of HCl (0.1 mol.L-1) was added. Then the solution was stirred overnight under N2 saturation. The solution was then titrated with NaOH (0.1 mol.L-1) under N2 saturation using a 905 Titrando automatic titrator (Metrohm, Switzerland) commanded with Tiamo® software V2.2. The pKa distribution of the surface functional groups was calculated by determination of the proton binding function, f(pKa), and the total surface charge, Q (mmol.L-1), was obtained by applying the method of Jagiello PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5KYWNlazwvQXV0aG9yPjxZZWFyPjE5OTQ8L1llYXI+PFJl
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ZE5vdGU+
ADDIN EN.CITE.DATA (Jagiello 1994; Jagiello et al. 2000; Jagiello et al. 1995).

2.3 Adsorption of MB and MO
Adsorption experiments were performed in batch experiments by adding 0.3 g of AC in 100 mL of MB and MO solutions at the desired concentration in the range 0.5 – 80 mg.L-1. The pH (2.5, 5 or 8) was adjusted by adding a small amount of either diluted HCl (0.1 mol.L-1) or diluted NaOH (0.1 mol.L-1) solution. The temperature (25, 35 or 50°C) was controlled with a thermostatic bath with an accuracy of ± 1°C. Suspensions of ACs in dye solutions were stirred with a magnetic device. Samples were investigated at different time intervals to measure dye removal and thus to perform kinetics studies. At each time increment, the residual dye concentration was determined by a Lambda 35 spectrophotometer (Perkin Elmer, USA) at a wavelength of 663 nm for MB and of 464 nm for MO (see again REF _Ref461543183 h Table 1). The amount of adsorbed dye at equilibrium, qe (mg.g-1), was calculated by application of equation 3:
qe=C0-Ce Vm ( SEQ _ * ARABIC 3)
where C0 and Ce (mg.L-1) are initial and equilibrium concentrations, respectively, V (L) is the volume of solution, and m (g) is the mass of AC in the suspension.

Thermodynamic parameters such as the variation of standard entropy, ?S°, of standard enthalpy, ?H°, and of standard free energy, ?G°, were calculated using a same method published in ADDIN EN.CITE <EndNote><Cite><Author>Enaime</Author><Year>2017</Year><RecNum>400</RecNum><DisplayText>(Enaime et al. 2017)</DisplayText><record><rec-number>400</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1502208791″>400</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Enaime, G,</author><author>Ennaciri, K,</author><author>Ounas, A,</author><author>Baçaoui, A,</author><author>Seffen, M,</author><author>Selmi, T,</author><author>Yaacoubi, A,</author></authors></contributors><titles><title>Preparation and characterization of activated carbons from olive wastes by physical and chemical activation: Application to Indigo carmine adsorption</title><secondary-title>Journal of Materials and Environmental Sciences</secondary-title></titles><periodical><full-title>Journal of Materials and Environmental Sciences</full-title><abbr-1>J Mater Environ Sci</abbr-1><abbr-2>J. Mater. Environ. Sci</abbr-2></periodical><pages>4125-4137</pages><volume>8</volume><number>11</number><dates><year>2017</year><pub-dates><date>13 Nov 2016</date></pub-dates></dates><urls><related-urls><url>https://www.jmaterenvironsci.com/Document/vol8/vol8_N11/434-JMES-2741-Enaime.pdf</url></related-urls></urls></record></Cite></EndNote>(Enaime et al. 2017).

2.3.1 Kinetic models
The pseudo-first order (PFO) kinetic model, also known as Lagergren’s model, is generally used to describe solid / liquid adsorption processes. This model assumes that the rate of solute uptake with time is proportional to the difference between the saturation concentration and the amount of adsorbed solute as a function of time. PFO model can be described by equation 4:
qt=qe,11-exp-k1 t( SEQ _ * ARABIC 4)
where qe,1 is the amount of adsorbed dye at equilibrium (mg.g-1), k1 (min-1) is the rate constant of pseudo-first order adsorption, and t (min) is the time. The initial adsorption rate, h1 ((mg/g)/min) PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5NaWFvPC9BdXRob3I+PFllYXI+MjAxNjwvWWVhcj48UmVj
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ADDIN EN.CITE.DATA (Miao et al. 2016; Lagergren 1898) is calculated by equation 5:
h1=k1qe,1( SEQ _ * ARABIC 5)
The pseudo-second-order model (PSO) kinetic model, also known as Ho and Mckay’s model, is widely used for describing adsorption dynamics. In this model, the adsorption process, rather than the particle mass transfer process, is considered as the rate-limiting factor ADDIN EN.CITE <EndNote><Cite><Author>Ho</Author><Year>1999</Year><RecNum>304</RecNum><DisplayText>(Ho,McKay 1999)</DisplayText><record><rec-number>304</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493213043″>304</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Ho, Y. S,</author><author>McKay, G,</author></authors></contributors><titles><title>Pseudo-second order model for sorption processes</title><secondary-title>Process Biochemistry</secondary-title></titles><periodical><full-title>Process Biochemistry</full-title><abbr-1>Process Biochem</abbr-1><abbr-2>Process. Biochem</abbr-2><abbr-3>Process Biochem</abbr-3></periodical><pages>451-465</pages><volume>34</volume><number>5</number><keywords><keyword>Kinetics</keyword><keyword>Sorption</keyword><keyword>Pseudo-second order</keyword></keywords><dates><year>1999</year><pub-dates><date>16 August 1998</date></pub-dates></dates><isbn>1359-5113</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/S0032959298001125</url></related-urls></urls><electronic-resource-num>http://doi.org/10.1016/S0032-9592(98)00112-5</electronic-resource-num></record></Cite></EndNote>(Ho,McKay 1999). The PSO kinetic model can be written according to equation 6
qt=qe,22 k2 t1 + qe,2 k2 t ( SEQ _ * ARABIC 6)
where qe,2 is the amount of adsorbed dye at equilibrium (mg.g-1) and k2 (g.mg-1.min-1) is the rate constant of pseudo-second order adsorption. h2 (mg.g-1.min-1) is the initial adsorption rate PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5IbzwvQXV0aG9yPjxZZWFyPjE5OTg8L1llYXI+PFJlY051
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ADDIN EN.CITE PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5IbzwvQXV0aG9yPjxZZWFyPjE5OTg8L1llYXI+PFJlY051
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ADDIN EN.CITE.DATA (Ho,McKay 1998, 1999) and is calculated as follows:
h2=k2 qe,22 ( SEQ _ * ARABIC 7)
The equation of Brouers-Sotolongo (BS) takes into account the adsorption process complexity PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAwNDwvWWVhcj48
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ADDIN EN.CITE.DATA (Brouers,Sotolongo-Costa 2006; Gaspard et al. 2006; Ben Hamissa et al. 2013; Kesraoui et al. 2016). The BS equation reads:
qn,?t=qe1-1+n-1t?n,??-1n-1 ( SEQ _ * ARABIC 8)
If we use the deformed n-exponential ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2006</Year><RecNum>294</RecNum><DisplayText>(Brouers,Sotolongo-Costa 2006)</DisplayText><record><rec-number>294</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493196368″>294</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F,</author><author>Sotolongo-Costa, O,</author></authors></contributors><titles><title>Generalized fractal kinetics in complex systems (application to biophysics and biotechnology)</title><secondary-title>Physica A: Statistical Mechanics and its Applications</secondary-title></titles><periodical><full-title>Physica A: Statistical Mechanics and its Applications</full-title><abbr-1>Phys A: Stat Mech Appl</abbr-1><abbr-2>Phys A: Stat. Mech. Appl</abbr-2><abbr-3>Phys A: Stat Mech Appl</abbr-3></periodical><pages>165-175</pages><volume>368</volume><number>1</number><keywords><keyword>Fractal kinetics</keyword><keyword>Complex systems</keyword><keyword>Non-extensive systems</keyword><keyword>Energy landscape</keyword><keyword>Levy distributions</keyword><keyword>Sorption in aqueous solutions</keyword></keywords><dates><year>2006</year><pub-dates><date>8/1/</date></pub-dates></dates><isbn>0378-4371</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/S0378437106001166</url></related-urls></urls><electronic-resource-num>http://doi.org/10.1016/j.physa.2005.12.062</electronic-resource-num></record></Cite></EndNote>(Brouers,Sotolongo-Costa 2006)
Expnx=1-n-1x-1n-1, ( SEQ _ * ARABIC 9)
the BSf model can therefore be written as shown in equation 10:
qt=qe 1-Expn-t?n,?? ( SEQ _ * ARABIC 10)
where ? is the fractal time exponent, n is a fractional (non-integer) reaction order, and qn,?(t) and qe,BS are the adsorbed amounts at time t and at saturation, respectively. ?c is the characteristic time of the complex kinetics, which depends on the initial concentration and on the two exponents n and ?. When ? = 1 and n = 1, the PFO equation is obtained, whereas ? = 1 and n = 2 leads to PSO. If ? ? 1 and n = 1, the Weibull distribution is obtained and reads:
q1,?t=qe,W 1-Exp-t?c?. ( SEQ _ * ARABIC 11)
For ? ? 1 and n = 2, the Hill equation is obtained and reads:
q2,?t=qe,H 1-1+t?c?-1. ( SEQ _ * ARABIC 12)
The “half reaction” time ?1/2, given by equation 14, is the necessary time to adsorb half of the initial concentration, and it can be derived from equation 13 using the deformed logarithm ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2014</Year><RecNum>306</RecNum><DisplayText>(Brouers 2014a)</DisplayText><record><rec-number>306</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493213508″>306</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F,</author></authors></contributors><titles><title>The fractal (BSf) kinetics equation and its approximations</title><secondary-title>Journal of Modern Physics</secondary-title></titles><periodical><full-title>Journal of Modern Physics</full-title><abbr-1>J Mod Phys</abbr-1><abbr-2>J. Mod. Phys</abbr-2><abbr-3>J Mod Phys</abbr-3></periodical><pages>1594</pages><volume>5</volume><number>16</number><keywords><keyword>Fractal Kinetics, Farmacokinetics, Cancer Research, Water Treatment, Adsorption, Porous</keyword><keyword>Materials, Activated Carbons</keyword></keywords><dates><year>2014</year><pub-dates><date>3 October 2014</date></pub-dates></dates><urls><related-urls><url>http://www.scirp.org/journal/jmp</url><url>http://dx.doi.org/10.4236/jmp.2014.516160</url></related-urls></urls><electronic-resource-num>doi.org/10.4236/jmp.2014.516160</electronic-resource-num></record></Cite></EndNote>(Brouers 2014a):
1+n-1t?C?-1n-1=12 ( SEQ _ * ARABIC 13)
?1/2=?c 2n-1-1n-11?. ( SEQ _ * ARABIC 14)
2.3.2 Adsorption isotherm models
In the present paper, six models were used to study the adsorption isotherms: Langmuir ADDIN EN.CITE <EndNote><Cite><Author>Langmuir</Author><Year>1918</Year><RecNum>72</RecNum><DisplayText>(Langmuir 1918)</DisplayText><record><rec-number>72</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1463931749″>72</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Langmuir, I</author></authors></contributors><titles><title>The adsorption of gases on plane surfaces of glass, mica, and platinum</title><secondary-title>Journal American of chemestry society</secondary-title></titles><periodical><full-title>Journal American of chemestry society</full-title><abbr-1>J American Chem Society</abbr-1><abbr-2>J. American. Chem. Society</abbr-2><abbr-3>J American Chem Society</abbr-3></periodical><pages>1361</pages><volume>40</volume><dates><year>1918</year></dates><urls></urls></record></Cite><Cite><Author>Langmuir</Author><Year>1918</Year><RecNum>72</RecNum><record><rec-number>72</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1463931749″>72</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Langmuir, I</author></authors></contributors><titles><title>The adsorption of gases on plane surfaces of glass, mica, and platinum</title><secondary-title>Journal American of chemestry society</secondary-title></titles><periodical><full-title>Journal American of chemestry society</full-title><abbr-1>J American Chem Society</abbr-1><abbr-2>J. American. Chem. Society</abbr-2><abbr-3>J American Chem Society</abbr-3></periodical><pages>1361</pages><volume>40</volume><dates><year>1918</year></dates><urls></urls></record></Cite></EndNote>(Langmuir 1918), Freundlich ADDIN EN.CITE <EndNote><Cite><Author>Freundlich</Author><Year>1906</Year><RecNum>73</RecNum><DisplayText>(Freundlich 1906)</DisplayText><record><rec-number>73</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1463932397″>73</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Freundlich, Herbert,</author></authors></contributors><titles><title>Over the adsorption in solution</title><secondary-title>Journal of Physical Chemistry</secondary-title></titles><periodical><full-title>Journal of Physical Chemistry</full-title><abbr-1>J Phys Chem</abbr-1><abbr-2>J. Phys. Chem</abbr-2><abbr-3>J Phys Chem</abbr-3></periodical><pages>385-471</pages><volume>57</volume><dates><year>1906</year></dates><urls></urls></record></Cite></EndNote>(Freundlich 1906), Brouers-Sotolongo (BS) ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2014</Year><RecNum>302</RecNum><DisplayText>(Brouers 2014b)</DisplayText><record><rec-number>302</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493212527″>302</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author></authors></contributors><titles><title>Statistical foundation of empirical isotherms</title><secondary-title>Open Journal of Statistics</secondary-title></titles><periodical><full-title>Open Journal of Statistics</full-title><abbr-1>Open J Stat</abbr-1><abbr-2>Open. J. Stat</abbr-2><abbr-3>Open J Stat</abbr-3></periodical><pages>687-701</pages><volume>4</volume><number>09</number><keywords><keyword>Adsorption Isotherms, Burr Functions, Adsorption Energy Distribution, Maximum Entropy</keyword></keywords><dates><year>2014</year><pub-dates><date>12 September 2014</date></pub-dates></dates><urls><related-urls><url>http://www.scirp.org/journal/ojs</url></related-urls></urls><electronic-resource-num>http://dx.doi.org/10.4236/ojs.2014.49064</electronic-resource-num></record></Cite></EndNote>(Brouers 2014b), Jovanovich ADDIN EN.CITE <EndNote><Cite><Author>Jovanovi?</Author><Year>1969</Year><RecNum>308</RecNum><DisplayText>(Jovanovi? 1969)</DisplayText><record><rec-number>308</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219319″>308</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jovanovi?, D. S.</author></authors></contributors><titles><title>Physical adsorption of gases</title><secondary-title>Kolloid-Zeitschrift und Zeitschrift für Polymere</secondary-title></titles><pages>1203-1213</pages><volume>235</volume><number>1</number><dates><year>1969</year></dates><isbn>1435-1536</isbn><label>Jovanovi?1969</label><work-type>journal article</work-type><urls><related-urls><url>http://dx.doi.org/10.1007/BF01542530</url></related-urls></urls><electronic-resource-num>10.1007/bf01542530</electronic-resource-num></record></Cite></EndNote>(Jovanovi? 1969), Hill-Sips (HS) ADDIN EN.CITE <EndNote><Cite><Author>Sips</Author><Year>1948</Year><RecNum>236</RecNum><DisplayText>(Sips 1948)</DisplayText><record><rec-number>236</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1476696756″>236</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Sips, Robert</author></authors></contributors><titles><title>The Structure of a Catalyst Surface</title><secondary-title>The Journal of Chemical Physics</secondary-title></titles><periodical><full-title>The Journal of Chemical Physics</full-title><abbr-1>J Chem Phys</abbr-1><abbr-2>J. Chem. Phys</abbr-2><abbr-3>J Chem Phys</abbr-3></periodical><pages>490-495</pages><volume>16</volume><number>5</number><dates><year>1948</year><pub-dates><date>MAY. 1948</date></pub-dates></dates><urls><related-urls><url>http://dx.doi.org/10.1063/1.1746922</url></related-urls></urls><electronic-resource-num>org/10.1063/1.1746922</electronic-resource-num></record></Cite></EndNote>(Sips 1948) and General Brouers-Sotolongo (GBS) PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAwNTwvWWVhcj48
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ADDIN EN.CITE.DATA (Brouers et al. 2005; Brouers 2014b). The first five models can be obtained by giving well-defined values to the parameters a and c in the GBS equation:
qe GBS=qe max 1-Expc-Ceba=qe max 1-1+cCeba-1c ( SEQ _ * ARABIC 15)
Thus, for c = 1 and a = 1, the Langmuir isotherm is recovered:
qeL=qemaxL 1-1+Ceb-1=qemaxL Ceb+Ce. ( SEQ _ * ARABIC 16)
For c = 0, we get the normal Brouers-Sotolongo (BS) isotherm:
qe BS=qe maxBS 1-Exp-Ceba, ( SEQ _ * ARABIC 17)
and at low concentration, Ce << b, one gets the Freundlich isotherm:
qe F=KF Cea ( SEQ _ * ARABIC 18)
For c = 0 and a = 1, we find the Jovanovich isotherm:
qe J=qe maxJ 1-Exp-Ceb, ( SEQ _ * ARABIC 19)
And finally, for c = 1, we get the HS isotherm:
qe HS=qe maxHS 1-1+Ceba-1. ( SEQ _ * ARABIC 20)
It was shown that the constant c should range between 0 and 1 ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015). However, if the isotherm does not reach saturation, the constant c can be higher than 1 and therefore the results have no physical meaning in a statistical approach. As it is difficult to choose between HS and BS isotherm (i.e., between c = 0 and 1), Brouers proposed to fit the adsorption isotherms with c = 0.5 ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015), leading to what is known as the Brouers-Gaspard (BG) isotherm, which reads:
qeGBS=qemax 1-Expc=0.5-Ceba=qemax 1-1+12 Ceba-2. ( SEQ _ * ARABIC 21)
The knowledge of a, b and c constants allows determining Ce or a given percentage of Ce; in particular, Ce12 (50% of Ce) was calculated with the following equations:
In the GBS case
Ce12=b 2c-1c1a, ( SEQ _ * ARABIC 22)
In BS case as:
Ce12=b(Ln2)1a, ( SEQ _ * ARABIC 23)
In Jovanovich case as:
Ce12=b Ln2, ( SEQ _ * ARABIC 24)
In HS and Langmuir case as:
Ce12=b , ( SEQ _ * ARABIC 25)
And finally in BG case as:
Ce12=b(22-2)1a. ( SEQ _ * ARABIC 26)
3. Results and discussion
3.1 Physicochemical characteristics of ACs
Elementary analysis, TGA/DTG, Raman spectra and XRD) was mentioned in to supplementary information part
Fig 1 shows the morphology of these ACs as seen by SEM. These images, obtained by using secondary electrons, demonstrate some important differences between materials in terms of grain morphology, surface roughness and open porosity. F200 ( REF _Ref464669679 h Fig 1a) has a heterogeneous surface with large pores of broad distribution of sizes. Regarding F300 (Fig 1b), the pores are much smaller and the surface appears to be very rough. Acticarbone® (Fig 1c) has a completely different morphology based on microspheres, somewhat similar to what is obtained by hydrothermal treatment of biomass ADDIN EN.CITE <EndNote><Cite><Author>Braghiroli</Author><Year>2015</Year><RecNum>310</RecNum><DisplayText>(Braghiroli et al. 2015)</DisplayText><record><rec-number>310</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219868″>310</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Braghiroli, Flavia L,</author><author>Fierro, Vanessa,</author><author>Parmentier, J,</author><author>Vidal, L,</author><author>Gadonneix, Philippe,</author><author>Celzard, Alain,</author></authors></contributors><auth-address>Institut Jean Lamour (IJL) Institut de Science des Matériaux de Mulhouse (IS2M)</auth-address><titles><title>Hydrothermal carbons produced from tannin by modification of the reaction medium: Addition of H+ and Ag+</title><secondary-title>Industrial Crops and Products</secondary-title></titles><periodical><full-title>Industrial Crops and Products</full-title><abbr-1>Ind Crops Prod</abbr-1><abbr-2>Ind. Crops. Prod</abbr-2><abbr-3>Ind Crops Prod</abbr-3></periodical><pages>364-374</pages><volume>77</volume><dates><year>2015</year><pub-dates><date>2015-12</date></pub-dates></dates><isbn>0926-6690</isbn><call-num>hal-01294574</call-num><urls><related-urls><url>https://hal.archives-ouvertes.fr/hal-01294574</url></related-urls></urls><electronic-resource-num>10.1016/j.indcrop.2015.09.010</electronic-resource-num><remote-database-name>Cnrs Univ-lorraine Inc-cnrs Ijl-ul</remote-database-name><research-notes>avec comité de lecture</research-notes><language>English</language></record></Cite></EndNote>(Braghiroli et al. 2015). Finally, Cecalite® (Fig 1d) presents a rather smooth surface with well-dispersed mesopores of rather equal sizes.

(a)
(b)
F300
F200

(d)
(c)
Cecalite
Acticarbone

Fig SEQ Figure * ARABIC 1 SEM pictures of activated carbons used here: (a) F200, (b) F300, (c) Acticarbone®, and (d) Cecalite®The adsorption-desorption isotherms of N2 and CO2 for the four activated carbons are shown in Fig 2a and Fig 2b, respectively. Fig 2c shows the corresponding PSDs obtained by application of the 2D-NLDFT method to both N2 and CO2 isotherms ADDIN EN.CITE <EndNote><Cite><Author>Jagiello</Author><Year>2013</Year><RecNum>318</RecNum><DisplayText>(Jagiello,Olivier 2013)</DisplayText><record><rec-number>318</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493540216″>318</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jagiello, Jacek</author><author>Olivier, James P</author></authors></contributors><titles><title>2D-NLDFT adsorption models for carbon slit-shaped pores with surface energetical heterogeneity and geometrical corrugation</title><secondary-title>Carbon</secondary-title></titles><periodical><full-title>Carbon</full-title><abbr-1>Carbon</abbr-1><abbr-2>Carbon</abbr-2><abbr-3>Carbon</abbr-3></periodical><pages>70-80</pages><volume>55</volume><dates><year>2013</year><pub-dates><date>27 December 2012</date></pub-dates></dates><isbn>0008-6223</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.carbon.2012.12.011</url></related-urls></urls><electronic-resource-num>10.1016/j.carbon.2012.12.011</electronic-resource-num></record></Cite></EndNote>(Jagiello,Olivier 2013). N2 adsorption isotherm at -196°C on Cecalite® was type Ia, characteristic of a microporous material with adsorption occurring by primary filling of microspores at very low relative pressure P/P0 ADDIN EN.CITE <EndNote><Cite><Author>IUPAC</Author><Year>2015</Year><RecNum>349</RecNum><DisplayText>(IUPAC 2015)</DisplayText><record><rec-number>349</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496755566″>349</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>IUPAC</author></authors></contributors><titles><title>International Union of Pure and Applied Chemistry; Korean Chemical Society</title><secondary-title>45th IUPAC World Chemestry Congress</secondary-title></titles><dates><year>2015</year><pub-dates><date>9-14 August 2015</date></pub-dates></dates><pub-location>Busan, Korea</pub-location><publisher>Elseviers</publisher><urls></urls></record></Cite></EndNote>(IUPAC 2015). The adsorption isotherms of F200 and F300 were type Ib, characteristic of microporous materials with micropores wider than those of type Ia present in Cecalite®. Cooperative filling as well as primary filling indeed takes place on a wider P/P0 range than for type Ia ADDIN EN.CITE <EndNote><Cite><Author>IUPAC</Author><Year>2015</Year><RecNum>349</RecNum><DisplayText>(IUPAC 2015)</DisplayText><record><rec-number>349</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496755566″>349</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>IUPAC</author></authors></contributors><titles><title>International Union of Pure and Applied Chemistry; Korean Chemical Society</title><secondary-title>45th IUPAC World Chemestry Congress</secondary-title></titles><dates><year>2015</year><pub-dates><date>9-14 August 2015</date></pub-dates></dates><pub-location>Busan, Korea</pub-location><publisher>Elseviers</publisher><urls></urls></record></Cite></EndNote>(IUPAC 2015). No horizontal plateau was clearly achieved, indicating pore widening; these isotherms also showed a type H4 hysteresis loop, characteristic of slit-shaped pores according to the IUPAC classification ADDIN EN.CITE <EndNote><Cite><Author>IUPAC</Author><Year>2015</Year><RecNum>349</RecNum><DisplayText>(IUPAC 2015)</DisplayText><record><rec-number>349</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1496755566″>349</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>IUPAC</author></authors></contributors><titles><title>International Union of Pure and Applied Chemistry; Korean Chemical Society</title><secondary-title>45th IUPAC World Chemestry Congress</secondary-title></titles><dates><year>2015</year><pub-dates><date>9-14 August 2015</date></pub-dates></dates><pub-location>Busan, Korea</pub-location><publisher>Elseviers</publisher><urls></urls></record></Cite></EndNote>(IUPAC 2015). Finally, the N2 isotherm on Acticarbone® was type IIb, with high adsorption at low P/P0, and a well-developed hysteresis loop, type H4, showing the simultaneous presence of micro and mesopores. The hysteresis loop of Acticarbone® was more pronounced than that of F300, which means that its mesoporous volume was more developed.

REF _Ref479946945 h Table 2 reports the textural characteristics obtained from N2 and CO2 isotherms. The values of ABET ranged from 582 m2.g-1 for Cecalite® to 1014 m2.g-1 for Acticarbone®. SDFT values were higher than those of ABET for F200, F300 and Cecalite®, meaning that these ACs have an important fraction of narrow microporosity in which only one single nitrogen monolayer can be adsorbed. On the contrary, ABET of Acticarbone® was higher than SDFT, meaning that the pore texture is highly developed with wider micropores wherein a supplementary nitrogen molecule could be accommodated between two layers of N2 adsorbed onto the pore walls. The volume of Dubinin-Raduskevich accessible to N2, VDR,N2, was always higher than VDR,CO2 except for Cecalite®; this means that there is also an important fraction of micropores wider than 1 nm because CO2 can probe pores narrower than 1 nm. In agreement with such finding, the average micropore diameter, L0, calculated by application of the DR model to the N2 isotherms, was indeed 0.7, 0.8, 1.1 and 0.6 nm for F200, F300, Acticarbone® and Cecalite®, respectively. Acticarbone® had the highest fraction of mesopororosity (29 % of total volume measurable by gas adsorption) while Cecalite® only had 3%. F200 and F300 had intermediate and nearly identical mesoporous fractions of 22 and 21 %, respectively. The higher supermicroporosity and mesoporosity of Acticarbone® is in good agreement with its higher value of ABET with respect to SDFT.

Fig SEQ Figure * ARABIC 2 Adsorption-desorption isotherms of: (a) N2 at -196°C, (b) CO2 at 0°C, and (c) PSDs obtained from N2 and CO2 adsorption data; (d) PSDs obtained by mercury intrusionTable SEQ Table * ARABIC 2: Textural characteristics of the four activated carbons obtained by adsorption-desorption of N2 at -196°C and of CO2 at 0°C, applying BET, DR and 2D-NLDFT methods.Materials ABET (m2.g-1) SDFT (m2.g-1) VDR, N2 (cm3.g-1) VDR,CO2 (cm3.g-1) V0.97
(cm3.g-1) Vµ,NLDFT (cm3.g-1) Vmes(cm3.g-1) Vmes (%)
F200 795 971 0.28 0.24 0.39 0.30 0.09 22
F300 884 1003 0.35 0.28 0.43 0.34 0.09 21
Acticarbone® 1014 967 0.35 0.18 0.56 0.38 0.16 29
Cecalite® 582 830 0.22 0.23 0.24 0.24 0.01 3
Fig 2d shows the PSDs obtained by mercury porosimetry. The highest porosity, i.e., the volume fraction of macropores mesopores available to mercury, corresponds to Acticarbone®, around 54.3 %. The porosity measurable with this technique for the other ACs was 39.7%, 35.3% and 38.0% for F200, F300 and Cecalite®, respectively. The mercury intrusion-extrusion curves are given in Fig SI 4 of the supplementary information, evidencing the entrapment of mercury when the pressure was decreased. This finding suggests the presence of a significant amount of ink bottle-shaped pores, but also that irreversible compression may have occurred under pressure. More information is also given in Table SI 2, suggesting that all ACs are different either in terms of macro/mesopore size (e.g. when F200 and Cecalite® are compared) or in terms of amounts of pores of similar sizes (e.g. when F300 and Acticarbone® are compared).

REF _Ref469179111 h Fig 3 shows the acidity distribution functions, f(pKa), of the four ACs. The resolution of this method was demonstrated by using solutions of organic acids with known pKa values ADDIN EN.CITE <EndNote><Cite><Author>Jacek</Author><Year>2000</Year><RecNum>354</RecNum><DisplayText>(Jagiello et al. 2000; Jagiello et al. 1995)</DisplayText><record><rec-number>354</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1499516098″>354</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jagiello, Jacek, </author><author>Bandosz, Teresa. J,</author><author>Schwarz, James. A,</author></authors></contributors><titles><title>Carbon surface characterization in terms of its acidity constant distribution</title><secondary-title>Letters to the editor / Carbon</secondary-title></titles><pages>1026-1028</pages><dates><year>2000</year></dates><urls></urls></record></Cite><Cite><Author>Jacek</Author><Year>1995</Year><RecNum>355</RecNum><record><rec-number>355</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1499516618″>355</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jagiello, Jacek, </author><author>Bandosz, Teresa. J, </author><author>Putyera, Karol, </author><author>Schwarz, James. A, </author></authors></contributors><titles><title>Determination of proton affinity distributions for chemical systems in aqueous environments using a stable numerical solution of the adsorption intergral equation </title><secondary-title>Journal of Colloid and Interface Science</secondary-title></titles><periodical><full-title>Journal of Colloid and Interface Science</full-title><abbr-1>J Colloid Interface Sci</abbr-1><abbr-2>J. Colloid. Interface. Sci</abbr-2><abbr-3>J Colloid Interface Sci</abbr-3></periodical><pages>341-346</pages><volume>172</volume><dates><year>1995</year><pub-dates><date>30 November 1994</date></pub-dates></dates><urls></urls></record></Cite></EndNote>(Jagiello et al. 2000; Jagiello et al. 1995). At least four important functional groups were evidenced, except for F300, which presented only two. Traditionally, pKa = 3-8 is the region of carboxylic acids while pKa = 8-10 corresponds to the phenolic region PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5CYW5kb3N6PC9BdXRob3I+PFllYXI+MTk5MzwvWWVhcj48
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aXRlPjwvRW5kTm90ZT4A
ADDIN EN.CITE.DATA (Bandosz et al. 1993; Benaddi et al. 2000). The pKa distribution curves showed the predominance of strongly basic species such as hydroxyl functionalities PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5aaGFuZzwvQXV0aG9yPjxZZWFyPjIwMTU8L1llYXI+PFJl
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RW5kTm90ZT5=
ADDIN EN.CITE PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5aaGFuZzwvQXV0aG9yPjxZZWFyPjIwMTU8L1llYXI+PFJl
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RW5kTm90ZT5=
ADDIN EN.CITE.DATA (Zhang et al. 2015; Seredych et al. 2016). The amount of very basic groups, pKa > 10, accounts for nearly half of all functional groups of all ACs. Therefore, all materials produced a pH slightly higher (1 to 1.4 pH unit) than that of the water in which they were suspended, as shown by the values of pHInitial presented in Table SI 3. F200 and Acticarbone® had similar total amounts of surface groups, 1.145 and 1.150 mmol.g-1, respectively, and also a very similar pHPZC, 7.20 and 7.36, respectively. F300 and Cecalite® showed much less functional groups, 0.605 and 0.787 mmol.g-1, respectively, and slightly higher pHPZC of 8.03 and 7.75, respectively. Table SI 3 shows the results of the potentiometric titration as well as pHPZC and pHInitial values for all ACs studied.

Fig SEQ Figure * ARABIC 3 Density of functional groups of all studied ACs3.2 Adsorption of dyes
In order to explain the adsorption kinetics and isotherms of MB and MO, the experimental data were non-linearly fitted by using the Levenberg Marquardt iteration algorithm supplied with OriginPro 2016 Software®. PFO, PSO and BSf models were used to fit the kinetic data. Langmuir, Freundlich, Jovanovich, HS, BS, BG and GBS models were used to fit the adsorption isotherms.

3.2.1 Effect of pH, temperature and thermodynamic analysis on MB and MO adsorption
Fig 4 shows the effect of pH and temperature on the removal of MB and MO by ACs. The removal efficiency was enhanced when the temperature increased from 25°C to 50°C, indicating the endothermic character of the adsorption ADDIN EN.CITE <EndNote><Cite><Author>Kesraoui</Author><Year>2016</Year><RecNum>307</RecNum><DisplayText>(Kesraoui et al. 2016)</DisplayText><record><rec-number>307</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493214363″>307</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kesraoui, Aida,</author><author>Selmi, Taher,</author><author>Seffen, Monig,</author><author>Brouers, François,</author></authors></contributors><titles><title>Influence of alternating current on the adsorption of indigo carmine</title><secondary-title>Environmental Science and Pollution Research</secondary-title></titles><periodical><full-title>Environmental Science and Pollution Research</full-title><abbr-1>Environ Sci Pollut Res</abbr-1><abbr-2>Environ. Sci. Pollut. Res</abbr-2><abbr-3>Environ Sci Pollut Res</abbr-3></periodical><pages>1-11</pages><volume>24</volume><number>11</number><keywords><keyword>Adsorption . Alternating current . Activated carbon . Anionic dye . Indigo carmine .Modeling</keyword></keywords><dates><year>2016</year><pub-dates><date>23 August 2016</date></pub-dates></dates><isbn>0944-1344</isbn><urls><related-urls><url>https://link.springer.com/article/10.1007/s11356-016-7201-4</url></related-urls></urls><electronic-resource-num>10.1007/s11356-016-7201-4</electronic-resource-num></record></Cite></EndNote>(Kesraoui et al. 2016). The removal efficiency also increased when the pH increased from 2.5 to 8 for MB adsorption, but decreased for MO adsorption. Rodríguez et al. found a similar result for the adsorption of MB and Orange II dyes on ACs ADDIN EN.CITE <EndNote><Cite><Author>Rodríguez</Author><Year>2009</Year><RecNum>325</RecNum><DisplayText>(Rodríguez et al. 2009)</DisplayText><record><rec-number>325</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493542250″>325</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Rodríguez, Araceli</author><author>García, Juan</author><author>Ovejero, Gabriel</author><author>Mestanza, María</author></authors></contributors><titles><title>Adsorption of anionic and cationic dyes on activated carbon from aqueous solutions: Equilibrium and kinetics</title><secondary-title>Journal of Hazardous Materials</secondary-title></titles><periodical><full-title>Journal of Hazardous Materials</full-title><abbr-1>J Hazard Mater</abbr-1><abbr-2>J. Hazard. Mater</abbr-2><abbr-3>J Hazard Mater</abbr-3></periodical><pages>1311-1320</pages><volume>172</volume><number>2–3</number><keywords><keyword>Adsorption</keyword><keyword>Activated carbon</keyword><keyword>Cationic and anionic dyes</keyword><keyword>Wastewater</keyword></keywords><dates><year>2009</year><pub-dates><date>12/30/</date></pub-dates></dates><isbn>0304-3894</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/S0304389409012710</url></related-urls></urls><electronic-resource-num>http://doi.org/10.1016/j.jhazmat.2009.07.138</electronic-resource-num></record></Cite></EndNote>(Rodríguez et al. 2009). This finding may be ascribed to the increase or decrease of repulsive and attractive forces between surface functional groups of ACs and anion of dyes used, which depend on pHPZC values. The anionic dyes are indeed favourably adsorbed at acidic pH < pHPZC, at which the surface is positively charged. On the contrary, cationic dyes are better adsorbed on anionic sites of the adsorbent surface when the latter is globally negatively charged for pH > pHPZC ADDIN EN.CITE <EndNote><Cite><Author>Kesraoui</Author><Year>2016</Year><RecNum>307</RecNum><DisplayText>(Kesraoui et al. 2016)</DisplayText><record><rec-number>307</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493214363″>307</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kesraoui, Aida,</author><author>Selmi, Taher,</author><author>Seffen, Monig,</author><author>Brouers, François,</author></authors></contributors><titles><title>Influence of alternating current on the adsorption of indigo carmine</title><secondary-title>Environmental Science and Pollution Research</secondary-title></titles><periodical><full-title>Environmental Science and Pollution Research</full-title><abbr-1>Environ Sci Pollut Res</abbr-1><abbr-2>Environ. Sci. Pollut. Res</abbr-2><abbr-3>Environ Sci Pollut Res</abbr-3></periodical><pages>1-11</pages><volume>24</volume><number>11</number><keywords><keyword>Adsorption . Alternating current . Activated carbon . Anionic dye . Indigo carmine .Modeling</keyword></keywords><dates><year>2016</year><pub-dates><date>23 August 2016</date></pub-dates></dates><isbn>0944-1344</isbn><urls><related-urls><url>https://link.springer.com/article/10.1007/s11356-016-7201-4</url></related-urls></urls><electronic-resource-num>10.1007/s11356-016-7201-4</electronic-resource-num></record></Cite></EndNote>(Kesraoui et al. 2016).

Thermodynamic parameters such as Gibbs free energy ?G° (kJ.mol-1), enthalpy ?H° (kJ.mol-1), and entropy ?S° (J.mol-1.K-1) were determined and listed in REF _Ref488312371 h Table 3. The positive value of ?H° for MB and MO in the presence of all ACs indicates the endothermic nature of the adsorption process. The positive values of ?S° correspond to an increase of disorder at the interface between dyes and the surface of all samples used PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Cb3VoYW1lZDwvQXV0aG9yPjxZZWFyPjIwMTY8L1llYXI+
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ADDIN EN.CITE.DATA (Bouhamed et al. 2016; Kesraoui et al. 2016), the absolute values reflecting the affinity of the adsorbents for those dyes. With only one exception, MO on Cecalite® at 25°C, the values of ?G° were always negative, showing the spontaneous nature of adsorption of MB and MO onto all ACs used ADDIN EN.CITE <EndNote><Cite><Author>Acosta</Author><Year>2016</Year><RecNum>326</RecNum><DisplayText>(Acosta et al. 2016)</DisplayText><record><rec-number>326</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493542379″>326</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Acosta, R.</author><author>Fierro, V.</author><author>Martinez de Yuso, A.</author><author>Nabarlatz, D.</author><author>Celzard, A.</author></authors></contributors><titles><title>Tetracycline adsorption onto activated carbons produced by KOH activation of tyre pyrolysis char</title><secondary-title>Chemosphere</secondary-title></titles><periodical><full-title>Chemosphere</full-title><abbr-1>Chemosphere</abbr-1><abbr-2>Chemosphere</abbr-2><abbr-3>Chemosphere</abbr-3></periodical><pages>168-176</pages><volume>149</volume><keywords><keyword>Waste tyre recycling</keyword><keyword>Tetracycline</keyword><keyword>Antibiotics removal</keyword><keyword>Wastewater treatment</keyword><keyword>Activated carbon</keyword><keyword>Adsorption modelling</keyword></keywords><dates><year>2016</year><pub-dates><date>4//</date></pub-dates></dates><isbn>0045-6535</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/S0045653516301084</url></related-urls></urls><electronic-resource-num>https://doi.org/10.1016/j.chemosphere.2016.01.093</electronic-resource-num></record></Cite></EndNote>(Acosta et al. 2016). ?G° indeed varied from -10.70 to -19.51 kJ.mol-1, suggesting the highly favourable adsorption of those molecules onto Acticarbone®, F300, and F200. In contrast, adsorption was only slightly favourable on Cecalite®, which exhibited absolute values of ?G° one order of magnitude lower than those observed for the other ACs. And especially, MO adsorption at pH 2.5 was even unfavourable at room temperature, as shown by the positive value of ?G° and the correspondingly lower values of ?H° and ?S° with respect to those of other activated carbons. A similar result was reported elsewhere ADDIN EN.CITE <EndNote><Cite><Author>Rodríguez</Author><Year>2009</Year><RecNum>325</RecNum><DisplayText>(Rodríguez et al. 2009)</DisplayText><record><rec-number>325</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493542250″>325</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Rodríguez, Araceli</author><author>García, Juan</author><author>Ovejero, Gabriel</author><author>Mestanza, María</author></authors></contributors><titles><title>Adsorption of anionic and cationic dyes on activated carbon from aqueous solutions: Equilibrium and kinetics</title><secondary-title>Journal of Hazardous Materials</secondary-title></titles><periodical><full-title>Journal of Hazardous Materials</full-title><abbr-1>J Hazard Mater</abbr-1><abbr-2>J. Hazard. Mater</abbr-2><abbr-3>J Hazard Mater</abbr-3></periodical><pages>1311-1320</pages><volume>172</volume><number>2–3</number><keywords><keyword>Adsorption</keyword><keyword>Activated carbon</keyword><keyword>Cationic and anionic dyes</keyword><keyword>Wastewater</keyword></keywords><dates><year>2009</year><pub-dates><date>12/30/</date></pub-dates></dates><isbn>0304-3894</isbn><urls><related-urls><url>http://www.sciencedirect.com/science/article/pii/S0304389409012710</url></related-urls></urls><electronic-resource-num>http://doi.org/10.1016/j.jhazmat.2009.07.138</electronic-resource-num></record></Cite></EndNote>(Rodríguez et al. 2009).

Fig SEQ Figure * ARABIC 4 Effect of: (a, b) pH at 25°C and (c, d) temperature, on the adsorbed amount at equilibrium of: (a, c) MB at pH 8, and (b, d) MO at pH 2.5, onto all ACs (C0 = 40 mg.L-1)Table SEQ Table * ARABIC 3: Thermodynamic parameters for the adsorption of MB and MO onto all activated carbons samples at different temperatures and C0 = 40 mg.L-1.Samples Dye pH ?H° (kJ.mol-1) ?S° (J.mol-1.K-1) ?G° (kJ.mol-1)
25°C 35°C 50°C
F200 MB 8 49.40 212.50 -14.30 -15.60 -19.50
MO 2.5 16.10 95.70 -12.50 -13.30 -14.90
F300 MB 8 16.60 100.70 -13.40 -14.30 -15.90
MO 2.5 63.20 247.60 -10.70 -12.90 -16.80
Acticarbone® MB 8 20.10 112.80 -13.20 -15.10 -16.00
MO 2.5 25.40 121.80 -11.20 -11.70 -14.20
Cecalite® MB 8 11.10 40.80 -1.20 -1.30 -2.20
MO 2.5 19.00 61.30 1.00 -0.20 -0.60
3.2.2 Kinetic models
REF _Ref488307254 h * MERGEFORMAT Table 4 shows the kinetics parameters obtained for initial dye concentrations equal to 10, 40, 60, and 80 mg.L-1 by application of the BSf model to the MB and MO adsorption data on Acticarbone®. Equivalent data for the other 3 ACs are given in Table SI 4, Table SI 5, Table SI 6 of the Supplementary Information. Fits were performed for values of n = 1, 1.5 and 2. Although the correlation parameter, R2, was always very high, the best fit for MB adsorption was achieved with the reaction order fixed at n = 1. Similar results were obtained when MB was biosorbed on Agave Americana fibres ADDIN EN.CITE <EndNote><Cite><Author>Hamissa</Author><Year>2013</Year><RecNum>295</RecNum><DisplayText>(Ben Hamissa et al. 2013)</DisplayText><record><rec-number>295</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493197243″>295</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Ben Hamissa, Aïcha Menyar,</author><author>Brouers, François,</author><author>Ncibi, Mohamed Chaker,</author><author>Seffen, Mongi,</author></authors></contributors><titles><title>Kinetic Modeling Study on Methylene Blue Sorption onto Agave americana fibers: Fractal Kinetics and Regeneration Studies</title><secondary-title>Separation Science and Technology</secondary-title></titles><periodical><full-title>Separation Science and Technology</full-title><abbr-1>Sep Sci Technol</abbr-1><abbr-2>Sep. Sci. Technol</abbr-2><abbr-3>Sep Sci Technol</abbr-3></periodical><pages>2834-2842</pages><volume>48</volume><number>18</number><dates><year>2013</year><pub-dates><date>2013/12/12</date></pub-dates></dates><publisher>Taylor &amp; Francis</publisher><isbn>0149-6395</isbn><urls><related-urls><url>http://dx.doi.org/10.1080/01496395.2013.809104</url></related-urls></urls><electronic-resource-num>10.1080/01496395.2013.809104</electronic-resource-num></record></Cite></EndNote>(Ben Hamissa et al. 2013), and when Indigo Carmine dye was adsorbed on activated carbon ADDIN EN.CITE <EndNote><Cite><Author>Kesraoui</Author><Year>2016</Year><RecNum>307</RecNum><DisplayText>(Kesraoui et al. 2016)</DisplayText><record><rec-number>307</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493214363″>307</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kesraoui, Aida,</author><author>Selmi, Taher,</author><author>Seffen, Monig,</author><author>Brouers, François,</author></authors></contributors><titles><title>Influence of alternating current on the adsorption of indigo carmine</title><secondary-title>Environmental Science and Pollution Research</secondary-title></titles><periodical><full-title>Environmental Science and Pollution Research</full-title><abbr-1>Environ Sci Pollut Res</abbr-1><abbr-2>Environ. Sci. Pollut. Res</abbr-2><abbr-3>Environ Sci Pollut Res</abbr-3></periodical><pages>1-11</pages><volume>24</volume><number>11</number><keywords><keyword>Adsorption . Alternating current . Activated carbon . Anionic dye . Indigo carmine .Modeling</keyword></keywords><dates><year>2016</year><pub-dates><date>23 August 2016</date></pub-dates></dates><isbn>0944-1344</isbn><urls><related-urls><url>https://link.springer.com/article/10.1007/s11356-016-7201-4</url></related-urls></urls><electronic-resource-num>10.1007/s11356-016-7201-4</electronic-resource-num></record></Cite></EndNote>(Kesraoui et al. 2016). Based on the correlation coefficient, and given the fact that ? describes better the heterogeneity of the surface when its value doesn’t exceed 1 ADDIN EN.CITE ;EndNote;;Cite;;Author;Brouers;/Author;;Year;2014;/Year;;RecNum;306;/RecNum;;DisplayText;(Brouers 2014a);/DisplayText;;record;;rec-number;306;/rec-number;;foreign-keys;;key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493213508″;306;/key;;/foreign-keys;;ref-type name=”Journal Article”;17;/ref-type;;contributors;;authors;;author;Brouers, F,;/author;;/authors;;/contributors;;titles;;title;The fractal (BSf) kinetics equation and its approximations;/title;;secondary-title;Journal of Modern Physics;/secondary-title;;/titles;;periodical;;full-title;Journal of Modern Physics;/full-title;;abbr-1;J Mod Phys;/abbr-1;;abbr-2;J. Mod. Phys;/abbr-2;;abbr-3;J Mod Phys;/abbr-3;;/periodical;;pages;1594;/pages;;volume;5;/volume;;number;16;/number;;keywords;;keyword;Fractal Kinetics, Farmacokinetics, Cancer Research, Water Treatment, Adsorption, Porous;/keyword;;keyword;Materials, Activated Carbons;/keyword;;/keywords;;dates;;year;2014;/year;;pub-dates;;date;3 October 2014;/date;;/pub-dates;;/dates;;urls;;related-urls;;url;http://www.scirp.org/journal/jmp;/url;;url;http://dx.doi.org/10.4236/jmp.2014.516160;/url;;/related-urls;;/urls;;electronic-resource-num;doi.org/10.4236/jmp.2014.516160;/electronic-resource-num;;/record;;/Cite;;/EndNote;(Brouers 2014a), the kinetics is also expected to be of order 1. For MO, excellent fits were also obtained with n = 1 although the highest values of R2 obtained when n =1.5 or n=2 were sometimes always with values of n of 1.5 or 2 (see the data for the other ACs in Table SI 4, Table SI 5, Table SI 6 of the Supplementary Information).

Since, on average, excellent fits were found for all ACs with n = 1, which is the value leading most frequently to the highest correlation coefficient, it was assumed that the adsorption of MB and MO dyes on all ACs follows first-order kinetics. Fixing the reaction order n = 1 was indeed necessary to compare the BSf kinetics to other models.

Table SEQ Table * ARABIC 4: Effect of reaction order n on BSf kinetics parameters obtained by non-linear fit of the adsorption of MB at pH 8 and MO at pH 2.5 on Acticarbone® at 25°C.Initial concentration of MB and MO (mg.L-1)
10 40 60 80
?c? R2 ?c? R2 ?c? R2 ?c? R2
MB n = 1 25.14 1.01 1.00 18.61 1.02 1.00 16.79 1.09 1.00 19.42 0.93 1.00
n = 1.5 20.09 1.32 0.998 14.98 1.36 0.999 13.69 1.47 1.00 15.45 1.26 0.999
n = 2 17.29 1.62 0.996 13.00 1.72 0.998 12.01 1.89 0.998 13.31 1.60 0.998
MO n = 1 15.70 0.95 1.00 22.52 1.11 0.999 23.00 1.03 1.00 22.09 0.84 0.998
n = 1.5 13.07 1.39 0.999 17.16 1.41 1.00 18.72 1.40 1.00 17.59 1.16 0.999
n = 2 11.86 1.91 1.00 15.15 1.84 0.999 16.46 1.81 0.999 15.30 1.5 1.00
REF _Ref479947663 h Fig 5 shows the experimental kinetic data and the calculated curves when n was fixed to 1 for MB and MO adsorption on Acticarbone®: a very good fit was obtained for all initial concentrations from 5 to 80 mg.L-1, for both dyes. Equally good fits were obtained for the other three ACs, and the corresponding results are given in Fig SI 5.

Fig SEQ Figure * ARABIC 5 BSf (1,?) kinetics model applied to the adsorption of: (a) MB at pH 8 and (b) MO at pH 2.5, onto Acticarbone® for different initial concentrations at 25°CFig SI 6 shows the non-linear fit of the adsorption data kinetics of MB at pH 8 and of MO at pH 2.5 onto Acticarbone® by the models listed in the experimental section (i.e., Pseudo-first-order (PFO), Pseudo-second-order (PSO) and Brouers-Sotolongo fractal (BSf(1,?))). The corresponding kinetic models’ parameters are summarised in Table 5. For selecting the best one, values of correlation coefficients, R2, and errors, 2, were examined. For example, for MO adsorption on F300 samples, R2 was found to be equal to 0.998, 0.992 and 1 for PFO, PSO and BSf(1,?), respectively, hence the Weibull equation (? ? 1 and n = 1) was the most relevant one for fitting the kinetic data.

Table SEQ Table * ARABIC 5: Kinetic parameters obtained by fitting the experimental data with PFO, PSO and BSf models (C0 =40 mg.L-1 of MO at pH 2.5 and of MB at pH 8, at 25°C).MB MO
Samples F200 F300 Acticarbone Cecalite F200 F300 Acticarbone Cecalite
qe,exp18.30 14.30 14.28 10.02 13.36 13.22 13.06 5.18
PFO
qe,1 18.37 14.26 14.26 11.11 13.25 13.15 13.07 4.72
k1 0.02 0.04 0.05 0.007 0.02 0.06 0.05 0.01
R2 0.992 0.999 1 0.988 0.996 0.998 1 0.954
2 0.224 0.009 0.002 0.138 0.055 0.019 0.002 0.101
h1 0.36 0.64 0.77 0.07 0.31 0.73 0.63 0.05
PSO
qe,2 21.35 15.31 15.13 15.27 14.73 13.74 13.74 5.70
k2 0.001 0.005 0.006 0.001 0.002 0.008 0.006 0.002
R2 0.974 0.979 0.976 0.999 0.995 0.992 0.981 0.982
2 0.728 0.243 0.253 0.114 0.082 0.091 0.220 0.038
h2 0.50 1.10 1.40 0.09 0.47 1.45 1.17 0.07
BSf(1,?)
qe,BS18.23 14.26 14.25 14.10 13.41 13.19 13.06 8.03
?C49.94 22.19 18.61 251.47 44.03 17.33 21.02 453.11
?1/2 16.73 6.60 5.71 53.73 10.54 4.24 6.61 43.03
? 1.10 0.99 1.02 0.78 0.84 0.85 1.04 0.51
R2 0.993 0.999 1 0.993 1 1 1 0.997
2 0.199 0.010 0.002 0.083 0.004 0.004 0.001 0.007
Relationships between fractal constants of BSf derived from the fits of this model to the adsorption data, ?c and ?, and their physicochemical properties of ACs, SDFT and surface functional groups, were looked for. Gaspard et al. ADDIN EN.CITE <EndNote><Cite><Author>Gaspard</Author><Year>2006</Year><RecNum>20</RecNum><DisplayText>(Gaspard et al. 2006)</DisplayText><record><rec-number>20</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1463750495″>20</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Gaspard, S.</author><author>Altenor, S.</author><author>Passe-Coutrin, N.</author><author>Ouensanga, A.</author><author>Brouers, F.</author></authors></contributors><auth-address>COVACHIMM, EA 3592 Universite des Antilles et de la Guyane, BP 250, 97157 Pointe a Pitre Cedex, Guadeloupe. [email protected] &lt;[email protected]ag.fr&gt;</auth-address><titles><title>Parameters from a new kinetic equation to evaluate activated carbons efficiency for water treatment</title><secondary-title>Water. Res.</secondary-title></titles><pages>3467-77</pages><volume>40</volume><number>18</number><keywords><keyword>Adsorption</keyword><keyword>Charcoal/*chemistry</keyword><keyword>Kinetics</keyword><keyword>*Models, Chemical</keyword><keyword>*Water Purification/methods</keyword></keywords><dates><year>2006</year><pub-dates><date>Oct</date></pub-dates></dates><isbn>0043-1354 (Print) 0043-1354 (Linking)</isbn><accession-num>16979694</accession-num><urls><related-urls><url>http://www.ncbi.nlm.nih.gov/pubmed/16979694</url><url>http://www.sciencedirect.com/science/article/pii/S004313540600426X</url></related-urls></urls><electronic-resource-num>10.1016/j.watres.2006.07.018</electronic-resource-num></record></Cite></EndNote>(Gaspard et al. 2006) already showed that BSf parameters are indeed correlated with the fractal dimension D for the adsorption of phenol and tannic acid onto three commercial ACs, and onto two ACs derived from Vetiveria zizanioides. REF _Ref469173444 h * MERGEFORMAT
Fig 6 shows the effect of SDFT and surface functional groups on the BSf(1,?) constants ?c and ? for the adsorption of MB and MO dyes. Certain correlation between ?MB, ?MO and SDFT and amount of functional groups can be seen. The constant ?c is also clearly correlated with SDFT and with the amount of functional groups. This Cconstant ? is always inferior to 1 (0.51 < ? < 0.99) for the adsorption of MB onto F300 and Cecalite® and for the adsorption of MO onto F200, F300 and Cecalite®, clearly suggesting fractal adsorption. In contrast, for the adsorption of MB onto Acticarbone® and F200 and for the adsorption of MO onto Acticarbone®, the kinetic is not clearly fractal because the constant ? is higher than 1 PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAwNTwvWWVhcj48
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Fig SEQ Figure * ARABIC 6 Effect of SDFT and amount of surface functional groups on the BSf(1,?) constants ?c and ? determined by adsorption of MB at pH 8 and of MO at pH 2.5 (C0 =40 mg.L-1, 25°C)On one hand, the increase of SDFT makes the initial speed of the adsorption increase, and consequently the characteristic time of adsorption ?c of MO and MB decreases. Therefore, the time of half-reaction also decreases. On the other hand, the increase of the amount of surface functional groups increases the characteristic time ?c towards an extremum before decreasing. Despite the increase of the amount of surface functional groups from 0.60 to 0.79 mmol.g-1 from F300 to Cecalite®, F300 being more porous than Cecalite®, the characteristic time increased from 22 to 251 min for MB adsorption, and from 17 to 452 min for MO adsorption. This shows that the effect of porosity is much more important than that of surface chemistry in agreement with previous studies PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAxNTwvWWVhcj48
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ADDIN EN.CITE.DATA (Brouers,Al-Musawi 2015; Jaramillo et al. 2012), and suggests that electrostatic interactions have a poor influence for retaining MB and MO at the surface of the ACs used. In other words, the surface functional groups have a significant impact on the kinetics of adsorption, but that of the nature of the porosity is even more important ADDIN EN.CITE <EndNote><Cite><Author>Mailler</Author><Year>2016</Year><RecNum>256</RecNum><DisplayText>(Mailler et al. 2016)</DisplayText><record><rec-number>256</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1480504224″>256</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Mailler, Romain</author><author>Gasperi, Johnny</author><author>Coquet, Yves</author><author>Buleté, Audrey</author><author>Vulliet, Emmanuelle</author><author>Deshayes, Steven</author><author>Zedek, Sifax</author><author>Mirande-Bret, Cécile</author><author>Eudes, Véronique</author><author>Bressy, Adèle</author></authors></contributors><titles><title>Removal of a wide range of emerging pollutants from wastewater treatment plant discharges by micro-grain activated carbon in fluidized bed as tertiary treatment at large pilot scale</title><secondary-title>Science of the Total Environment</secondary-title></titles><periodical><full-title>Science of the Total Environment</full-title><abbr-1>Sci Tot Environ</abbr-1><abbr-2>Sci. Tot. Environ</abbr-2><abbr-3>Sci Tot Environ</abbr-3></periodical><pages>983-996</pages><volume>542</volume><keywords><keyword>Emerging pollutants</keyword><keyword>Pharmaceuticals and hormones</keyword><keyword>Adsorption</keyword></keywords><dates><year>2016</year></dates><isbn>0048-9697</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.scitotenv.2015.10.153</url></related-urls></urls><electronic-resource-num>org/10.1016/j.scitotenv.2015.10.153</electronic-resource-num></record></Cite></EndNote>(Mailler et al. 2016).

The fractal constant ? increases with the specific surface area, SDFT, due to the difference of porosity between the materials. This suggests that the fractal character of the surface increases due to its higher geometrical heterogeneity ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015). Moreover, this result was also confirmed by the increase of ? with the number of functional surface groups, leading to chemical heterogeneity ADDIN EN.CITE <EndNote><Cite><Author>Kesraoui</Author><Year>2016</Year><RecNum>307</RecNum><DisplayText>(Kesraoui et al. 2016)</DisplayText><record><rec-number>307</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493214363″>307</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kesraoui, Aida,</author><author>Selmi, Taher,</author><author>Seffen, Monig,</author><author>Brouers, François,</author></authors></contributors><titles><title>Influence of alternating current on the adsorption of indigo carmine</title><secondary-title>Environmental Science and Pollution Research</secondary-title></titles><periodical><full-title>Environmental Science and Pollution Research</full-title><abbr-1>Environ Sci Pollut Res</abbr-1><abbr-2>Environ. Sci. Pollut. Res</abbr-2><abbr-3>Environ Sci Pollut Res</abbr-3></periodical><pages>1-11</pages><volume>24</volume><number>11</number><keywords><keyword>Adsorption . Alternating current . Activated carbon . Anionic dye . Indigo carmine .Modeling</keyword></keywords><dates><year>2016</year><pub-dates><date>23 August 2016</date></pub-dates></dates><isbn>0944-1344</isbn><urls><related-urls><url>https://link.springer.com/article/10.1007/s11356-016-7201-4</url></related-urls></urls><electronic-resource-num>10.1007/s11356-016-7201-4</electronic-resource-num></record></Cite></EndNote>(Kesraoui et al. 2016). The degree of fractality ? increase when the SDFT and the number of surface functional groups increase the adsorption becomes faster and the characteristic time ?c becomes lower.

3.2.3 Adsorption isotherms
A stochastic analysis of physicochemical reactions in complex systems ADDIN EN.CITE <EndNote><Cite><Author>Stanislavsky</Author><Year>2013</Year><RecNum>329</RecNum><DisplayText>(Stanislavsky,Weron 2013)</DisplayText><record><rec-number>329</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493543586″>329</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Stanislavsky, A.</author><author>Weron, K.</author></authors></contributors><titles><title>Is there a motivation for a universal behaviour in molecular populations undergoing chemical reactions?</title><secondary-title>Physical Chemistry Chemical Physics</secondary-title></titles><periodical><full-title>Physical Chemistry Chemical Physics</full-title><abbr-1>Phys Chem Chem Phys</abbr-1><abbr-2>Phys. Chem. Chem. Phys</abbr-2><abbr-3>Phys Chem Chem Phys</abbr-3></periodical><pages>15595-15601</pages><volume>15</volume><number>37</number><dates><year>2013</year></dates><publisher>The Royal Society of Chemistry</publisher><isbn>1463-9076</isbn><work-type>10.1039/C3CP52272E</work-type><urls><related-urls><url>http://dx.doi.org/10.1039/C3CP52272E</url></related-urls></urls><electronic-resource-num>10.1039/C3CP52272E</electronic-resource-num></record></Cite></EndNote>(Stanislavsky,Weron 2013) showed that the exponent c is related to clustering or agglomeration at the surface of the adsorbent (i.e., the constant c gives an idea of the degree of heterogeneity of the adsorbent since the value of c is inversely proportional with the surface heterogeneity). REF _Ref489698804 h Table 6 shows the effect of changing the constant c in the GBS model on the quality of the fits, seen through the resultant changes of correlation coefficient, R2. The fits were performed with c = 0, 0.5 and 1 (i.e., correspond to BS, BG and HS isotherm models, respectively), and the best ones were obtained with the BS model (c = 0) for the adsorption of MB on all ACs except F300, best described by HS model (c = 1). Similar results were obtained when Pb2+ was biosorbed onto algal biomass ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015). The isotherms of MO adsorption onto F200 and Acticarbone® were best described by BG isotherm (c = 0.5). MB and MO adsorption isotherms onto F300 were very well described by the HS model, indicating a poorly heterogeneous surface. In contrast, MB and MO isotherms adsorption onto Cecalite® were best fitted by the BS model, indicating the high heterogeneity of the surface. Generally, MB and MO adsorption isotherms onto all samples were adequately described by HS, BG and BS models, indicating the presence of active sites with heterogeneous sorption interactions ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015).Table SEQ Table * ARABIC 6: Effect of constant c of GBS isotherm model obtained by non-linear fit of adsorption data of MB at pH 8 and MO at pH 2.5 onto all ACs at 25°C.GBS R2
F200 F300 Acticarbone® Cecalite®
MB c = 0 0.998 0.961 0.983 0.970
c = 0.5 0.997 0.976 0.971 0.969
c = 1 0.996 0.983 0.979 0.970
MO c = 0 0.998 0.989 0.997 0.997
c = 0.5 0.998 0.993 0.998 0.995
c = 1 0.998 0.995 0.997 0.994
In order to compare the stochastic isotherm models to more classical models, and to avoid deciding between BS and HS models whose fits are equally good, the BG model was used ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015) (i.e., with the constant c fixed at 0.5). REF _Ref465325229 h * MERGEFORMAT Fig 7 shows the fit of the adsorption data of MB at pH 8 and MO at pH 2.5 on Cecalite® and Acticarbone® by the models listed in the experimental section (i.e., Freundlich, Langmuir, Jovanovich, BS, HS, and BG). The corresponding models parameters and the R2 values are listed in REF _Ref488307532 h Table 7 and Table SI 7. Based on R2, Freundlich, Jovanovich and Langmuir models (Table SI 7) were not appropriate for fitting MB and MO adsorption isotherms. HS and BG models seemed to be more adequate, but the BS model give the best fits for all pH and temperatures tested (see REF _Ref488307532 h Table 7). Brouers and Al-Musawi ADDIN EN.CITE <EndNote><Cite><Author>Brouers</Author><Year>2015</Year><RecNum>309</RecNum><DisplayText>(Brouers,Al-Musawi 2015)</DisplayText><record><rec-number>309</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493219446″>309</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Brouers, F</author><author>Al-Musawi, Tariq J</author></authors></contributors><titles><title>On the optimal use of isotherm models for the characterization of biosorption of lead onto algae</title><secondary-title>Journal of Molecular Liquids</secondary-title></titles><periodical><full-title>Journal of Molecular Liquids</full-title><abbr-1>J Mol Liq</abbr-1><abbr-2>J. Mol. Liq</abbr-2><abbr-3>J Mol Liq</abbr-3></periodical><pages>46-51</pages><volume>212</volume><keywords><keyword>Isotherm</keyword><keyword>Algae</keyword><keyword>Model</keyword><keyword>Nonlinear</keyword><keyword>Burr function</keyword></keywords><dates><year>2015</year></dates><isbn>0167-7322</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.molliq.2015.08.054</url></related-urls></urls><electronic-resource-num>10.1016/j.molliq.2015.08.054</electronic-resource-num></record></Cite></EndNote>(Brouers,Al-Musawi 2015) explained that this finding is related to the presence of active sites on a physically and chemically heterogeneous surface. Chemical heterogeneity is the result of different functional groups such as carbonyls, carboxyls, phenols, lactones, amines, aldehydes, as well as delocalised electrons determining the more or less acidic / basic nature of ACs, as seen from the potentiometric titration technique ADDIN EN.CITE <EndNote><Cite><Author>Jacek</Author><Year>1994</Year><RecNum>166</RecNum><DisplayText>(Jagiello 1994)</DisplayText><record><rec-number>166</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1471078201″>166</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jagiello, Jacek,</author></authors></contributors><titles><title>Stable Numerical Solution of the Adsorption Integral Equation Using Splines</title><secondary-title>Langmuir</secondary-title></titles><periodical><full-title>Langmuir</full-title><abbr-1>Langmuir</abbr-1><abbr-2>Langmuir</abbr-2><abbr-3>Langmuir</abbr-3></periodical><pages> 2778-2785</pages><volume>10</volume><number>8</number><dates><year>1994</year><pub-dates><date>May 18, 1994</date></pub-dates></dates><urls><related-urls><url>http://www.cheric.org/research/tech/periodicals/view.php?seq=193769</url></related-urls></urls><electronic-resource-num>10.1021/la00020a045</electronic-resource-num></record></Cite></EndNote>(Jagiello 1994). Geometrical heterogeneity is due to the pores of different sizes and morphologies ADDIN EN.CITE <EndNote><Cite><Author>Jagiello</Author><Year>2013</Year><RecNum>318</RecNum><DisplayText>(Jagiello,Olivier 2013)</DisplayText><record><rec-number>318</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493540216″>318</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Jagiello, Jacek</author><author>Olivier, James P</author></authors></contributors><titles><title>2D-NLDFT adsorption models for carbon slit-shaped pores with surface energetical heterogeneity and geometrical corrugation</title><secondary-title>Carbon</secondary-title></titles><periodical><full-title>Carbon</full-title><abbr-1>Carbon</abbr-1><abbr-2>Carbon</abbr-2><abbr-3>Carbon</abbr-3></periodical><pages>70-80</pages><volume>55</volume><dates><year>2013</year><pub-dates><date>27 December 2012</date></pub-dates></dates><isbn>0008-6223</isbn><urls><related-urls><url>http://dx.doi.org/10.1016/j.carbon.2012.12.011</url></related-urls></urls><electronic-resource-num>10.1016/j.carbon.2012.12.011</electronic-resource-num></record></Cite></EndNote>(Jagiello,Olivier 2013).

Fig SEQ Figure * ARABIC 7 Non-linear fits of isotherm data at 25°C by several isotherm models for the adsorption of: (a, c) MB at pH 8, and (b, d) MO at pH 2.5 on (a, b) Cecalite® and (c, d) Acticarbone®
Table SEQ Table * ARABIC 7: Isotherm parameters of Brouers-Sotolongo, Hill-Sips and Brouers-Gaspard models fitted to the adsorption data of MB and MO at pH 8 and 2.5, respectively, and at 25°C.Dyes MB MO
Samples F200 F300 Acticarbone Cecalite F200 F300 Acticarbone Cecalite
Brouers-Sotolongo
qe max 29.42 22.82 20.07 12.31 36.79 29.23 29.37 21.97
a 0.99 1.80 1.20 0.61 0.73 0.95 1.05 0.82
b 2.39 2.66 4.28 11.08 7.89 2.96 3.56 10.70
Ce1/2 0.71 1.36 1.57 1.55 1.51 0.84 1.13 2.49
2 0.18 2.43 1.93 0.55 0.26 1.20 0.41 0.20
R2 0.998 0.973 0.971 0.963 0.998 0.992 0.997 0.998
Hill-Sips
qe max 35.04 23.96 21.11 12.39 48.86 32.35 31.99 26.78
a 1.14 2.49 1.69 0.87 0.79 1.20 1.34 0.93
b 2.15 2.19 3.16 5.29 8.95 2.24 2.69 9.96
Ce1/2 2.15 2.19 3.16 5.29 8.9 2.24 2.69 9.96
2 0.39 1.13 1.11 0.80 0.21 0.61 0.40 0.36
R2 0.997 0.987 0.983 0.946 0.999 0.996 0.997 0.995
Brouers-Gaspard
qe max 32.09 23.37 20.54 41.48 42.82 30.67 30.80 24.24
a 1.08 2.15 1.46 0.43 0.76 1.09 1.17 0.89
b 2.22 2.37 3.56 434.33 8.36 2.50 3.03 10.03
Ce1/2 1.86 2.17 3.13 280.36 6.53 2.10 2.58 8.12
2 0.29 1.58 1.40 0.40 0.23 0.80 0.20 0.30
R2 0.997 0.976 0.971 0.969 0.998 0.993 0.998 0.995
A correlation between the constants a and b, derived from a stochastic model of GBS isotherm adsorption in the case c = 0.5 (i.e., BG model), and SDFT and amount of surface functional groups. Constants a and b were calculated by fitting the isotherm data of MB and MO adsorption on all ACs. Fig 8 shows the effect of SDFT and surface functional groups on the BG constants a and b for the adsorption of MB and MO dyes. Correlations were indeed observed: the constant a initially decreased with the amount of functional groups then increased, clearly proving that the reaction of adsorption was initially fast. It can be concluded that there is a close relationship between the exponent a and the fractal character of the surface due to its heterogeneity ADDIN EN.CITE <EndNote><Cite><Author>Hamissa</Author><Year>2007</Year><RecNum>330</RecNum><DisplayText>(Ben Hamissa et al. 2007)</DisplayText><record><rec-number>330</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1493545593″>330</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Ben Hamissa, Aïcha Menyar,</author><author>Brouers, François,</author><author>Borhane, Mahjoub,</author><author>Seffen, Mongi,</author></authors></contributors><titles><title>Adsorption of Textile Dyes Using Agave Americana (L.) Fibres: Equilibrium and Kinetics Modelling</title><secondary-title>Adsorption Science &amp; Technology</secondary-title></titles><periodical><full-title>Adsorption Science &amp; Technology</full-title><abbr-1>Adsor Sci Technol</abbr-1><abbr-2>Adsor. Sci. Technol</abbr-2><abbr-3>Adsor Sci Technol</abbr-3></periodical><pages>311-325</pages><volume>25</volume><number>5</number><dates><year>2007</year></dates><urls><related-urls><url>http://journals.sagepub.com/doi/abs/10.1260/026361707783432533</url></related-urls></urls><electronic-resource-num>doi:10.1260/026361707783432533</electronic-resource-num></record></Cite></EndNote>(Ben Hamissa et al. 2007). When the surface heterogeneity decreased, the exponent constant a increased.

Fig SEQ Figure * ARABIC 8 Effect of SDFT and amount of surface functional groups on the GBS (c = 0.5) constants a and b determined from adsorption at 25°C of MB at pH 8 and of MO at pH 2.5Higher porosity of ACs increased the adsorption capacity and made the reaction faster, as observed by the increase of the constant a. Such increase is the result of geometrical heterogeneity, which is due to differences of pores’ sizes and shapes, and to cracks and pits PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5OY2liaTwvQXV0aG9yPjxZZWFyPjIwMDg8L1llYXI+PFJl
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ADDIN EN.CITE.DATA (Ncibi et al. 2008; Francois,Francisco 2016; Altenor et al. 2012). The constant a of BG mentioned in REF _Ref488307532 h Table 7 and the rate constants of initial kinetics of adsorption, h1 and h2 ( REF _Ref488307710 h Table 5), increase with the porosity of ACs (Table 2). However, when a ; 1, a slow initial sorption kinetics is observed and all sites don’t have the same energy as assumed by Langmuir ADDIN EN.CITE <EndNote><Cite><Author>Langmuir</Author><Year>1918</Year><RecNum>72</RecNum><DisplayText>(Langmuir 1918)</DisplayText><record><rec-number>72</rec-number><foreign-keys><key app=”EN” db-id=”99wz2p9tqes92re0zpsv9trhsrsrsxzar9zs” timestamp=”1463931749″>72</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Langmuir, I</author></authors></contributors><titles><title>The adsorption of gases on plane surfaces of glass, mica, and platinum</title><secondary-title>Journal American of chemestry society</secondary-title></titles><periodical><full-title>Journal American of chemestry society</full-title><abbr-1>J American Chem Society</abbr-1><abbr-2>J. American. Chem. Society</abbr-2><abbr-3>J American Chem Society</abbr-3></periodical><pages>1361</pages><volume>40</volume><dates><year>1918</year></dates><urls></urls></record></Cite></EndNote>(Langmuir 1918), whereas when a > 1, a fast initial sorption kinetics occurs and there is probably more than one molecule sorbed by active site. It can be concluded that the exponent a expresses the fractal properties of a heterogeneous system and of its related adsorption mechanisms PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Ccm91ZXJzPC9BdXRob3I+PFllYXI+MjAxNDwvWWVhcj48
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ADDIN EN.CITE.DATA (Brouers 2014a; Kesraoui et al. 2016; Brouers,Al-Musawi 2015; Francois,Francisco 2016).

Regarding the constant b, REF _Ref465575742 h Fig 8 shows that the increase of the porosity from Cecalite® to Acticarbone® decreased the constant b from 10.05 to 0.15 mg/L for MO, and from 436.00 to 2.00 mg.L-1 for MB. However, there is no clear effect of the amount of functional surface groups on the constant b, making difficult to establish whether porosity has a predominant role over surface functions in that case PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5KYXJhbWlsbG88L0F1dGhvcj48WWVhcj4yMDEyPC9ZZWFy
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4. Conclusions
In the present study, four micro/mesoporous activated carbons (ACs: Acticarbone® and Cecalite® from CECA Company, and F300 and F200 from Calgon Corporation) were thoroughly characterised. From the many different techniques that were used, the main porous, structural, nanotextural and physicochemical features of those materials could be accurately determined.

Their adsorption properties with respect to methylene blue (MB) and methyl orange (MO) were investigated at different pH and temperatures. For all ACs, both MB and MO uptakes increased with temperature. However, due to the different cationic / anionic natures of those dyes, the adsorption of MB increased with pH, whereas that of MO decreased with pH.The thermodynamics studies revealed that the adsorption process is spontaneous and endothermic, and the kinetic studies showed that, in all cases, the Brouers-Sotolongo fractal kinetic model (BSf) was the best for describing the adsorption process. Although changing the reaction order n had a low impact on the quality of the fits, suggesting that this parameter is not very important, the best results were most of the time obtained with n = 1. The fractal time parameter ? of the BSf kinetic model increased with both surface area and amount of surface functional groups of the AC, due to the correspondingly increased geometrical and chemical heterogeneity.

The adsorption isotherm studies showed that both Brouers-Sotolongo (BS) and Hill-Sips (HS) models fitted the experimental data quite well. For comparing those stochastic models with more classical ones, and as it was difficult to decide between BS and HS isotherms, we fixed the parameter c = 0.5, thus corresponding to the Brouers-Gaspard (BG) equation. It was seen that the constant a of the model decreased with the amount of surface functional groups, but increased with the surface area. Now, a < 1 suggests a slow initial sorption and that all the sites do not have the same energy, whereas a > 1 corresponds to a fast initial sorption and to more than one molecule sorbed by active site. The behaviour experimentally observed for a thus indicates that the latter is a measure of the scaling (fractal) properties of the AC surface. The parameter c of the same model is related to the agglomeration and clustering of AC particles, or to the fractal distribution of mesopores. The constants of the BG isotherm could also be correlated with physicochemical characteristic of the ACs.

To sum up, the stochastic and fractal models of Brouers-Sotolongo, which are non-empirical complex models established from a probabilistic calculation, are the most adequate to describe the adsorption of dyes on ACs, and provide a meaningful physicochemical explanation of the different adjustable parameters. However, more investigations based on microscopy observations are needed to confirm the relationships between the constant c and agglomeration of AC particles or fractal distribution of pores. Another point to be studied is the determination of the adsorption energy distributions. These objectives will be fulfilled in the near future, as well as the search for possible correlations between pH and temperature and the stochastic parameters.

Supplementary information
SI includes results on ACs characterization by thermogravimetric analysis in air, Raman spectroscopy, XRD, Chemical composition by elemental analysis, potentiometric titration and intrusion-extrusion of mercury. We included some curves and tables of kinetic and isotherms as SI.
Acknowledgements
The Tunisian group gratefully acknowledges the financial support of the EU-METALIC Erasmus Mundus project, and of the Tunisian Ministry of Higher Education and Scientific Research. The French group gratefully acknowledges the financial support of the CPER 2007-2013 “Structuring the Competitiveness Fibre Cluster”, through local (Conseil Général des Vosges), regional (Région Lorraine), national (DRRT and FNADT) and European (FEDER) funds.

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Figure captions
TOC
h z c “Figure” Fig 1 SEM pictures of activated carbons used here: (a) F200, (b) F300, (c) Acticarbone®, and (d) Cecalite®Fig 2 Adsorption-desorption isotherms of: (a) N2 at -196°C, (b) CO2 at 0°C, and (c) PSDs obtained from N2 and CO2 adsorption data; (d) PSDs obtained by mercury intrusionFig 3 Density of functional groups of all studied ACsFig 4 Effect of: (a, b) pH at 25°C and (c, d) temperature, on the adsorbed amount at equilibrium of: (a, c) MB at pH 8, and (b, d) MO at pH 2.5, onto all ACs (C0 = 40 mg.L-1)Fig 5 BSf (1,?) kinetics model applied to the adsorption of: (a) MB at pH 8 and (b) MO at pH 2.5, onto Acticarbone® for different initial concentrations at 25°CFig 6 Effect of SDFT and amount of surface functional groups on the BSf(1,?) constants ?c and ? determined by adsorption of MB at pH 8 and of MO at pH 2.5 (C0 =40 mg.L-1, 25°C)Fig 7 Non-linear fits of isotherm data at 25°C by several isotherm models for the adsorption of: (a, c) MB at pH 8, and (b, d) MO at pH 2.5 on (a, b) Cecalite® and (c, d) Acticarbone®Fig 8 Effect of SDFT and amount of surface functional groups on the GBS (c = 0.5) constants a and b determined from adsorption at 25°C of MB at pH 8 and of MO at pH 2.5

List of tables
TOC
h z c “Table” Table 1: Main characteristics of dyes used in the present work.Table 2: Textural characteristics of the four activated carbons obtained by adsorption-desorption of N2 at -196°C and of CO2 at 0°C, applying BET, DR and 2D-NLDFT methods.Table 3: Thermodynamic parameters for the adsorption of MB and MO onto all activated carbons samples at different temperatures and C0 = 40 mg.L-1.Table 4: Effect of reaction order n on BSf kinetics parameters obtained by non-linear fit of the adsorption of MB at pH 8 and MO at pH 2.5 on Acticarbone® at 25°C.Table 5: Kinetic parameters obtained by fitting the experimental data with PFO, PSO and BSf models (C0 =40 mg.L-1 of MO at pH 2.5 and of MB at pH 8, at 25°C).Table 6: Effect of constant c of GBS isotherm model obtained by non-linear fit of adsorption data of MB at pH 8 and MO at pH 2.5 onto all ACs at 25°C.Table 7: Isotherm parameters of Brouers-Sotolongo, Hill-Sips and Brouers-Gaspard models fitted to the adsorption data of MB and MO at pH 8 and 2.5, respectively, and at 25°C.

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