Multiscale Analysis of Transport in Porous Media

Home , Tortuosity

AIX-MARSEILLE UNIVERSITY THESIS Submitted with the view of obtaining the degree of DOCTOR OF AIX-MARSEILLE UNIVERSITY Discipline: Material Science Doctoral school: Physics and Material Science

Khac-Long NGUYEN Thesis defense: May 22th 2019

MULTISCALE ANALYSIS OF TRANSPORT IN POROUS MEDIA

Supervisors: Dr. Renaud DENOYEL Dr. Véronique WERNERT

JURY Dr. Anne Galarneau, ICGM, Montpellier Reviewer Dr. Benoit Coasne, LIPHY, Grenoble Reviewer Prof. Pascaline Pré, IMT Atlantique Examiner Prof. Frédéric Dallemer, Aix-Marseille University Examiner Dr. Renaud Denoyel, Aix-Marseille University Thesis Director Dr. Véronique Wernert, Aix-Marseille University Thesis Co-Director

To my parents

To my wife, Lien NGUYEN and my son Anh-Vu NGUYEN

ACKNOWLEDGEMENT

First and foremost, I would like to express my sincere gratitude to my supervisors Dr. Renaud Denoyel and Dr. Véronique Wernert for their support and the valuable discussion during my PhD work. Special thanks to Dr. Renaud Denoyel for giving me this opportunity to pursue a doctoral degree at Aix-Marseille University. Great thanks should be given to Dr. Véronique Wernert who devoted a great time for my calculation and experiments. I will forever be indebted to them for the patience, inspiration, encouragement and motivating comments effective and crucial in my academic life. I would like to thank the Ministry of Education and Training of Vietnam for the financial support and the opportunity to study in France. I would like to express my deep gratitude to ANR TAMTAM for the support with my conference travel and the experimental apparatus. I wish to express my acknowledge to Dr. Isabelle Beurroies for supporting me in all the discussions and the training course in adsorption that helped me to be an independent user of the experimental instrument for vapor adsorption. I would also like to express my thanks to the reviewers, Dr. Anne Galarneau and Dr. Benoit Coasne and all the members of the committee, Prof. Pascaline Pré and Prof. Frédéric Dallemer, for taking the time to review this dissertation. My special thanks would give to all the lab mates in MADIREL: Wei, Tracy, Lobna, André, Hailong, Ritu, Pingping, Weiliang, Paul, Vinsensia, Pierre-Henry, Rifan, Ephrem, Girish and Damien for the instructions with mercury porosimetry and my friends: Tung, Hai, Tan, Hung, Thinh, Nam for spending a lot of time together, for their friendship letting me feel happy in the foreign country. Most importantly, I would like to sincerely thank my family, especially my beloved wife, Lien Nguyen, and son, Anh-Vu Nguyen, also to my parents and brother, sister for their love and patience. They give me encouragement, support, strength all these years. I would like to express my wholehearted regards to my family.

ABSTRACT

The correlation of the structural parameters with the transfer properties of a fluid through a porous media is a significant subject in physics, chemistry, geology, and engineering. The architectural parameters such as porosity and pore size distribution do not describe the complexity of most porous organizations consisting of labyrinths of interconnected pores with random shapes and cross-sections. This complexity is described by a parameter called tortuosity. Classical methods such as measurements of nitrogen adsorption isotherms, Hg porosimetry and determination of the tortuosity by electrical measurements are used together to better model transport properties. The materials are also characterized by inverse size exclusion chromatography. The apparent total and particle tortuosities are determined by the analysis of the peak shape of chromatographic probes. In the latter case, the particle tortuosity of silica is calculated from effective intraparticle diffusion coefficient determined by modelling the chromatographic peak broadening of polystyrenes obtained either in dynamic or in static conditions under non-adsorbing conditions by using the solvent tetrahydrofuran (THF). In dynamic conditions, the constant term in the van Deemter equation is a combined contribution of eddy diffusion and polydispersity of the polystyrenes and depends on the size of the molecule. The broad pore size distribution of totally porous silica contributes also to the spreading of the peak. The transport of polystyrenes through silica columns has also been studied in adsorbing conditions by changing the solvent. With the mixture of n-heptane and THF, one obtains many peaks for a polystyrene sample due to the polydispersity of the polystyrene. In fact, the adsorption increases with the molecular weight of the polystyrenes. The surface diffusion of polystyrene decreases with an increase in the retention factor or molecular size. Keywords: porous media; porosity; tortuosity; morphology; transport; liquid chromatography; diffusion; adsorption

RÉSUMÉ

La corrélation entre les propriétés structurales des matériaux et les propriétés de transport d’un fluide à travers les matériaux poreux intervient dans de nombreux procédés en physique, chimie, géologie et ingénierie. Les propriétés telles que la porosité et la distribution de taille de pore ne reflètent pas la complexité du réseau poreux qui consiste en un réseau de pores interconnectés irrégulier et de différentes sections. La complexité peut être décrite par un paramètre appelé la tortuosité. Les matériaux sont caractérisés avec les méthodes classiques de caractérisation comme les isothermes d’adsorption de l’azote, la porosimétrie au mercure et la détermination de la tortuosité par des mesures électriques afin de modéliser les propriétés de transport. Les tortuosités de silices ayant différentes morphologies (particules poreuses, particules de type cœur-coquille et monolithe) ont également été déterminées par chromatographie liquide. En chromatographie liquide la tortuosité intraparticulaire est calculée à partir du coefficient de diffusion intraparticulaire de polystyrènes déterminés à partir de l’élargissement des pics obtenus en mode dynamique et en mode statique en conditions non-adsorbantes avec le solvant tétrahydrofurane (THF). En mode dynamique, dans l’équation de van Deemter, le terme constant dépend de la diffusion d’eddy et de la polydispersité des polystyrènes. La silice poreuse Si100 présente une distribution de taille des pores assez large ce qui entraîne l’élargissement des pics chromatographiques. Le transport de polystyrènes à travers les silices en conditions adsorbantes a également été étudié en modifiant le solvant. En conditions adsorbantes, avec un mélange de THF et d’heptane, pour un polymère de taille donné, plusieurs pics sont obtenus en raison de la polydispersité du polystyrène. L’adsorption augmente avec la masse molaire du polystyrène. La diffusion de surface diminue lorsque le facteur de rétention ou la masse molaire augmentent. Mots clés: milieu poreux; porosité; tortuosité; morphologie; transport; chromatographie en phase liquide ; diffusion; adsorption

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TABLE OF CONTENTS

ACKNOWLEDGEMENT ...... i

ABSTRACT ...... ii

RÉSUMÉ ...... iii

TABLE OF CONTENTS ...... iv

LIST OF FIGURES ...... vii

LIST OF TABLES ...... xi

NOMENCLATURE ...... xii

1 INTRODUCTION ...... 1

1.1. Research objectives ...... 1

1.2. Structure of this thesis ...... 2

1.3. Transport properties in porous media ...... 3

1.3.1. The relationship between tortuosity and porosity in porous media ...... 3

1.3.2. Determination of diffusion coefficients by liquid chromatography ...... 8

1.3.2.1. Fundamental theory of dynamic method...... 10

1.3.2.2. Peak parking method ...... 16

1.3.3. Eddy diffusion ...... 19

1.3.4. Adsorption and surface diffusion ...... 22

1.4. Literature survey ...... 24

1.4.1. Silica topology analysis ...... 24

1.4.2. Eddy diffusion in liquid chromatography ...... 27

1.4.3. Adsorption and diffusion on silica surfaces ...... 29

1.5. Motivation for the thesis ...... 30

2 CHARACTERIZATION TECHNIQUES ...... 32

2.1. Inverse size exclusion chromatography ...... 32

2.1.1. Principles...... 32

2.1.2. Equipment ...... 35

2.1.3. Columns ...... 36

2.1.4. Samples and solutions ...... 36

2.2. Nitrogen adsorption ...... 39

2.2.1. Theory ...... 39

2.2.2. Apparatus ...... 43

2.3. Mercury porosimetry ...... 43

2.3.1. Principles...... 43

2.3.2. Apparatus ...... 46

2.3.3. Method analysis for pore size determination ...... 47

2.4. Determination of tortuosity by conductivity measurements ...... 48

2.4.1. Theory ...... 48

2.4.2. Equipment ...... 50

3 CHARACTERIZATION OF POROUS MATERIALS ...... 52

3.1. Determination of the porosities and mean pore size ...... 52

3.1.1. Characterization by ISEC ...... 52

3.1.2. Characterization by gas adsorption ...... 55

3.1.3. Characterization by Mercury porosimetry ...... 56

3.1.4. Comparison of the porosities and mean pore size obtained by different methods ...... 57

3.2. Determination of tortuosity by electrical measurements ...... 59 4 TRANSPORT PROPERTIES IN MULTISCALE MATERIALS IN NON-ADSORBING CONDITIONS ...... 62

4.1. Transport properties in the dynamic condition ...... 62

4.2. Diffusion in static conditions by peak parking (PP) method ...... 68 4.2.1. Effective diffusion coefficient and apparent total tortuosity determined by the PP method...... 68 4.2.2. Determination of effective intraparticle diffusion coefficient of polystyrenes in non- adsorbing conditions ...... 74

4.3. Study of eddy diffusion ...... 83

4.3.1. Van Deemter, Knox and Giddings plots ...... 83

4.3.2. B and C terms...... 86

4.3.3. Eddy diffusion term and polydispersity ...... 88

4.4. Conclusion ...... 96

5 TRANSPORT PROPERTIES IN ADSORBING CONDITIONS ...... 98

5.1. Adsorption isotherm of toluene on silica from heptane ...... 98

5.2. Band broadening of polystyrenes in adsorbing conditions ...... 103

5.3. Diffusion in adsorbing conditions ...... 108

5.3.1. Determination of effective diffusion coefficients in adsorbing conditions ...... 108

5.3.2. Determination of surface diffusion ...... 111

5.4. Conclusion ...... 115

6 CONCLUSIONS AND PERSPECTIVES ...... 116

REFERENCES ...... 118

LIST OF FIGURES

Figure 1.1. Transport properties in porous media ...... 1

Figure 1.2. Typical structure of a) totally porous particle, b) core-shell particle and c) monolith .. 3

Figure 1.3. Illustration of a tortuous path through a porous media ...... 4 Figure 1.4. Three retarding effects in the flow of large molecules through porous media: i) steric hindrance; ii) relationship between tortuosity and probe size; iii) friction [27] ...... 9

Figure 1.6. An example of a chromatographic peak and a Gaussian fit ...... 12 Figure 1.7. Height equivalent to a theoretical plate (HETP) as a function of interstitial velocity based on Van Deemter equation ...... 13 Figure 1.8. Different exchange processes due to the flow velocity inequalities in the column [48] ...... 20

Figure 1.9. Surface diffusion of sample molecules on the stationary phase [80] ...... 23

Figure 2.1. A principle of the inverse size exclusion chromatography ...... 33

Figure 2.2. The typical scheme of a chromatographic analysis system ...... 35 Figure 2.3. Molecular diffusion coefficients obtained by TDA for the small polymers in the mixture of THF and n-heptane ...... 39

Figure 2.4. The six main types of physisorption isotherm [108] ...... 40

Figure 2.5. Two typical intrusion–extrusion cycles of mercury [98] ...... 44

Figure 2.6. A view of mercury penetrometer cell ...... 46

Figure 2.7. Overview of electrical measurement for porous particle and Nyquist impedance plot 51 Figure 3.1. a) Chromatograms and b) evolution of retention time to the molecular diameter of all the probes at the flow rate of 0.5ml.min-1 through Chromolith column ...... 52 Figure 3.2. Total porosity of polystyrenes as a function of the cubic root of the molecular weight of the polystyrenes for Chromolith, Poroshell and Si100 columns ...... 53

Figure 3.3. Particle porosity of polystyrenes as a function of the molecular size rm for Chromolith Si, Poroshell and Si100 columns ...... 54 Figure 3.4. ISEC for Si 100, Chromolith Si and Poroshell columns -a) Experimental and calculated distribution coefficient and b) Pore size distribution ...... 55 Figure 3.5. Nitrogen adsorption for Si 100, Chromolith and Poroshell –a) Adsorption-desorption isotherms and b) Pore size distribution (NLDFT) ...... 56 vii

Figure 3.6. Hg porosimetry measurements for Chromolith Si, Si 100 and Poroshell samples -a) Intrusion and extrusion cumulative pore volume curves and b) Pore size distributions in whole range and mesopore range...... 57 Figure 3.7. The relation of experimental total tortuosity (square point) and calculated total tortuosity by equation 2.30 (solid line) for-a) Si 100 and b) Poroshell 120 ...... 59 Figure 4.1. Chromatograms for toluene at the flow rate of 0.5 mL.min-1 for Si 100, Chromolith and Poroshell columns ...... 62 Figure 4.2. HETP curves with and without the contribution of extra volume through Si 100 column for a) a small molecule as toluene, b) and c) molecules with intermediate size (P3 and P6), and d) a molecule excluded from the pore of the particle like P10 ...... 63 Figure 4.3. Comparison of corrected HETP data through three columns: Si 100, core-shell and Chromolith with a) toluene, molecules with intermediate size as b) P4 and c) P6 and a big molecule (P10) ...... 64 Figure 4.4. Corrected HETP of molecules having sizes smaller than pore size as a function of interstitial velocity for Si 100 column ...... 65 Figure 4.5. Evolution of the ratio of the effective intraparticle diffusion coefficient to the molecular diffusion coefficient of the intermediate molecules through Si 100 column as a function of molecule/pore size ratio ...... 66 Figure 4.6. Experimental HETPs of some polystyrene standards measured on a) Poroshell and b) Chromolith columns ...... 67 Figure 4.7. Comparison of a) normalized peaks and b) evolution of peak variance versus velocity, for zero-volume column and studied columns...... 67 Figure 4.8. Representation of the obtained peak shape by the peak parking method in THF for a low molecular weight molecule (toluene) in a) Si 100, b) Poroshell, c) Chromolith columns and an intermediate size molecule (polymer P5) with d) Si 100, e) Poroshell and f) Chromolith columns ...... 70 Figure 4.9. Plots of the peak variances of peak parking experiments as a function of parking time for a) Si 100, b) Poroshell and c) Chromolith columns ...... 72 Figure 4.10. Plots of the ratio of the experimental effective diffusion coefficient and the molecular eff diffusion coefficient (D /Dm) versus the ratio of molecular size and mesopore size (rm/rp) through Si 100, Poroshell and Chromolith columns ...... 73

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Figure 4.11. Plots of the apparent total tortuosity of Si 100, Poroshell and Chromolith solids obtained by PP method versus the ratio of molecular size and mesopore size (rm/rp) ...... 74

eff eff Figure 4.12. D /Dm acquired experimentally and D /Dm estimated by RTW or EMT models as a function of molecule/pore size ratio for Si 100 column a) model with τp =1.4, b) model with τp =2 and c) model with a variable τp obtained by Weissberg equation with p=1.5...... 77

eff eff Figure 4.13. Comparison of the D /Dm achieved experimentally and the D /Dm assessed by the Maxwell model versus the ratio of molecular size and mesopore size of for Si 100, Poroshell and Chromolith columns ...... 78 Figure 4.14. Plots of the ratio of the effective particle diffusion to the molecular diffusion versus the molecule/pore size ratio ...... 79 Figure 4.15. Particle tortuosity obtained from the PP method and by the Weissberg equation as a function of particle porosity ...... 82 Figure 4.16. Comparison of reduced experimental HETP and the one obtained by different models through silica columns: Si 100, Poroshell and Chromolith with a) toluene, molecules with small size b) P2 and intermediate size as c) P5 and d) P6 ...... 86 Figure 4.17. Comparison of reduced b term for oluene and polystyrene from P1 to P8 in dynamic and static conditions through silica columns a) data obtained by PP method and Van Deemter equation, b) data evaluated by PP method and Giddings equation ...... 87 Figure 4.18. Evolution of C term for toluene and a series of polystyrene from P1 to P8 as a function of the molecule/ pore size ...... 88 Figure 4.19. Calibration curve of the molecular weight to the retention time of intermediate samples ...... 89

Figure 4.20. Relationship of the mean retention time tr and the molecular weight through the Si column at 0.5ml.min-1 with some intermediate molecules ...... 91 Figure 4.21. Chromatograms of P2 and different fractions of P2 having different number of units at 1.2ml.min-1 through Si 100 column ...... 92

Figure 4.22. Evolution of A +Hpoly term for oluene and polystyrenes from P1 to P8 as a function of velocity (a,b,c) and as a function of the molecule/pore size ratio (d) for the three columns ...... 93 Figure 4.23. A plot of the reduced total contribution of trans column eddy diffusion and polydispersity as a function of the molecule/ pore size by different models for toluene and polystyrenes from P1 to P8 ...... 95

Figure 5.1. Equilibrium adsorption isotherm of toluene in n-heptane with silica ...... 99

Figure 5.2. Chromatograms of toluene and P1 in THF and n-heptane at 0.5 ml/min through Chromolith column ...... 100 Figure 5.3. Chromatograms of P2 in THF and n-heptane at 0.5 ml/min with Chromolith column ...... 101 Figure 5.4 Chromatograms of P2 at 0.5ml/min in four different eluent compositions through Chromolith column ...... 102

Figure 5.5. The calculated retention factors of P2 for Chromolith column in ...... 103 the different mixtures of n-heptane and THF ...... 103 Figure 5.6. Chromatograms obtained for polystyrenes in the mixture of n-heptane and THF at the flow rate of 1ml.min-1 a) P2 with Si 100 column and b) P3 with Poroshell column ...... 104 Figure 5.7 The distribution of each fraction of the sample P2 in the mixture of THF and n-heptane through Si 100 column ...... 106 Figure 5.8. Plots of the retention time of a series of polystyrenes as a function of the molecular weight through the chromatographic columns ...... 107 Figure 5.9. Retention factor versus the number of units for the 3 chromatographic columns in the case of P2 and P3 ...... 107 Figure 5.10. Chromatograms recorded during the peak parking experiments a) for P2 and b) for the fraction of 6 units of P2 through Si 100 column at the flow rate of 1 ml.min-1 ...... 109

Figure 5.11. A plot of peak variance as a function of the parking time for P2 in Si 100 column . 109 Figure 5.12. Comparison of the peak variance for the fraction of 7 units and 10 units for P2 and P3 for the Si100 column ...... 110 Figure 5.13. Ratio between experimental effective diffusion coefficient and molecular diffusion coefficient as a function of retention factor ...... 111 Figure 5.14. Evolution of the effective intraparticle diffusion coefficient to the molecular weight of a series of polystyrenes in non-adsorbing conditions through Si100 column ...... 112 Figure 5.15. Relationship between the ratio of the pore diffusion coefficient to the molecular diffusion coefficient with the number of units for P2 and P3 ...... 112 Figure 5.16. Ratio of the intraparticle diffusion and surface diffusion coefficients to the molecular diffusion coefficient as a function of retention factor for P2 and P3 ...... 114

LIST OF TABLES

Table 1.1. The effective diffusion models for the chromatographic columns ...... 18

Table 2.1. Specifications of the columns made of pure silica (data from manufacturer) ...... 36 Table 2.2 Molecular weights, polydispersity index, number of units, bulk diffusion coefficient in THF and in the mixture of THF and n-heptane and hydrodynamic radii rm ...... 38 Table 3.1. Surface area, porosity and pore size obtained by ISEC, Hg porosimetry and N2 adsorption for Si 100, Chromolith Si and Porosell 120 column ...... 58

Table 3.2. Tortuosities obtained by electrical measurements ...... 61

Table 4.1. Values of internal porosity used in the calculation of Deff by PP method ...... 72 Table 4.2. Estimation of the internal, external and total tortuosities as well as the topological factor (p) by the PP and electrical measurements for Si 100, Poroshell and Chromolith columns ...... 83

Table 4.3. Hpoly from Knox model for various molecules in the different chromatographic columns at 0.5ml.min-1 ...... 90

Table 4.4. hpoly for P2 and P3 from experiment and model of the sum of n Gaussian functions .... 96 Table 5.1. Calculated results for some fractions with different numbers of units for P2 in adsorbing conditions through Si100 column ...... 113

NOMENCLATURE

Equation variables and their units A A-term of the Van Deemter equation, m

2 -1 As surface area, m .g a reduced eddy diffusion a1,a2 empirical parameters for the evolution of diffusion coefficient to molar mass a’ Landauer model parameter a.u absorbance, mAu B B-term of the Van Deemter equation, m2.s-1 b1,b2 empirical parameters for the evolution of retention time to molar mass b reduced longitudinal diffusion coefficient C C-term of the Van Deemter equation, s

-1 Ci initial concentration, mol.l

-1 Ce Equilibrium concentration, mol.l c reduced mass transfer coefficient

eff 2 -1 Da effective internal stationary phase diffusivity in adsorbing conditions, m .s

eff 2 -1 D effective diffusion coefficient, m .s

2 -1 Dm molecular diffusion coefficient, m .s

2 -1 DL axial dispersion coefficient, m .s

2 -1 Dm molecular diffusion coefficient, m .s

eff 2 -1 Dp effective intraparticle stationary phase diffusivity, m .s

eff 2 -1 Dpz effective internal stationary phase diffusivity in the shell, m .s

2 -1 Ds surface diffusion coefficient, m .s

2 -1 Dr average radial dispersion coefficient, m .s

d’p diameter size either of the cylindrical element of the stationary phase skeleton in monolith or of the stationary phase spherical particles, m

Es activation energy, J f volumetric flow rate, m3.s f (r) the relative volume fraction of pores xii

Hlong plate height due to the longitudinal diffusion, m

Hpoly contribution of the polymer polydispersity, m h reduced plate height heddy reduced eddy diffusion hIC reduced short-range interchannel eddy diffusion hpoly reduced diffusion due to polydispersity hTC reduced transcolumn eddy diffusion hTS reduced transchannel eddy diffusion J transport flux K partition coefficient

Ka adsorption equilibrium constant

-1 KH Henry constant, l.g

-1 KL Langmuir equilibrium constant, l.g k' retention factor k1 zone retention factor kf friction factor kd distribution coefficient km external mass transfer coefficient L column length, m le distance traved, m

-1 Mw molecular weight of the analyte, g.mol

Mn the number average molecular weight m mass of adsorbent, g mp cumulative mass, g

NA Avogadro constant, na amount adsorbed, mol.g-1

a -1 nm monolayer capacity, mol.g p topological parameter p’ the number of units of polystyrenes P e Péclet number xiii

-1 Qst isosteric heat of adsorption, kJ.mol

-1 qe amount of adsorbed adsorbate at equilibrium, mol.g

-1 qm the theoretical monolayer saturation capacity, mol.g r column radius, m

Ra resistivity of the free electrolyte in the blank column

Rc capillary internal diameter, m

Reff the resistivity of the pore saturated with the conductive solution, Ω

Ro the resistivity of the free electrolyte, Ω rk cylindrical pore radius, m rm molecule hydrodynamic radius, m rp radius of the stationary phase, m S cross-section of the monolith, m2 S’ the reciprocal of the slope of calibration curve of lnMw to the retention time

2 So cross-section of the blank column, m t thickness of the adsorbed film tr retention time, min ti retention time without column, min to retention time with a non-adsorbing compound, min tp parking time, min u interstitial velocity of the mobile phase, m2.s V the volume of the solution, l

3 Vc column volume, m

3 Ve external volume, m

Vi initial volume of a electrolyte,

3 Vm micropore volume, m

3 Vo the volume of the capillaries, m

3 Vp particle volume, m

3 Vs volume of the solid, m

3 Vt total volume , m v reduced velocity xiv w peak width, s2

Greek letters α empirical parameter of the Knox equation β Maxwell model parameter γ surface tension of mercury

γ1 labyrinth factor (obstruction factor)

γ2 packing characterization factor due to the eddy diffusion contribution

γr an adjustable parameter

2 σpoly space variance of the sample due to the dispersity

2 2 ∆σt peak variance in a unit of time, s

2 2 ∆σz peak variance in a unit of length, m ε porosity

εe external porosity

εo internal porosity for toluene

εpz intraparticle porosity of the pores either inside the particles for fully-porous particles or in the skeleton with monolith or in the shell for the core-shell particles

εt total porosity

λi structural parameter due to flow exchange mechanism for the contribution i of eddy diffusion θ contact angle ξ Torquato model parameter ρ ratio of the solid core to the core-shell

-3 ρs density of silica, g.cm

-1 σeff effective conductivity, S.m

2 2 σi peak variance without column, s

-1 σ0 bulk conductivity, S.m

-1 σp internal conductivity, S.m

-1 σpz internal particle conductivity inside the shell, S.m

2 2 σr peak variance, s xv

τ tortuosity

τe external tortuosity

τp internal particle tortuosity

τpz internal particle tortuosity inside the shell

τt total tortuosity Ω ratio of the internal effective diffusivity to the molecular diffusivity

ωi structural parameter due to diffusive exchange mechanism for the contribution i of eddy diffusion

Abbreviations BET Brunauer-Emmett-Teller CT Computed tomography DFG Driving force gradient DLS Dynamic light scattering EMT Effective medium theory GRM General rate model HETP Height equivalent to a theoretical plate HPLC High performance liquid chromatography ISEC Inverse size exclusion chromatography NMR Nuclear magnetic resonance NLDFT Non-local density functional theory PDI Polydispersity index PFG-NMR Pulsed field gradient nuclear magnetic resonance PP Peak parking RTM Residence time weighted TDA Taylor dispersion analysis THF Tetrahydrofuran

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______Introduction

1 INTRODUCTION

1.1. Research objectives The porous material morphology plays a significant role on the transport properties of fluids covering a broad range of specialties (geology, engineering, chemistry and physics) and having important applications such as catalysis, separation, cement chemistry, etc. Porous media are generally characterized by their porosity, which is the ratio of the volume of the pores to the total volume, i.e. the volume of the matrix surrounding the pores and the pore volume. Another characteristic of porous media is the pore size distribution. Depending on the size of the pores, porous media can be divided into macroporous materials like rocks with pore sizes greater than 50 nm, mesoporous materials such as numerous silica having the pore size in the range of 2 to 50 nm and microporous materials such as zeolite with pores smaller than 2 nm [1]. In liquid chromatography, mesoporous silicas with different morphologies are often used to separate molecules. An example of a chromatographic column filled with spherical mesoporous particles is presented in Fig.1.1. The molecules are transported through the column by the fluid flow. It is generally admitted that transport is due to convection, depending on the fluid velocity, and diffusion processes in the macropores around particles whereas only to diffusion through the mesopores inside particles.

Figure 1.1. Transport properties in porous media

______Introduction

Transport depends on many material structural parameters, i.e porosity, pore size and surface area of the materials, which are generally determined by classical methods such as gas adsorption or mercury porosimetry [1,2]. However, these dimensional parameters do not describe the complexity of most porous materials consisting of labyrinths of interconnected pores with random shapes and cross-sections. This complexity has a direct influence on the transport properties through porous media. One of the parameter reflecting the complexity of porous networks is called tortuosity [3]. In its simplest definition, it is defined as a ratio of way lengths in the presence and in the absence of porous material, but, as we shall see, other definitions are generally used. In liquid chromatography, tortuosity is used to calculate the intraparticle diffusion coefficients of the molecules through a column but it is often taken as a constant adjustable value ranging between 1.4 and 2 whatever the size of the molecule [4–8]. In this thesis we propose, as one of the objectives, to determine tortuosity at different scale: the total tortuosity at the column scale, the tortuosities in the macropore domains and in the mesopore domain by using liquid chromatography or electrical measurements. The apparent tortuosity seen by a particular molecule will be studied as a function of its size in order to better model the effective diffusion coefficient of the molecules through columns packed with porous silica materials presenting different architectures. The effective diffusion model will be applied to porous media made of fully porous particles, core-shell particles or monoliths, which received a lot of interest from many authors [9–14]. The core-shell particles consist of a solid core and a porous shell surrounding the solid core. Monoliths have a rod shape with a through-pore system around a prous skeleton. The aim of this thesis is to study the transport properties of molecules of different sizes through columns filled with silica having different morphologies in non-adsorbing and adsorbing conditions, taking into account the interaction between the molecules and the surface of the solid.

1.2. Structure of this thesis This thesis is divided into five chapters. The current chapter is a bibliographic review section offering some background information about the relationship between tortuosity and porosity in porous media, the fundamental theory of liquid chromatography, the determination of diffusion coefficients by liquid chromatography, the eddy diffusion, the adsorption and surface diffusion. In chapter 2, the techniques used here for characterizing the porous materials will be presented. In chapter 3, the physical properties of porous materials used in this study will be presented, which are necessary to interpret the experimental results in the following chapters.

______Introduction

In chapter 4, the transport properties in non-adsorbing conditions will be investigated in static and dynamic conditions by liquid chromatography. In chapter 5, transport will be studied in adsorbing conditions and the role of surface diffusion will be analysed. Finally, chapter 6 contains the conclusions and recommendations for further work. 1.3. Transport properties in porous media 1.3.1. The relationship between tortuosity and porosity in porous media The porous materials used in this study are chromatographic columns filled with mesoporous silica having different morphologies including totally porous particles, core-shell particles or monoliths. The typical cross-sections of the full-porous particle, core-shell and monolithic columns are given in Fig.1.2. Monoliths made by sol-gel methods consist of monolithic rods made of highly porous silica skeleton and through-pores. Contrary to porous spherical particle, the external porosity in monoliths is high, producing a highly linked system of channels with high permeability. The particles in core-shell consists in a solid core surrounded by a porous shell (also called porous zone). The ratio of the size of the solid core to the size of the particle is expressed by the parameter ρ. All these columns are well controlled hierarchical materials with several porosity scales from mesopores to macropores and different topologies. The macropores around the particles in the case of porous spherical particles and core-shell particles or around the skeleton in the case of monoliths allow the mobile phase to flow rapidly at low pressure through the porous media. The mesopores in totally porous particles, in the shells of core-shell particles or in the skeletons of monoliths form the fine porous structure and provide a large uniform surface area on which adsorption easily occurs. a) b) c)

Figure 1.2. Typical structure of a) totally porous particle, b) core-shell particle and c) monolith

______Introduction

As already mentioned, porosity and tortuosity are among the most important parameters to characterize the structure of porous media in view of transport understanding. In general, porosity is determined as the ratio between the empty space and the total volume of porous materials.

When studying a chromatographic column, the internal porosity (εpz) referring to only the mesopores inside particles or skeleton has to be distinguished from the total porosity of the packed bed (εt) and the external porosity of the interparticle space or interskeleton (εe) which corresponds to macropores. The presence of solid walls in a porous system causes the diffusion paths of species to deviate significantly from straight lines, which leads to introduce the concept of tortuosity. Tortuosity is a structural parameter, it describes the complexity of the conducting network such as sinuosity, connectivity, constriction, death volume, curvature of pores, etc. Tortuosity is an unspecified concept; it has not yet a universally admitted definition. It is used to illustrate the restriction effect on transport of complicated topology [3,15,16]. An illustration of this parameter is shown in Fig.1.3.

Figure 1.3. Illustration of a tortuous path through a porous media

Mathematically, the tortuosity can be defined as the ratio of the length of the tortuous path, le,which probes must follow through the porous medium, and the straight length of the sample, L: l (1.1) τ = e L The value of tortuosity is larger than 1.. The definition is straightforward but it is practically very difficult to determine the flow paths in which the fluid percolates through the porous medium. This unidimensional definition fails in the description of the complexity of porous systems and in the modeling of the transport properties in three-dimensional networks. Experimental, empirical and theoretical approaches have been made to assess the tortuosity depending on the application in question as electrical measurements [3,15–17], pulse field gradient nuclear magnetic resonance

______Introduction

(NMR) [18–22], simulation [23–25], gas chromatography [4,6] or liquid chromatography [26–28]. From these studies it appears useful to define the tortuosity factor as a ratio of distances to the square [15]. Carman [3] suggested using the conductivity measurements to determine the total tortuosity. The porous materials in conductivity measurements are filled with an electrolyte, which has the bulk conductivity σo. The effective conductivity of a porous medium saturated with the electrolyte is lower than the electrolyte conductivity by a factor depending on the total porosity t and the total tortuosity t as: (1.2) εt . σo σeff = τt The value of total tortuosity obtained by this equation depends on the sinuousness of the flow trajectory in pores and also on complex geometric factors like the connectivity and the constrictivity of pores. By this method, one can easily measure the total tortuosity in either a packed or fluidized bed. A chromatographic bed has an external and internal porosity as well as an external and internal tortuosity factor respectively. To be modeled and better understood, the effective conductivity in the whole system should be deduced from the conductivity contribution of each domain (macropores and mesopores). This can be done by assuming that domains are either in series or in parallels but such models cannot describe correctly complex porous materials. Effective Medium theory (EMT) models are proposed to describe the properties of heterogeneous systems. The Maxwell equation is one of these models. In the work of Barrande et al [16], the intraparticle tortuosity of porous spheres is determined by a comparison of the data of the electrical conductivity measurements and a Maxwell equation for a dilute system of spheres. It should be noted that the absolute tortuosity should be a topologic characteristic of the porous network independent of the size of the probe molecule introduced to measure it. When the size of the molecule increases the term “apparent tortuosity” should be preferably used. The tortuosity determined by electrical conductivity measurements should be close to the absolute tortuosity since ions small as compared to pore sizecan be used as conductors within the electrolyte. Other methods of tortuosity evaluation in a direct manner are based on the determination of the ratio of the value of the bulk diffusion coefficient Dm to its apparent value in the porous network,

Deff instead of using conductivity:

(ε [r ]). D (1.3) Deff = t m m τt

______Introduction where: - Dm is the bulk diffusion coefficients of the molecules, which can be experimentally determined by dynamic light scattering (DLS) or Taylor dispersion analysis (TDA) [27],

- εt[rm] is the total porosity accessible to a molecule of size rm The diffusion coefficient of the probes through porous media can be obtained by different techniques such as liquid chromatography [26,29], Wicke-Kallenbach diffusion cells [30] or diffusion nuclear magnetic resonance (NMR) [19,22]. The relation between diffusivity and tortuosity depends on the way diffusion coefficients are measured, this is why we wrote porosity between parenthesis in equation (1.3) In peak parking method or in NMR approaches, the molecule gradient concentration is measured inside the porosity so the apparent tortuosity is eff directly equal to the ratio Dm /D . In NMR techniques, tortuosity in porous media is obtained by the measurement of the restricted self-diffusion coefficient at so-called long measurement times. At these times, the displacement of the diffusing molecules is much larger than the characteristic length-scale of the pore space. The measured restricted diffusion coefficient reaches an asymptotic value independent of the measurement time and directly related to the tortuosity of the porous material. The restricted self- diffusion coefficient achieved by NMR techniques is directly related to tortuosity. Carman [3] also recognized that diffusion and electrical conductivity are closely related phenomena, both formulated by the Laplace equation. Based on the relationship presented by Clennel [15] between the tortuosity derived from conductivity and from effective diffusion coefficient, Barrande et al [16] proposed a generic equation for tortuosity definition as follows:

J eff (ε) J (1.4) ( ) = ( ) DFG τ DFG o where J/DFG is the ratio between a parameter J characterizing the transport flux (flow of matter, velocity, electrical current, etc.) and the driving force gradient DFG, responsible for the transport. The indexes “eff” and “0” indicate measurements with and without the porous medium of porosity ε, respectively. This is not a geometric definition, but the tortuosity defined through diffusion measurements, conductivity measurements is expected to reflect the topological properties of the pores network. Theoretical approaches are often based on a specific model of the structure of the porous medium [31,32]. These models generally do not contain any adjustable parameters, but they are applied to idealized structures. In empirical approache models, adjustable parameters are used to fit the experimental data. However, those adjustable parameters depend on the pore geometry and vary significantly from a 6

______Introduction material to the other [33]. Hence, there have been various attempts to determine empirical relationships between porosity and tortuosity [33]. Among these, Commiti and Renaud (1989) [34] conducted experiments on the flow through beds packed with non-porous spherical or cubic particles. The relationship between the tortuosity and the porosity was suggested as follows:

τ = 1 − plnε (1.5)

This equation satisfies the boundary condition =1 for ε=1 and may be used for various types of packed beds. The topological parameter, p depends on particle shape. p values about 0.53 have been found for cubes and from 0.86 to 3.2 for plates. Later, Mauret and Renaud (1997) [35] obtained the value of p about 0.49 for fluidized beds of spheres. This value was in good agreement with the value of 0.5 given by Weissberg [36]. Weissberg related the porosity of packed bed of non-porous spherical particles to a parameter which is connected to the ‘tortuosity factor’ discussed by Carman [3] by a demonstration of Eq.1.5. This parameter evaluates the decrease of diffusion efficiency as a result of confinement. Wyllie and Gregory [37], studying the fluid flow through unconsolidated porous aggregatesof cubic particles, found p = 0.63. Matyka et al. [38] used a numerical method to study the relationship between porosity and tortuosity in a microscopic model of a porous medium. The lattice Boltzmann method was applied to solve the flow equations in low Reynolds number regime, then the flow streamlines were found and the tortuosity of the flow with cubic particles was finally determined. The obtained result is closely related to Eq.1.5. Tortuosity can also be directly calculated by random walk simulation using, for example, three-dimensional X-ray computed chromatography (CT) images of packed beds [39]. In the long- time limit, the random walkers fully explore the high tortuosity of porous media, and the diffusion coefficient reaches a constant value. Hence, the geometrical tortuosity of the porous media can then be calculated as the ratio of the diffusion coefficient in bulk fluid to this constant value for the lattice walk in a simple cubic lattice. This method is more interesting than the lattice Boltzmann method because the required computation time is shorter. Barrande et al. [16] suggested that the apparent internal particle tortuosity of porous particles,

τp, can be obtained by using the Weissberg model [36]:

τp = 1 − plnεpz (1.6)

where εpz is the particle porosity (pz=porous zone) and p is a topological factor. When the size of the molecule increases the accessible porosity decreases, the friction with the pore walls increases but the effect on the apparent tortuosity is not clear. In most of the publications, the tortuosity is taken as a constant value. The porosity of the silica particles may be 7

______Introduction seen as a distribution of nonporous particles creating a particle with porosity εpz. The Weissberg equation can thus be used. Another assumption is that the evolution of tortuosity with probe size rm is given by the same equation applied to the accessible particle porosity εpz[rm] [27]:

τp[rm] = 1 − pln εpz[rm] (1.7)

1.3.2. Determination of diffusion coefficients by liquid chromatography The transport properties of molecules of different sizes are studied by high performance liquid chromatography (HPLC) by the determination of the diffusion coefficients of the molecules through the chromatographic columns which are packed with the porous materials. Chromatography with a liquid mobile phase is a widely applied method in separation sciences. It is based on different rates of migration of sample components through the column. In HPLC, columns with internal diameter size in the range of 2 to 8 mm are used. These columns are generally packed with particles having an average diameter lower than 50 μm. Size exclusion chromatography (ISEC) is a well-known separation method where the main retention mechanism is the size exclusion effect. The method is applied in non-adsorbing conditions. The stationary phase of an SEC method is always a mechanically stable porous media with a defined narrow distribution of pore sizes. Molecules with different sizes carried along the column by the fluid mobile phase are transported from the bulk fluid phase to the external surface of the packed bed and then diffuse within the particles. The molecules with the sizes greater than those of the pores cannot explore the interior of the solid and are thus more rapidly transported through the column than the smaller molecules that enter the pores [40]. SEC is generally used to characterize the size of polymers. In Inverse Size Exclusion Chromatography, known polymers are used to characterize the material. The porosities (total, external and particle) of the packed bed and the pore size distribution curves are then identified through the retention time of each probe along the column [41]. The diffusion coefficients of probe molecules can be obtained by the analysis of their chromatographic peak. When molecule size is no more negligible as compared to pore size, three retarding effects have to be considered as represented in Fig. 1.4.

______Introduction

Figure 1.4. Three retarding effects in the flow of large molecules through porous media: i) steric hindrance; ii) relationship between tortuosity and probe size; iii) friction [27]

In case i), the steric hindrance corresponds to a decrease of the accessible particle porosity when the size of the molecule increases because the center of the molecule is excluded from the vicinity of the wall on a distance equal to the radius of the molecule rm. Topologically, it can be seen in a first approximation as an isomorphic constriction of the pores, but this could depend on the pore structure and organization. In case ii) the accessible tortuosity depends on the probe size. As seen in Fig.1.4, the structure seen by a molecule depends on its size because the possible pathways depend on its size. The large probe can only explore a fraction of the total pore volume. The apparent tortuosity depends on the molecule size. In case iii) there is a friction between the molecule moving within the pore and the wall. This friction increases with an increase of molecule size. Generally, a friction factor kf[rm] depending on the ratio of molecular size (rm) to the pore size (rp) is introduced to correct the bulk diffusion coefficient from this wall drag effect. Several models like the Renkin equation can be found in the literature and applied to liquid chromatography.

The extent of all these contributions depends on the ratio, λ, between probe size (rm) and the pore size (rp). To evaluate these contributions, liquid chromatography is used here to study the transport properties of probes of various sizes in non-adsorbing and adsorbing conditions. Two methods used to characterize transport in a porous medium by modeling the band broadening of the chromatograms are the dynamic and static (or peak parking) methods. In dynamic method, the effective intraparticle diffusion coefficients are obtained by using the band broadening of chromatographic peaks in combination with Van Deemter equation and general rate model (GRM) [4,27,42]. In static method the band broadening is analyzed at zero flow rate by keeping the eluent for a given time in column, which is described as arrested elution or peak parking (PP) method initially set up by Giddings and Knox [4,43–45].

______Introduction

1.3.2.1. Fundamental theory of dynamic method. In liquid chromatography, a solute is distributed in the mobile phase and the stationary phase. When the mobile phase moves relatively to the stationary phase under the fluid flow, the separation occurs. When the concentration of the solute in the mobile is equivalent to its concentration in the stationary phase, the equilibrium can be approached. In fact, this equilibrium is never reached under the continuous flowing movement of the mobile phase But a near equilibrium state for a mass rapid mass exchange is considered [46]. To support this near equilibrium condition, the concept of the theoretical plate is adopted. The column is considered to contain several layers in which each layer is equivalent to one theoretical plate or the column height equivalent to a theoretical plate (HETP) introduced by Martin and Synge in 1941 [47] as shown in Fig 1.5. Each plate has a specific length and the solute will spend a finite time which is enough to achieve equilibrium with the two phases. A small plate means that the equilibrium will be achieved faster and the exchange of components between the two phases is more efficient. HETP is calculated by the ratio of the length of the column to a number of the theoretical plates.

Figure 1.5. Theoretical plates in a chromatographic column [48]

The plate concept is useful for the description of the broadening of the band during its movement along a chromatographic column and also for the idea that each plate represents a type of residence time [46]. Along with the notion of residence time, the properties of the solute in the mobile and stationary phases and the partition coefficient K give a better view of the operating conditions. The partition coefficient K is the ratio of the concentration of the solute in the stationary phase (Cs) to that of the solute in the mobile phase (Cm), K = Cs/Cm. The retention time, tr is counted from injection to the maximum concentration at the output of the column. It is a contribution of the total residence times of the solute in the mobile phase (tm) and in the stationary phase (ts), respectively. It is also determined from the specific average local residence times, τm and τs in the two-phase system, tr = nt (τm +τs) = tm + ts whereas nt is the number of transfers between the stationary phase and mobile phase. The step or transfer length (l=L/nt), a ratio of the length of the column to the number of the transfer is similar to the HETP [46].

______Introduction

On the other hand, one uses the notion of retention volume, Vr which is the mobile phase volume needed to transport the solute from the injection time, through the column and to the detector (corresponding to the highest point of the chromatographic peak). According to the plate theory, the retention volume is determined as Vr =f tr = Vm + KVs in which Vm denotes the total pore volume occupied by the mobile phase, Vs is the total volume of stationary phase and f is the flow rate of the mobile phase [46].

2 Admitting that the peak dimensionless variance is equal to the number of the step σ = nt , the peak variance in unit of distance can be written [46]:

2 2 σz = nt l = l. L = HETP. L (1.8a)

2 whereas L is the length of column, σz is the peak variance in units of distance. When measured in 2 2 2 units of time (σz = σr u푅), HETP is derived from the characteristic parameters of the chromatographic peaks as follows:

2 2 2 σr uR σr (1.8b) HETP = = L 2 L tr 2 where σr is the peak variance in units of time, uR is the overall velocity (also defined as superficial velocity) and tr is the mean retention time which corresponds to the maximum of the peak of a given probe. In practice, the plate model is used to describe all zone spreading phenomena which takes into account the nonequilibrium processes in a column and longitudinal diffusion effect [48]. HETP is always used to quantify column efficiency which describes how well a column separates components in a mixture. Column efficiency increases with a decrease in the value of HETP or a decrease in the value of peak variance. From Eqs.1.18a and1.18b, one can see that the peak variance or the width of the peak increase with the distance or the time of the solute traveled in the column [46]. When a column is packed with porous particles, the column volume can be divided into the interstitial volume surrounding the particles and the intraparticle volume. The solutes are eluted through the column under the control of both convection and diffusion in the region outside the particles but only of diffusion in the pores [49]. The interstitial velocity which corresponds to the flow in the interstitial pores between particles is related to the overall velocity by u=uR/

When the chromatographic peak has a symmetrical peak, a Gaussian function is used to fit the chromatographic peak. If the peaks are not symmetrical the theory of the moments is preferred. An example of a Gaussian fit of an elution peak in the fully-porous particle packed column is shown in Fig.1.6. 11

______Introduction

140

120

100 Peak 80 Gaussian fit

a.u 60

0 2 3 4 5 6 7 8 time (min)

Figure 1.6. An example of a chromatographic peak and a Gaussian fit The contribution of band spreading in the extra-column volume can be reduced or even completely eliminated by using a zero dead volume connector connecting the inlet to the outlet capillaries of the column [27]. The extra-column volume corresponds to the dead volumes (i) in injection system, (ii) between injector and column, (iii) between column and the detector, and (iv) in detector itself. The spreading of the peak is due to the low diffusion of sample molecules in the liquid mobile phase during the experiments [40]. The corrected HETP is written as:

2 2 (1.9) (σr − σi ) HETP = L 2 (tr − ti) where σi and ti are the standard deviation and the retention time of the peak for a zero dead volume connector, respectively. In this work, the values of mean retention time and of the variance of a peak were analysed by fitting the peaks with Gaussian function [27]. Van Deemter [42] proposed an equation to describe the relation between HETP and interstitial velocity, u as: B (1.10) HETP = A + + Cu u where A is the eddy diffusion term, B is the longitudinal diffusion term and C is the mass transfer term. The interstitial velocity u is calculated by: f (1.11) u = 2 πr εe 12

______Introduction where the f is the flow rate, r the radius of the column (r) and εe the external porosity of the column. The typical profile of HETP curve based on Van Deemter equation is given in Fig.1.7.

Figure 1.7. Height equivalent to a theoretical plate (HETP) as a function of interstitial velocity based on Van Deemter equation

The value of HETP is under the control of three main different contributions: (i) eddy dispersion referring to effects of packing inhomogeneity of the flow pattern in the column (A term), (ii) longitudinal or axial diffusion accounting for the natural diffusion of sample molecules due to the differences of their concentration gradient in the solution (B term), and (iii) the overall solid-liquid mass transfer resistance including trans-particle or trans-skeleton mass transfer and external film mass transfer explaining the retardation in achieving local equilibrium between the bulk eluent and the stationary phase (C term) [50]. The A term depends on the structural homogeneity of the column, the fluid path is different in the center and near the walls, which causes a band broadening. It is expected to be a constant contribution to the overall HETP whaterver the molecule. The B term is a result of the axial molecular diffusion of molecules in the porous bed. It has an important contribution to the band dispersion, mainly at low velocity. This

______Introduction contribution decreases with an increase in the velocity. The B term can also be determined by the peak parking method [4,26]. The mass transfer of a probe between the mobile and stationary phase (C term) is supposed to correspond to a non-equilibrium partition of the solute concentration between the mobile phase and the stationary phase. When the mobile phase velocity increases, the rate of equilibration becomes slower, thus broadening the eluting peak. The overall mass transfer consists of the film mass transfer resistance, the adsorption-desorption kinetics, and the internal diffusion inside the mesopores. Since the convection inside particle is assumed to be negligible as compared to that outside particle, the probe in the porous particle will primarily move by molecular diffusion. This contribution to mass transfer is proportional to the velocity. Gritti and Guiochon [51] showed that the effective intraparticle diffusion is constant with velocity: it depends on the pore structural properties of the stationary phase. In the case of columns packed with fully porous spheres, the HETP for a probe having the size rm under non-adsorbing conditions was modelled by the general rate model (GRM) established by Kucera and extended to other morphologies by Miyabe and other authors [4,52–54]: 2 2 (1.12) 2DL εpz[rm] d′p d′p HETP = + 2εe (1 − εe ) ( ) [ + eff ] u u εt[rm] 6km 60Dp [rm] where: - DL is the axial dispersion coefficient, - u is the interstitial velocity,

- εe is the external porosity of the column,

- εpz[rm] is the particle porosity accessible to a molecule of size rm,

- εt[rm] is the total porosity accessible to a molecule of size rm,

- d’p is particle diameter,

eff - Dp [rm] is the effective intraparticle diffusion coefficient of a probe of rm size,

- km is the external mass transfer coefficient.

The value of km can be derived by Wilson-Geankoplis equation [55] 1/3 (1.13) Dm 1.09 εe d′p u km = ′ [ ] dp εe Dm

The axial dispersion coefficient, DL was derived with the assumption that axial dispersion encompasses molecular diffusion and eddy diffusion [42]:

______Introduction

DL = γ1Dm + γ2d′pu (1.14) where Dm is the molecular diffusion coefficient, γ1 is a coefficient (often defined as the obstructive factor which is proportional to the reverse of tortuosity) which should correspond to the diffusion transport efficiency through the external porous network and γ2 is a packing characterization factor due to the eddy diffusion contribution [42]. At very small velocity, making the C term negligible, the peak broadening is a result of the diffusion both in external and internal pores. In this case, the value of γ1 is expressed by [27]:

εt[rm] (1.15) γ1 = εeτt[rm] where τt[rm] is the apparent tortuosity of the whole accessible porosity of a molecule of size rm. Comparing Eq.1.10 and Eq.1.12, one can obtain the expression of the mass transfer C term for a fully porous particle in non-adsorbing conditions as:

2 2 εpz[rm] d′p d′p (1.16) C = 2εe (1 − εe ) ( ) [ + eff ] εt[rm] 6km 60Dp [rm] The mass transport for intermediate sized molecules through columns packed with the porous particle is clearly under the control of C term and an almost linear behavior is observed for the plots of the experimental HETP curves versus interstitial velocity [27]. The values of C term were evaluated from the linear section of the HETP plots at high flow rates. Hence, the effective intraparticle diffusion coefficient in porous particle was determined. Under adsorbing conditions, the HETP value for a column packed with totally porous particle is determined by Eq.1.16 [4,56,57]. The HETP equation for a column packed with core-shell particles or monolithic column is slightly different. 2 2 (1.17) 2DL 2εe k1[rm] d′p d′p HETP = + ( ) [ + eff ] u u (1 − εe) 1 + k1[rm] 6km 60Da [rm]

eff where: - Da is the effective intraparticle diffusion coefficient in adsorbing conditions for a molecule of size rm,

- k1[rm] is the zone retention factor accessible to a probe of size rm defined as:

(1 − εe) (1.18) k1[rm] = (εpz[rm] + (1 − εpz[rm])Ka) εe

- Ka is the equilibrium constant of the analyte between the adsorbent and the mobile phase inside the porous particles.

The value of Ka is expressed by [50,58]: 15

______Introduction

εt[rm] tr − to (1.19) Ka = (1 − εt[rm]) to

- tr and to are the mean retention time of an elution peak in adsorbing and in non-adsorbing conditions, respectively.

1.3.2.2. Peak parking method In static conditions, the effective diffusion coefficient through a chromatographic column is obtained by the peak parking (PP) method or arrested elution method which was initially introduced in gas chromatography by Knox and McLaren [4], and then in liquid chromatography [43]. This method has been widely used in gas and also in liquid chromatography with columns packed with spherical particles or monoliths [27,43,50,51,59–66]. In peak parking method, the flow is stopped when the molecule is at a position near the middle of the column and the molecule can diffuse through the porous media for a given time called parking time tp. Then the pump is restarted at the same flow rate and the molecule is moved to the detector where the peak broadening is measured. The experiment is repeated for a series of parking times. In agreement with Einstein’s diffusion equation for a unidimensional symmetry of diffusion, 2 the band variance in a unit of length, ∆σz, is given by [26,43,50]:

2 eff ∆σz = 2D tp (1.20) where: - Deff is the effective diffusion coefficient of the solute in the column,

- tp is the parking time The peak variance in liquid chromatography is measured in the time unit. The conversion between 2 2 2 unit length and unit time is obtained by using the velocity of the molecule uR (σz = σt uR). The velocity of the molecule is given by the length of the column divided by the mean retention time of the molecule (uR=L/tr). The retention time of a sample in the column is written as [9,50,56]: L (1.21) t = (1 + k [r ]) r u 1 m where k1[rm] is the zone retention factor of the polymer with size rm and is determined by Eq.1.18.

In non-adsorbing conditions, the value of k1[rm] depends only on the porosity parameters which have been achieved by ISEC method.

2 From the slope of the plot of the total peak variance in the time unit, ∆σt , versus parking time, eff tp, the value of D can be calculated by[43]:

______Introduction

2 2 (1.22) eff 1 ∆σt u D = 2 2 tp (1 + k1[rm])

The longitudinal diffusion of the probe during its transfer along the column, Hlong, is defined in the book of Giddings [48] by:

∆σ2 (1.23) H = z long L To have a better view when comparing the column efficiency of columns with different particle sizes, the reduced values depending only on structural parameters are used [45,48]. The reduced longitudinal diffusion coefficient term independent of bulk diffusion can be written as [50]: (1.24) Hlong b = u Dm Therefore, combining Eqs.1.20, 1.22, 1.23, 1.24 provides:

Deff (1.25) b = 2 (1 + k1[rm]) Dm The reduced b term could be measured by peak parking method through the effective diffusion coefficient, Deff. Finally, the reduced longitudinal diffusion coefficient b for any type of column can be directly obtained from this equation [50]:

1 ∆σ2 u2 (1.26) b = t Dm tp (1 + k1[rm]) To calculate the effective intraparticle diffusion coefficient of an eluent through a chromatographic column from the effective diffusion coefficient obtained from the peak parking method, a model is needed. The parallel-zone or residence time weighted (RTW) model and effective medium theory (EMT) models (Maxwell, Torquato, and Landauer) different by their degrees of accuracy and applicability can be used. The RTW model proposed by Knox and Scott [43] is based on the additivity of the mass density fluxes in a mobile and stationary phase. Other models developed from the effective medium theory are more accurate but require different parameters. The dispersion of different phase in Landauer model is assumed strictly irregular [67]. It is not suited to represent the diffusion in particulate media. The Torquato model is based on a probabilistic theory [68]. Among the explicit models, the Maxwell model is the simplest model and it is unnecessary to use higher order terms as in the Torquato model. The Maxwell equation [31] is widely accepted in the field of chemical engineering to represent the diffusion in packed

______Introduction

bed columns. The Maxwell model can be extended to the core-shell particles [62,69,70] and monoliths [62,68,70]. Table 1.1 summarizes the different models used to derive the effective intraparticle diffusion coefficient for different types of chromatographic columns. In these models, the effective intraparticle diffusion coefficient is expressed under the parameter Ω, as a ratio of effective intraparticle diffusion coefficient in the porous particles (with the fully porous and core-shell eff particles) or the porous skeleton (for the monolithic column), Dp (r m), to the bulk diffusion

coefficient Dm[63,71]:

′eff eff εpz[rm]Dp [rm] Dp [rm] (1.27) Ω = = Dm Dm

eff whereas D′p [rm] is the local effective intraparticle diffusion coefficient which depends only on the friction with the pore wall and the particle tortuosity in the peak parking method where the molecule is still present inside the pore. In the case of core-shell particles, due to the presence of the solid core in the spherical particles, the porosity of the porous shell around the solid core will be higher than the apparent porosity of the whole particle by a factor of (1-ρ3), in which ρ is the core-to-particle diameter ratio. The eff effective intraparticle diffusion coefficient of a probe of size rm in the porous shell, Dpz [rm] is eff distinguished from the one of the particle, Dp [rm] by this expression [62,70]:

ρ3 (1.28) Deff[r ] = Deff[r ](1 + ) pz m p m 2 Table 1.1. The effective diffusion models for the chromatographic columns

Model Case of spheres (with porous particle ρ= 0 Case of cylinder and, with core shell ρ>0

eff RTW D 1 εe = [ + (1 − εe)Ω] (1.29) whereas τe is the external tortuosity Dm εe (1+k1[rm]) τe

eff eff Maxwell D 1 1+2 β(1−ε ) D 1 1+ β(1−εe) = e (1.30) = (1.32) Dm εe (1+k1[rm]) 1−β(1−εe) Dm εe (1+k1[rm]) 1−β(1−εe)

Ω−1 Ω−1 with β = (1.31) with β = (1.33) Ω+2 Ω+1

eff 2 eff 2 Torquato D 1 1+2 β(1−εe)−2εe ξ2β D 1 1+ β(1−εe)−εe ξ2β = 2 (1.34) = 2 (1.35) Dm εe (1+k1[rm]) 1−β(1−εe)−2εe ξ2β Dm εe (1+k1[rm]) 1−β(1−εe)−εe ξ2β

where ξ2 is the so-called three-point with ξ2= 0.11 for monoliths [73,74].

parameter. ξ2= 0.3277 for spheres [72]

______Introduction

Landauer √ ′2 Ω Deff a′+ a + ⁄2 = (1.36) Dm εe (1+k1[rm])

1 with a′ = [3ε − 1 + (2 − 3ε )Ω] (1.37) 4 e e

1.3.3. Eddy diffusion Eddy diffusion is one of the main contributions controlling the broadening of a compound band during its migration along a column. The value of this term of HETP equations is often considered as constant in liquid chromatography and estimated to varie from three to five times the particle diameter [75]. As proposed by Giddings [48], the velocity inequalities of the flow pattern in the center and near the wall of the column leading to zone spreading has several origins as presented in Fig 1.8. The different contributions are: transparticle (the interchange occurring through the stagnant mobile phase in the porous solid particles), transchannel (the contribution due to the exchange of molecules between near particles or inside throughpores), short-range interchannel (the effect as a result of a significant velocity differential occurring between large, open channel and the smaller surrounding ones with a characteristic length over a distance of one particle diameters), long-range interchannel (variations in the average velocity for a characteristic length of a hundred particle diameters) and transcolumn (velocity differences with a characteristic length equal to the column radius).

______Introduction

Figure 1.8. Different exchange processes due to the flow velocity inequalities in the column [48] Giddings also proposed that eddy diffusion is the result of these velocity inequalities. The overall eddy diffusion is a combination of these different contributions. Hence, the A term from Van Deemter equation corresponds to this overall eddy diffusion. The contribution of the long- range interchannel eddy diffusion to the overall column efficiency was estimated small so this term was neglected in the calculation of the overall eddy diffusion. The transparticle effect is taken into account in the C term corresponding to the mass transfer between the liquid and the solid phase of the van Deemter equation. Finally, Only three types of such biases seems to contribute effectively to the overall eddy diffusion inside the column: transchannel, short-range interchannel and trans-column velocity biases [48]. Van Deemter et al [42] assumed that the eddy diffusion contribution is independent of the interstitial velocity of the mobile phase. By empirical approach in a different way, Knox [45] presented the equation for the dependence of the reduced plate height on reduced velocity by

______Introduction taking into account the contributions from the three dispersion mechanisms in which the value of total eddy diffusion depends on the velocity as follows: b (1.38) h = a v1/3 + + cv v where: -h is the reduced plate height, obtained by the ratio of HETP to particle size d’p

(h=HETP/d’p) -v is the reduced velocity calculated by:

d′p f d′p (1.39) v = u = 2 Dm πr εe Dm - f is the flow rate of a mobile phase, - r is the radius of a column, - a,b,c are the reduced eddy diffusion, longitudinal diffusion and mass transfer terms, respectively. Giddings [48] interpreted two exchange mechanisms, i.e. flow and diffusive mechanisms, allowing the transfer of molecules from one eluent streamline to another. The first mechanism is under the control of the complex flow pattern of these streamlines through the three-dimensional structure of the packed column. When transferring from one streamline to another, the velocity of a molecule changes. This mechanism applies mainly at high velocities but the possibility for molecules to be transferred between near streamlines by diffusion is neglected. The second exchange mechanism applies mainly at low velocities when the transfer of the molecules by diffusion is faster than that by flow exchange. Based on this coupling theory of eddy diffusion developed from the random walk approach, Giddings [48] assumed the eddy diffusion resulting from a combination of flow and diffusive mechanisms. Then the reduced HETP is written as: 3 (1.40) 1 b h = ∑ ( ) + + cv 1/(2λi) + 1/(ωiv) v i=1 The first term of the right-hand side of Eq.1.40 describes eddy diffusion as the sum of three main contributions including transchannel, short-range interchannel and transcolumn eddy diffusion in which 2λi is the reduced plate height due to the flow exchange mechanism among velocity extremes and ωiv is the reduced plate height as a result of the diffusive exchange mechanism. The parameters λi, ωi for a given inequality i depend only on the geometrical structure of the packing. These structural parameters may be regarded as nearly constant for the particular category i in the velocity inequality classification [48]. At high velocities, the reduced eddy

______Introduction diffusion approaches the constant value of 2λi. At low velocities, the reduced eddy diffusion converges on the reduced plate height due to the diffusive exchange mechanism and is proportional to velocity. It is generally assumed that the value of the eddy dispersion term should be the same for all probes, if excluded or not from the mesopore volume of the chromatographic column and whether the compound is adsorbed or not [44,76]. By total pore blocking experiments, Gritti and Guiochon [77] challenged this assumption on the essence of the important differences between the HETP of a column measured before and after blocking solute access to mesopores. It was shown that the contribution of eddy diffusion to the efficiency of packed columns depends on whether access to the mesoporous volume of the particles is possible or not. The overall eddy dispersion term of a retained solute entering in the mesopore volumes was two to three times less than that of a molecule which had no access to the internal porous volumes. As seen in packed columns and in monolithic columns, an origin of band broadening may come from the effect of transcolumn eddy diffusion [75].

1.3.4. Adsorption and surface diffusion The mass transfer steps in a chromatographic columns are sorted out into two types: (1) mass transfer of molecules without the control of physico-chemical interactions with the stationary phase surface (axial dispersion, external mass transfer, and effective intraparticle diffusion) and (2) kinetic processes concerning adsorptive interactions (surface diffusion and adsorption– desorption kinetics) [78]. Adsorption is a surface phenomenon in which molecules interact with the interface between two media. This process creates a layer of the adsorbate on the surface of the adsorbent. There have been two general mechanisms in this process: physical and chemical adsorption. Physical adsorption is the result of Van der Waals force and electrostatic force between adsorbate molecules and the atoms which compose the surface of an adsorbent. Chemical adsorption involves a chemical reaction between the adsorbent surface and the adsorbate. In comparison with physical adsorption in which the adsorption may have multi-layer models under certain conditions, the strong interaction in chemical adsorption creates new types of electronic bonds and only occurs in a single layer. The adsorbents can be characterized by surface properties such as surface area and polarity. Gas adsorption measurement is a useful method to obtain these characterized properties. The adsorption isotherm is defined as a plot which expresses the amount adsorbed on a surface of

______Introduction adsorbent against either bulk concentration for liquid adsorption or partial pressure in the case of gas at a fixed temperature. In order to estimate the practical or dynamic adsorption capacity of an adsorbent, it is essential to have information on adsorption equilibrium. The kinetic analyses have to be performed based on rate processes. The intraparticle diffusion is the most typical of the rate limiting steps in solid adsorbents [79]. Adsorption happens when there is a contact between an adsorbent with the surrounding fluid of a certain composition. After a sufficiently long time, the adsorbent and the surrounding fluid reach equilibrium. In this state, one can determine the amount of the component adsorbed on the surface of the adsorbent from the variation of concentration in solution. This is the so called solution depletion method. In liquid chromatography, the reaction rate of physical adsorption is assumed to be fast enough so the influence of adsorption–desorption kinetics is usually neglected. The retention mechanism relates to an adsorption phenomenon in which there is an equilibrium state between the amount of adsorbate in solution and the amount of adsorbate adsorbed on the surface of the adsorbent. Chromatographic data are interpreted on the basis of coupling between adsorption and other transport phenomena such convection and bulk diffusion. Moreover, adsorption introduces itself a new transport phenomenon, i.e. surface diffusion [78]. Fig.1.9 illustrates the adsorption and surface diffusion phenomena. The isosteric heat of adsorption can be released when a sample molecule is adsorbed from the mobile phase on the solid phase. This adsorbed molecule can migrate if it can surpass the energy barrier between the surrounding adsorption sites known as the activation energy of surface diffusion.

Figure 1.9. Surface diffusion of sample molecules on the stationary phase [80] Generally, the effective diffusion coefficient of the porous media in adsorbing conditions is usually interpreted by assuming the parallel contribution of intraparticle diffusion of the molecule in the pore filled with mobile phase (term obtained in non-adsorbing conditions) and surface

______Introduction diffusion of the analyte adsorbed on the surface of the pores [73,79,81,82]. Effective intraparticle eff diffusion coefficient, Dp , is the diffusion of the sample molecules in the porous media without interaction with the stationary phase surface. Molecules are transported by diffusion due to their concentration gradients. The adsorbed molecules can also migrate by surface diffusion, Ds. Because the sample molecules diffuse in accordance with the amount adsorbed gradient, the mass flux by surface diffusion is proportional to the equilibrium constant, Ka. Hence the effective eff intraparticle diffusion coefficients in adsorbing condition, Da is often expressed by this relationship:

eff eff Da = Dp + (1 − εpz)KaDs (1.41)

1.4. Literature survey In this part we focus more precisely on results of the literature obtained with systems close to ours. 1.4.1. Silica topology analysis Silica is widely used in industrial applications especially in separation sciences as a chromatographic column. A lot of works had been published on the determination of the topology of different types of columns packed with silica. Tallarek et al [19] applied the pulsed field gradient nuclear magnetic resonance (PFG-NMR) to a direct and detailed experimental study of topological and dynamic aspects in chromatographic columns packed with spherical-shaped, porous C18 silica particles. The translational displacement of the intraparticle fluid molecules in a packed bed of porous C18 silica particles was determined from the displacement probability distribution, allowing to extract an effective intraparticle diffusion coefficient. Ultimately, the intraparticle tortuosity factor obtained from the steady-state pore diffusion measurement based on the averaged propagator formalism for nuclear spin displacements, arrived at the value of 2.2. Nakashima and Watanabe [39] estimated the transport properties of porous silica bead by X-ray computed tomography (CT) and random walk simulation using the digital image data of a pore structure made of packed monodisperse glass beads with a diameter of 2.11 mm creating a pore size of 0.4 mm. This monosized spherical bead pack is an analogue of natural sandy sediments involving mainly silica with percentage of 73 % in mass. The diffusion coefficient in three- dimensional space was defined either by using the time derivatives of the mean-square displacement from PFG-NMR laboratory experiments or by monitoring the diffusive migration of

______Introduction iodine by X-ray CT. Hence, the value of tortuosity of the bead pack was measured as the determined ratio of the diffusion coefficient of iodide ion in bulk water to that in the water- saturated porous media by a medical X-ray CT system. A value of 1.53 was obtained, which is comparable with the value presented by Latour et al [83]. The tortuosity was also calculated numerically by random walk simulation leading to a value of 1.6 in agreement with the one obtained by the KI solution diffusion experiment. Gritti and Guiochon [26] investigated the dispersion of an almost unretained compound, thiourea in a mixture of methanol and water (25/75, v/v) on six different stationary phases packed with porous silica particles with different C18 surface densities. The peak parking method was applied to provide accurate values of the effective diffusion coefficient. The effective diffusion coefficient was also expressed by the sum of parallel diffusion fluxes in the solutions contained in the external and the internal porosities of the bed, and the surface diffusion of the solid adsorbent. In this expression, the apparent diffusion coefficient is dependent on both the ratio of the frequency factors of surface and bulk diffusion and the internal obstructive parameter of the pores. To account for the data of internal obstruction, an assumption of the ratio of the frequency factor for the surface to the frequency factor for bulk molecular diffusion between 15 and 40 was made. The internal tortuosity factor of the silica particles, which is inversely proportional to the internal obstructive factor, was then obtained between 1.1 for neat silica and 1.4 for the grafted silica. Barrande et al [16] used electrical measurements and Maxwell equation to determine the tortuosity of porous particles. The total tortuosity of different porous particles suspensions was calculated from the total porosity and conductivity factors measured directly from the electrical measurement. By using a dilution method where the conductivity of the suspension is determined as a function of its porosity, the conductivity of particles was obtained by making a least-squares fit of the value of total conductivity versus porosity, which can be expressed rigorously as a function of particle conductivity by the Maxwell equation at high dilution of the suspension. From the porous particle conductivity, its tortuosity can be calculated. Finally, the particle tortuosity of porous silica particles range between 1.3 and and 1.5 for the samples selected in this paper. Another interesting finding is the assessment of the value of the topological parameter p = 0.49 for assembly of spheres.

Miyabe et al [56] studied the transport of butylbenzene through a C18-silica monolithic column in reversed-phase liquid chromatography systems. A mixture of methanol and water (30/70, v/v) was used as a solvent. The intraparticle diffusion coefficient of butylbenzene was calculated by the porosities and the obstructive factor in the internal volumes occupied by the mobile phase from

______Introduction the peak parking-moment analysis measurements. The parallel pore model was also applied to estimate the effective intraparticle diffusion coefficient in relation to the external porosity, the tortuosity factor and the hindrance parameter. Ultimately, the value of tortuosity was calculated at 3.4 when the value of the hindrance parameter was obtained by the equation proposed by Satterfield et al [84] in which the hindrance factor only depends on the ratio of the diameter of the solute molecule to the average pore diameter. When using the equation proposed by Brenner and Gaydos [85] to calculate the hindrance parameter, a similar value of the tortuosity factor was determined. Hlushkou et al [74] used the physical reconstruction of the interskeleton macropore space of a capillary silica monolith by confocal laser scanning microscopy for direct pore scale numerical simulations of fluid flow by the lattice-Boltzmann method and convective diffusive mass transport in the calculated flow fields by a random-walk particle-tracking technique on a high-performance computing platform. By the analysis of transient diffusion with increasing observation time and diffusion length, the time-dependent diffusion coefficient was seen as an approach of the asymptotic value of the diffusion coefficient. Hence, the tortuosity of the interskeleton macroporosity of silica monolith was determined by the ratio of the molecular diffusion to this asymptotic value in the long-time limit at around 1.37. Daneyko et al [24] examined the influence of microscopic order on transverse dispersion in the external pore of chromatographic beds by the numerical simulations for polydisperse random sphere packings (computer-generated with particle size distributions of modern core–shell and sub-2 μm fully porous particles), the macropore space morphology of a physically reconstructed silica monolith, and computer-generated regular pillar arrays. The lattice-Boltzmann method was used for simulation of fluid flow and the random walk particle-tracking technique was used to simulate the mass transport in the polydisperse sphere packings, in the monolith, and in the pillar arrays. The external obstruction factor was determined by monitoring the long-time limit of the diffusion coefficient, analogous to that ratio of the preasymptotic diffusion coefficient in packing to the diffusion coefficient in the bulk fluid. The values of external tortuosity were then derived of 1.493 for the sub-3 μm core–shell particles, of 1.488 for the sub-2 μm fully porous particles and of 1.37 for the macropore space of the monolith. For the pillar arrays, the tortuosity was 1.639 and 1.314 with the external porosity of 0.4 and 0.7, respectively. Reich et al [86] investigated the overall diffusive hindrance factor expression of solutes for random mesoporous silica by using electron tomography to physically describe the mesopore space of three macro-mesoporous silica monoliths. The geometric tortuosity was determined from

______Introduction geodesic distance propagation for each reconstruction at around 1.13. A model of overall diffusive hindrance factor was given as a function of the ratio between tracer size and pore size. The porosity and particle tortuosity are explored by a point-like tracer in self-similar macro- mesoporous silica monoliths. This global factor was independent of the topological parameter, p. The authors gave the suggestion that the Weissberg equation could only be applied when the pore networks are self-similar. The diffusive tortuosity was calculated from pore-scale simulations of the diffusion of point-like tracers in the reconstructions. The values of tortuosity decrease when mesopore size and porosity of the reconstructed mesopore spaces increase, from 1.67 for a sample with a mesopore size of 12.3nm, to 1.45 for a sample with a mesopore size of 21.3nm and to 1.35 for a sample with a mesopore size of 25.7nm.

1.4.2. Eddy diffusion in liquid chromatography Eddy diffusion has got a lot of interest in liquid chromatography. Most of the work focused on the ways to obtain eddy diffusion, the influences of eddy dispersion on the mass transfer of a chromatographic column and the evolution of eddy diffusion as a function of interstitial velocity or probe size. Knox and Scott [43] investigated the B and C terms in the Van Deemter equation by liquid chromatography. By the peak parking method for simple aromatic solutes in methanol and methanol-water mixtures with octadecyl silyl (ODS) silia gel as packing material, the longitudinal diffusion coefficient was evaluated then the B term. C values are obtained by measuring the band- dispersion of the same solutes at high reduced velocities. The model of reduced plate height was proposed as shown in Eq.1.38 in which the constant eddy diffusion in Van Deemter equation [42] was replaced by the eddy diffusion dependent on the velocity as a product of A term and the cubic root of reduced velocity. Then this model was fitted with the reduced plate height calculated from the band broadening to get the value of A = 2.5. Gritti et al [87] studied the mass transfer kinetics of butylbenzoate in the mixture of methanol and water (65:35, v /v) through a silica monolithic column. The axial dispersion in the mobile phase due to the irregularity of the bed structure was determined by the coupling theory of Giddings which assumes that the plate height is the combination of the flow and diffusive mechanisms. When neglecting the effect of trans-column eddy diffusion, the eddy diffusion was only caused by the flow mechanism and the value was equal to the average characteristic distance in the skeleton of chromatographic monolithic bed (3.7µm). The achieved data show that axial

______Introduction dispersion controls HETP at low and moderate velocities. Band dispersion during this range of velocities was also controlled by the eddy diffusion and axial diffusion in the macropores. Gritti and Guiochon [51] analyzed the mass transfer of a series of polystyrenes with different molecular weights in THF through a 4-µm porous silica column. The ISEC and peak parking methods were used. The eddy dispersion term in the overall HETP equation was calculated by the coupling theory of eddy dispersion established by Giddings. For conventional chromatographic columns (length 15 cm or less, inner diameter 4.6 mm), the eddy dispersion was assumed to be constituted of three main contributions: the transchannel, the short-range interchannel and the long-range interchannel. The approximate magnitude of the flow and diffusion parameters was given according to Giddings [48]. At low flow rates, the values of eddy diffusion obtained for all samples increased along with the interstitial velocity. They approached the constant values when the velocity was over than 0.25cm/s. Gritti and Guiochon [50] measured all the parameters of the mass transfer kinetics in a chromatographic column. The mass transfer properties of different small molecules through C18 shell kinetex column in a mixture of water and acetonitrile (35/65, v/v) were studied. The B term (longitudinal diffusion) was obtained from effective diffusion obtained by peak parking method. The effective intraparticle diffusion was obtained by applying the effective diffusion into the parallel diffusion model, then the C term was obtained through Eq.1.16. The eddy diffusion term including the transchannel velocity biases, the short-range interchannel velocity biases, and the transcolumn velocity biases was determined either by invasive methods and simplified 2D structures or by non-invasive approaches. In the case of invasive techniques, local electrochemical detection method with micro-electrodes was used to determine the transcolumn eddy diffusion. The total pore blocking method, which consists in filling the porous particles with n-nonane, was applied to obtain the sum of the transchannel and short-range interchannel eddy diffusion terms. The idealized 2D systems was used to calculate the transchannel eddy diffusion term. With non- invasive methods, the overall A terms of packed and monolithic columns in liquid chromatography were calculated either by the subtraction of the previously measured mass transfer terms (B and C) from the experimental data of HETP or by the reconstruction of the structure of the column bed. The obtained results through C18 shell kinetex column showed that the reduced eddy diffusion term decreased with increasing retention factor. Daneyko et al [88] studied the influence of the particle size distribution (PSD) on hydraulic permeability and eddy dispersion in bulk packings. The narrow-PSD and wide-PSD random computer-generated packings based on the experimental PSDs of sub-3μm core-shell and sub-2μm

______Introduction fully porous particles were constructed by scanning electron microscopy. The reduced eddy diffusion was expressed by the coupling theory of Giddings with the neglect of trans-column contribution. The structural parameters characterizing the eddy dispersion contribution to band broadening originating from flow inhomogeneities at the transchannel scale and at the short-range interchannel scale were obtained by fitting the reduced plate height data normalized by the surface-mean diameter to a condensed form of the Giddings equation. These structural parameters depend on the bed porosity.

1.4.3. Adsorption and diffusion on silica surfaces The adsorption and diffusion of polystyrene on silica were studied by some authors. They focused on the adsorption isotherm of polystyrene and the surface diffusion on a solid surface. The effect of pore size of silica on the adsorption of polystyrenes has also received some interest in the literature [80,89–92]. Eltekov and Nazansky [93] investigated the gel permeation chromatography and adsorption of polystyrenes (PS) from different solvents on microporous silicas. Carbon tetrachloride, toluene, ethylbenzene, and n-heptane were used as solvents. The adsorption isotherms were calculated by the solution depletion method. The adsorption isotherm showed that adsorption of polystyrenes changed from positive for hydroxylated Silochrom C-2 to negative for silanized Silochrom C-3 when adsorbed from carbon tetrachloride and ethylbenzene. The PS adsorbed from toluene was adsorbed negatively as compared with adsorption from ethylbenzene. The chromatographic peaks of the PS standards and n-heptane obtained were wide and asymmetrical due to the adsorption of PS on the surface of silica. Miyabe and Suzuki [94] used the moment analysis of the chromatographic peaks in high performance liquid chromatography technique to study the surface diffusion of p-tert- octylphenol in the mixture of methanol and water (70/30, vol.) on an octadecylsilyl-silica gel (ODS). The adsorption equilibrium constant (or Henry constant as commonly used in chemistry) was obtained from the retention time calculated by first absolute moment. The intraparticle diffusion coefficients were determined from the intercept of the linear plots between the plate height defined by the second moment and a multiplicative inverse of the interstitial velocity. The surface diffusion coefficient was then evaluated by correcting the contribution of pore diffusion to the intraparticle diffusion as shown in Eq.1.41. The contribution of surface diffusion to the intraparticle diffusion was found to be 90 - 95%. The results show that the value of the surface diffusion coefficient increases by a factor of about 2 to 3 with increasing adsorbed amount.

______Introduction

Kawaguchi et al [89] performed the kinetic studies of individual adsorption of polystyrenes (PS) from cyclohexane at different polymer concentrations on a porous silica. When a PS chain easily diffuses inside the pores of silica, the time to achieve an equilibrium state was less than 10h whereas an adsorption equilibrium time of 35h was attained for the larger PS penetrating into the pores with much deformation. It was found that the adsorbed amount of the larger PS is higher than that of shorter PS at short adsorption time with an increase in concentration. The difference in adsorbed amounts may be due to the increase of anchorage point by polymer chain when their length increases. El’tekova and El’tekov [90] studied the adsorption of polystyrene on porous silica and presented data on the liquid chromatography. Carbon tetrachloride (CCl4) was used as a solvent. Polystyrene in CCl4 solutions was adsorbed primarily on hydroxylated silica and a monolayer was formed on the surface of silica at equilibrium concentrations larger than 3mg.mL-1. In the low-concentration range, the adsorption isotherm was linear allowing the calculation of Henry constant. It was also calculated from the retention volume of a polymer in the column, the volume of the mobile phase and the weight of the sorbent in the column by the liquid chromatographic experiment at 250C. In 2009, Kim et al [95] studied the adsorption of polystyrene oligomers in the mixture of THF and n-hexane at different compositions on a pure silica surface with normal phase liquid chromatography technique. The chromatograms of polystyrene with different separated peaks were obtained. The degree of polymerization of the oligomer peaks was identified by using mass spectrometry. When the elution volume of unretained compounds was seen as an adjustable parameter based on Martin’s rule which expressed the relationship between the retention factor and the degree of polymerization of a polymer in liquid chromatography, the retention factor was proportional to the degree of polymerization.

1.5. Motivation for the thesis As described earlier, a lot of efforts has been done to determine the tortuosity by NMR [19– 22,39] or by electrical measurements [3,15–17] using probes having a size negligible as compared to the pore size or by liquid chromatography [26–28]. Generally, the particle tortuosity was taken as a constant value ranging between 1.4 and 2 whatever the size of the molecule. In this thesis, the total and particle tortuosities of different mesopore structures are determined by impedance spectroscopy and by using the Maxwell equation. The particle tortuosity is also expressed by Weissberg equation in which the particle tortuosity depends on porosity and on a parameter, p,

______Introduction depending on the material topology. It is also used to predict the evolution of apparent tortuosity as a function of molecular size. The eddy dispersion term of HETP equations was usually expected to be a constant contribution to the overall HETP [27,87]. The evolution of eddy diffusion to the molecular size is still questionable [48,50]. The source of the differences of eddy diffusion of different molecules may be due to the transcolumn contribution [75]. Therefore, in this research, the eddy diffusion is determined by subtraction of longitudinal diffusion and mass transfer term from HETP. The source of the change in eddy diffusion with different molecules will be shown. The adsorption on silica was investigated in solvents like carbon tetrachloride [93] or cyclohexane [89]. Another solvent which is less toxic and gives a better view of the influence of surface diffusion on band spreading should be considered. The knowledge and understanding of how the surface diffusion influences elution in liquid chromatographic methods is crucial.

______Characterization techniques

2 CHARACTERIZATION TECHNIQUES

Depending on the size of the pore inside porous materials, different types of characterization methods are used. Inverse size-exclusion chromatography is known as a porosity analysis method used for characterization of porous materials by using series of probes with different molecular sizes giving information in the mesopore range. Gas adsorption measurement is principally applicable to the pores in the range of 0.4nm to 100nm when using nitrogen as an adsorptive [96].Mercury porosimetry is used to characterize porous materials having pores between about 400µm and 3nm [1,2,97,98]. These three approaches give complementary information and their comparison helps in defining pore structures. The electrical measurement was also performed in this work to evaluate the tortuosity parameter of materials.

2.1. Inverse size exclusion chromatography 2.1.1. Principles The inverse size exclusion chromatography (ISEC) method is used to determine the porosity, pore size, pore size distribution of porous materials. In this method, a series of polymers of different sizes are used in non-adsorbing conditions. The solution of each polymer is successively injected and eluted through the chromatographic column. For each solution, the retention time tr is measured. Smaller probe molecules penetrate more into pores, have a longer path and therefore lead to larger retention time. From the mean retention time and the flow rate, one can assess the pore volume seen by each molecule and then its accessible porosity. This method is different from size exclusion chromatography in which the separation of molecules by different sizes occurs in stationary phases with a defined pore size in order to analyse polymer size. The principle of ISEC method is shown in Fig.2.1. Small molecules see all the pore volume Vp, intermediate molecules explore a fraction of the pore volume and large molecules are excluded from pores. The assumptions in the calculation of the pore size distribution of the packing material are that the probe molecules are spheres and the shape of the pores are presumed cylinder as in mercury porosimetry and nitrogen sorption method. Unlike the methods of characterization of the porous materials based on capillary phenomena, assumptions about contact angle or surface tension are not necessary for this approach [1].

______Characterization techniques

Figure 2.1. A principle of the inverse size exclusion chromatography

For characterization of the chromatographic column, it is important to first define the types of volumes used in this method. The total porous volume of the column, Vt is defined as the sum of the volume external to the particles (or skeletons), Ve and the volume of the pores within the particles, Vp. The total accessible volume of a solute of rm size, Vt[rm], is the sum of the porous volume and a portion of the internal pore volume. This leads to the general equation: (2.1a) Vt [rm] = Ve + kd [rm]Vp

The distribution coefficient of a probe of rm radius can be derived as:

Vt [rm] − Ve (2.1b) kd [rm] = Vt − Ve and by dividing by the column volume: (2.2) εt [rm] − εe kd [rm] = εt − εe with: - Vt[rm] is the total accessible volume to a solute, easily obtained from the flow rate of the pump, f, and the retention time of a solute of rm size, tr[rm] by the relationship:

Vt [rm] = f tr [rm] (2.3)

- εt[rm] is the total porosity accessible to a molecule of radius rm, derived from:

Vt [rm] − Vo (2.4) εt [rm] = Vc where Vo is the volume of the capillaries, obtained from the experiments done without column, Vc 2 is the geometrical volume of the chromatographic column, Vc = πr L with r and L are the radius and length of the column, respectively.

- εe is the external porosity of a column, obtained with the molecules excluded from the porosity,

______Characterization techniques

- εt is the total porosity of the column determined with the smallest probe.

Kd is 0 for totally excluded molecules and Kd=1 for the smallest molecule. Depending on the difference of solute distribution between the internal and external space, the internal volume of the column, Vp, is calculated from the pore volume of the smallest probe exploring all the pore volume and the interparticle volume of the column, Ve, is evaluated from the elution volume of the probes which cannot see all the pores in bead. The accessible total porosity to a given solute ranges between the total porosity of the column which corresponds to the total porosity of the smallest molecule entering every the internal and external pores and the external porosity corresponding the pores seen by the molecules excluded from the internal pores. Finally, the accessible particle porosity for the probe of rm radius, εpz[rm], is calculated by the following equation [99]: (2.5) εt [rm] − ε ε [r ] = e pz m 3 (1 − εe )(1 − ρ ) where εe is the external porosity and ρ is the ratio of the solid core to the shell particle diameter. =0 for totally porous particles and monolith columns and ρ=0.625 for the core-shell particles used in this study.

For a spherical molecule with the size rm penetrating a cylindrical pore of radius r, the accessible pore volume to the mass center of the molecule is reduced by the size of the solute to a cylinder of radius r-rm (see Fig.1.4.i). The local distribution coefficient is thus given geometrically as [100,101]:

(r − r )2 πl r (2.6) k = m = (1 − m)2 d r2 πl r where l is the length of the pore. For an assembly of cylinders with a continuous distribution of radii, the total pore volume accessible, Va, to a solute of size rm is taken as equivalent to the volume of pores having a size larger than that of the solute with this equation [41,102]:

∞ 2 (2.7) Va [rm] = ∫ f(r)(1 − rm/r) dr rm where rm is the radius of the probe assumed to be spherical and f(r)dr is the volume of pores with the radius in the range of r and r+dr.

The theoretical overall distribution coefficient, kd[rm], to a particular solute of rm radius will be the sum of contributions from each pore size and is expressed by the integral [100,103]:

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∞ f(r)(1 − r /r)2 dr (2.8) ∫r m k [r ] = m d m ∞ ( ) ∫0 f r dr where f(r) is the relative volume fraction of pores as a function of pore size, the differential pore size distribution in other words. It could be calculated from a Gaussian distribution of pore sizes with a mean value rp and a standard deviation of the distribution contribution σ as [41]:

1 1 2 (2.9) f(r) = exp [− (lnr − lnrp) ] √2πσ2 2σ2 In this study, the Solver program in Excel was used to fit the model with the experimental kd[rm] values to obtain f(r). The mean pore size, rp is determined by using the following equation:

∞ ∫ rf(r)dr (2.10) r = 0 p ∞ ( ) ∫0 f r dr

2.1.2. Equipment The inverse size exclusion chromatography (ISEC) measurements were performed using the 1200 HPLC system (Agilent Technologies), having a quaternary gradient pump with a multi-diode array UV-VIS detector, an automatic sample injector with a 100µL loop, an autosampler and a thermostated column compartment. The maximum pressure which can be reached by this apparatus is 400bar. The injection volume was set at 1µL and all experiments were conducted at 25 °C, fixed by the column thermostat. The concentration of the solutes samples at the outlet was recorded using the diode array detector at 262 nm. The system is controlled by the Chemstation software.

Figure 2.2. The typical scheme of a chromatographic analysis system

______Characterization techniques

The schematic overview of a chromatographic analysis system is shown in Fig.2.2. The solvent was injected through the column by the quaternary gradient pump. The injector autosampler introduce the sample into the mobile phase stream which carries the sample into the chromatographic column packed with the porous materials. A UV detector is used to follow the concentration of the molecule when solvent elutes through the HPLC column. The mobile phase exits the detector and can be sent to waste. 2.1.3. Columns Three silica columns from different commercial companies with different structural characteristics are used in this thesis. The totally porous spherical particle column, Lichrospher Si 100 and the monolithic silica column, Chromolith Performance Si 100-4.6 are provided by Merck (Darmstadt, Germany). The core-shell column, Poroshell 120 HILIC is bought from Agilent (United States). The manufacturer specifications of used porous materials are shown in Table 2.1. Table 2.1. Specifications of the columns made of pure silica (data from manufacturer)

Name Silica type Producer Column’s Particle size d’p, Mesopore dimension length µm size rp, [mm] * I.D. [mm] nm Lichrospher Totally porous Merck 250*4 5 5 Si 100 particles

Poroshell Core-shell Agilent 150*4.6 4 6

120 Particles (ρ=dsolid core/d’p= 2.5/4=0.625) Chromolith Monoliths Merck 100*4.6 1 (skeleton) 6.5 Si

2.1.4. Samples and solutions

Tetrahydrofuran (THF) with the chemical formula of C4H8O used as a mobile phase in non- adsorbing conditions was purchased from Carlo Erba Reagents (SDS). In adsorbing conditions, a mixture of n-heptane (n-C7H16) and THF (97/3, v/v) is used to study the transport properties in adsorbing conditions.

Toluene was purchased from Aldrich. A series of polystyrenes (-CH[C6H5]-CH2-)n with various molecular weights and molecular sizes were provided from Polymer Standards Service (Mainz, Germany). Solutions of the polymers and toluene are prepared at a concentration of 1 gL-1. For the polymerization of polystyrene, n-Butyllithium (C4H9Li) is added to styrene monomer then it reacts with another styrene radical in the next step and so on. At the end of this stage, the terminating

______Characterization techniques agent proton H+ is added to remove Lithium at a given time. So, the molecular weight of the polystyrenes is written as:

Mw=104p’+58 (2.11) whereas p’ is the number of units of polystyrenes.

The molecular weight, Mw and polydispersity index, PDI of samples are given by the manufacturer. The values of molecular diffusion coefficient, Dm in non-adsorbing conditions for the smallest Polymers (Toluene, P1, P2, P3 and P4) are obtained by Taylor dispersion analysis and by Dynamic light scattering (DLS) measurements for the other Polymers (P5-P12). The

kB .T hydrodynamic radii rm are calculated with the Stokes-Einstein equation, rm = where T is 6πηp Dm the absolute temperature, kB is the Boltzmann constant, and ηp is the viscosity of the mobile phase

(for THF, ηp = 0.46 cP at 298 K). The data of molecular diffusion coefficient and probe radius obtained from DLS measurements for polystyrene from P5 to P12 in the solvent THF were taken from Wernert et al 2010 [27] and are presented in Table 2.2. Taylor dispersion analysis (TDA) measurements [27,104–107] were used to obtain the diffusion coefficients of P1, P2, P3 and P4 both in THF and in the mixture of n-heptane and THF. Solutions of these polymers are prepared at a concentration of 1 gL-1. The measurements are also carried out on the Agilent 1200 Series HPLC replacing the column by a stainless steel tube (0.876 mm i.d., length 1.20 m). Polystyrenes are eluted through the tube at the flow rate of 0.01 ml/min. The molecular diffusion coefficients of polystyrenes are obtained by using this relationship:

R2(t − t ) (2.12) D = c r i m 2 2 (σr − σi ) 2 where Rc is the capillary internal diameter, σr is the peak variance and tr is the mean retention time corresponding to the peak of a given probe and σi, ti is the standard deviation and the retention time of the peak of a sample measured without capillary, respectively. This equation is verified as long as the two following conditions are fulfilled [104]. First, the dimensionless 2 residence time Dmtr/Rc should be higher than 1.4 in order to make sure that molecules diffuse radially from the center of the capillary to its edges; second, the Péclet number (Pe = uRc/Dm) should be higher than 70, i.e. in this case the contribution of the axial dispersion on the mass transfer is negligible. The measured bulk diffusion coefficient obtained by Eq.2.12 for these small polymers are reported in Table 2.2.

______Characterization techniques

Table 2.2 Molecular weights, polydispersity index, number of units, bulk diffusion coefficient in THF and in the mixture of THF and n-heptane and hydrodynamic radii rm

(1) Polymer Molecular PDI Molecular diffusion Dm in the Probe radius Number code weight, coefficient in THF, mixture of in THF, of units (1) 2 -1 Mw Dm/m s THF and n- rm heptane, /g mol-1 /nm 2 -1 Dm/m s

Toluene 92 (2,350,11).10-9 (2) ~ 0,2050,00 ~ -9 (2) -9 (2) P1 162 1.00 (1,850,11).10 1.83. 10 90,26 0,02 1 P2 690 1.09 (7,090,20).10-10 (2) 8.57.10-10 (2) 0,680,02 6 P3 1380 1.05 (5,160,21).10-10 (2) 5.47.10-10 (2) 0,930,04 13 P4 3250 1.05 (3,210,28). 10-10 (2) 3.01.10-10 (2) 1,500,13 31 P5 8900 1.03 (2,0360,005).10-10 (3) ~ 2,360,01 85 P6 19100 1.03 (1,3270,006).10-10 (3) ~ 3,620,02 183 P7 33500 1.03 (8,5200,005).10-11 (3) ~ 5,6330,00 322 -11 (3) P8 96000 1.04 (5,0830,008).10 ~ 49,44 0,02 923 P9 243000 1.03 (3,1940,009).10-11 (3) ~ 15,020,04 2336 P10 546000 1.02 (2,1010,009).10-11 (3) ~ 22,840,10 5249 P11 827000 1.08 (1,6500,008).10-11 (3) ~ 29,090,15 7951 P12 1850000 1.05 (1,120,01).10-11 (3) ~ 42,900,40 17788 (1) Given by supplier (2) TDA measurements (3) DLS measurements [27]. The diffusion coefficients for each fraction of polystyrenes in the mixture of THF and n- heptane with different units can be estimated from a general equation relating diffusion coefficient to molar mass, Mw as following:

−a2 Dm = a1 (Mw) (2.13) where a1 and a2 are the empirical parameters depending on both the polymer and the solvent. In this study, only the small molecules (P1, P2, P3 and P4 were studied) in adsorbing condition. As seen in Fig.2.3, the diffusion coefficients of these molecules in the mixture of THF and n- heptane at 25°C obtained by TDA experiments may be fitted by a power equation (R2=0.9944) as:

−8 −0.599 Dm = 4.021 x 10 (Mw) (2.14)

______Characterization techniques

2,5E-09

2E-09

TDA analysis /s

2 1,5E-09 Proposed model eq.2.14

1E-09 Dm, m Dm,

5E-10

0 0 1000 2000 3000 4000 Mw, g/mol

Figure 2.3. Molecular diffusion coefficients obtained by TDA for the small polymers in the mixture of THF and n-heptane

2.2. Nitrogen adsorption 2.2.1. Theory Adsorption is the enrichment of one or more components in an interfacial layer between the gas and solid phase. Physisorption occurs whenever an adsorbable gas (the adsorptive) is equilibrated with the surface of a solid (the adsorbent). The interfacial layer consists of two regions: the surface layer of the adsorbent (called the adsorbent surface) and the adsorption space in which enrichment of the adsorptive occurs. In the case where the molecules of the adsorptive penetrate the surface layer and enter the structure of the bulk solid, the term absorption is used. The other phenomenon inversely related to adsorption is desorption in which the amount adsorbed decreases. Usually, both adsorption and desorption are studied. For a solid that would contain all types of pores, the physisorption process occurs initially in the adsorption space of micropore (with the pore size less than 2nm) regarded as micropore filling different from surface coverage which takes place on the walls of open macropores (pores with the size larger than 50nm) or mesopores (pore size in the range of 2 to 50nm) in which the monolayer-multilayer adsorption and capillary condensation occur [8]. All the adsorbed molecules are in contact with the surface layer of the adsorbent in monolayer adsorption and not in direct contact in multilayer adsorption. In capillary condensation the pore space remaining after multilayer adsorption has occurred is filled with condensate at a pressure that depends on pore size. Capillary condensation is often accompanied by hysteresis which means that the adsorption and desorption curves do not coincide. During the gas adsorption process, at a constant temperature, the relationship between the amount of adsorbed substance and the equilibrium pressure of the gas is known as adsorption

______Characterization techniques isotherm. The amount of gas adsorbed may be determined by measuring either the amount of gas removed from the gas phase or the uptake of the gas by the adsorbent. The shape of the adsorption isotherm is classified into six types described in Fig.2.4

Figure 2.4. The six main types of physisorption isotherm [108] The adsorption isotherm is expressed by the relationship between the amount adsorbed (in mol.g-1) and the relative pressure (p/p°) of the pressure p to the saturation pressure p0 at a constant temperature. Type I isotherms are seen with microporous solids having relatively small external surfaces (e.g. activated carbons, molecular sieve zeolites, and certain porous oxides). The limiting uptake is controlled by the accessible micropore volume rather than by the internal surface area. Type II isotherm is the typical form of isotherm for a non-porous or macroporous adsorbent. The Type II isotherm symbolizes unrestricted monolayer-multilayer adsorption. Point B, the beginning of the almost linear middle section of the isotherm, indicates the stage at which the statistical monolayer coverage finishes and the multilayer adsorption is about to start. Type III isotherm is convex to the p/p° axis over its entire range and hence does not show a point B. There are a few systems for this type (e.g. nitrogen on polyethylene). In such cases, adsorbate-adsorbate interaction has an important role. This is encountered with water on hydrophobic surfaces. Type IV isotherm exhibits the characterization of many mesoporous solids with the hysteresis loop, which is associated with capillary condensation occurring in mesopore structures and has a limiting uptake over a range of high p/p°. The initial part of the curve in this type is attributed to monolayer-multilayer adsorption since it follows the same profile as the corresponding part of a

______Characterization techniques

Type II isotherm obtained with the given adsorptive on the same surface area of the adsorbent in a non-porous form. Type V isotherm is unusual; it is related to the Type III isotherm where the adsorbent-adsorbate interaction is weak. The hysteresis loop is also seen in this type, for example water on microporous carbons. Type VI isotherm shows a series of multilayer adsorption steps on a uniform non-porous surface. The sharpness of the steps depends on the system and the temperature. The step-height now corresponds to the monolayer capacity for each adsorbed layer and, in the simplest case, remains nearly constant for two or three adsorbed layers (e.g. argon or krypton on graphitized carbon blacks at 77K). The lowest relative pressure where the desorption branch of adsorption hysteresis seen in type IV and type V, join the adsorption branch is almost independent of the nature of the porous adsorbent but depends mainly on the nature of the adsorptive (e.g. for nitrogen at its boiling point of 77K p/p° =0.42). Some exceptions to this rule are observed, especially systems containing micropores.Brunauer-Emmett-Teller (BET) gas adsorption methods is a standard procedure applied to determine the surface area of finely-divided and porous materials [108]. The BET equation is expressed as the linear form: p 1 (C − 1)p (2.15) a o = a + a o n (p − p) nmC nmCp

a a -1 where n is the amount adsorbed at the relative pressure p/p°and nm(mol.g ) is the monolayer capacity. C is constant dependent adsorption energy.

2 -1 The surface area As(m .g ) as the area of solid surface per unit mass of material is obtained a from the BET monolayer capacity nm and the average area, am (molecular cross-sectional area in nm2), occupied by the adsorbate molecule in the complete monolayer by Eq. 2.16

a As = nmNA am (2.16)

2 whereas NA is the Avogadro constant, am = 0.162nm for N2 at 77K The theory underlying the BET method for determination of surface area incorporating multilayer coverage is developed from an oversimplified extension to multilayer adsorption of the Langmuir mechanism which determines surface area founded on a monolayer coverage of the a solid surface by the adsorptive. To obtain a reliable value of nm, it is necessary that the knee of the isotherm be fairly sharp (i.e. the BET constant C is not less than about 50). A very low value of C (< 20) is related to an appreciable overlap of monolayer and multilayer adsorption; for that case, the application of the BET analysis may be not safe [96]. 41

______Characterization techniques

Referring to the accessibility of pore space to the adsorptive, the particle porosity is the ratio of the volume of open pores to the total volume of the particle. The total pore volume, Vp, is often got from the amount of vapour adsorbed at a relative pressure close to unity on the assumption that the pores are then filled with condensed adsorptive in the normal liquid state. Therefore, the particle porosity εpz is determined by this relationship:

Vp (2.17) εpz = Vp + Vs where Vs is the volume of the solid that can be obtained from the solid density. Determination of the mesopore size distribution is generally obtained by using methods based on the Kelvin equation. The radius of curvature of the adsorptive condensed in the pore, rk (in nm) for nitrogen at 77K is expressed as: −0.953 (2.18) r = k ln p/ po

Since condensation is regarded to occur after an adsorbed layer has formed on the pore walls, it is necessary to consider the thickness of this adsorbed film. In the case of cylindrical pores, the cylindrical pore radius, rp(in nm), is obtained by using the equation by Barret et al [109]: (2.19) rp = rk + t where t is the thickness of the adsorbed film. The thickness of the nitrogen multilayer adsorbed at boiling point, t (in A0) depending essentially on the equilibrium pressure and temperature has been evaluated by application of the equation by de Boer [110] for the adsorbents like siliceous materials in this form:

13.99 1/2 (2.20) t = ( o ) 0.034 − log10(p⁄p ) The method of Barrett et al [109] is a procedure for the determination of pore size distributions from experimental isotherms using the Kelvin model of pore filling. It applies only to the mesopore and macropore. With the non-local density functional theory (NLDFT) approach [111,112], a more reliable pore size distribution for the range of micro and mesopores is calculated on the assumption that the adsorbent surface is homogeneous and the pores have a regular shape (cylinders or slits).The pore size distribution is obtained by solving the integral isotherm equation, which correlates the o o experimental isotherm Nexp(p/p ) with a series (or kernel) of the theoretical isotherms Ntheo(p/p ,w) as follows :

______Characterization techniques

wmax o o (2.21) Nexp(p/p ) = ∫ Ntheo(p/p , w)f(w)dw wmin where: - w is the pore width,

o - Nexp(p/p ) is a point of the experimental adsorption isotherm,

o - Ntheo(p/p ,w) is the amount adsorbed on the model adsorption isotherm on a single pore of width w at the same relative pressure - f(w) is the pore size distribution function. 2.2.2. Apparatus The gas adsorption isotherms were determined by using the Micromeritics ASAP. Alternative gases can also be used, such as krypton (at 77 K), argon (at 87 K), or carbon dioxide (at 273 K). Prior to the measurement of an adsorption isotherm, the sample needs to be prepared by a degassing and heating step at 1200C. During this process, where the surface of the adsorbent is exposed to a high vacuum at elevated temperatures, all the physisorbed species are removed [108]. After outgassing, the sample is protected from the atmosphere. The quantity of gas adsorbed on a surface is recorded as a function of the relative pressure of the adsorptive. The adsorption isotherm is obtained by measuring the difference between cumulative introduced gas amount and the amount remaining unadsorbed. If the isotherm exhibits low pressure hysteresis (i.e. at p/p° <0.4, with nitrogen at 77 K) the technique should be checked to establish the degree of accuracy and reproducibility of the measurements [108]. 2.3. Mercury porosimetry 2.3.1. Principles Mercury porosimetry technique is widely used for the measurements of pore size distribution and other porosity-related parameters of materials. Due to its non-wetting property, the liquid- solid contact angle θ is larger than 900, mercury will intrude capillaries only under pressure. The inverse relationship between the outwardly applied pressure P and the diameter of the pore d in which mercury has intruded is expressed by Washburn equation as: d = -4γcosθ /P. (2.22) where γ is the surface tension of mercury. Generally, the surface tension of pure mercury at 250C is assumed to be 485 mN.m-1. The value of the contact angle θ depends on the solid surface. Nevertheless, in most practical cases, most researchers use a fixed value regardless of the detailed sample material, eg 1300 or 1400 [1,97]. 140° is a value recommended for silica [1].

______Characterization techniques

After each externally applied pressure increment, the pore size distribution is determined by measuring the volume of mercury entering the porous sample. Total porosity is obtained from the total intruded volume.

Figure 2.5. Two typical intrusion–extrusion cycles of mercury [98] The relation between the intruded volume of mercury and the applied pressure is shown by the two typical intrusion-extrusion cycles in Fig.2.5 [98,113]. Region (a) is due to the compaction of the powder bed, and the region (b) corresponds to the intrusion of mercury into the empty space between particles (interparticle volume), followed by the filling of the intraparticle pores in the region (c), and for some materials (reversible) compression is then possible at higher pressures (d). Hysteresis between the intrusion (increasing pressure) and extrusion (decreasing pressure) is observed, and extrusion (e) occurs at different pressures from that of intrusion. At the end of the first intrusion–extrusion cycle, some mercury usually kept by the sample precludes the loop from closing (f). Hysteresis is also seen at the second intrusion–extrusion cycle after the first one (g) but eventually, the loop closes, showing that there is no further entrapment of mercury. On most samples, the loop closes after just the second cycle. Hysteresis and entrapment are in principle of different origin. The comprehensive pore size analysis is obtained by having an understanding of the hysteresis and entrapment phenomena [98]. There are different mechanisms proposed to explain intrusion-extrusion hysteresis. The hysteresis explained by the single-pore mechanism is a basic property of the intrusion-extrusion process because of an energetic barrier correlated with the formation of the vapour–liquid interface during extrusion or to a difference in advancing and receding contact angles. The hysteresis is also explained by ink-bottle theory in which during extrusion process, a large amount of mercury is still kept in the pore network because of the difference of sizes between the entrance of the pore and its core size. The network models consider not only ink-bottle but also percolation effects in 44

______Characterization techniques pore networks. The pore blocking occurring on the intrusion branch is fairly similar to the percolation effects included in gas desorption from porous networks. In disorganized pore networks, pore blocking is the main mechanism, and the pore size distribution can only be calculated from the intrusion branch by applying complex network models based on percolation theory. With certain systems, a limited amount of architectural information from the intrusion– extrusion hysteresis loop can be obtained by applying such models [114]. The general profile of a mercury intrusion–extrusion hysteresis loop is often close to that of the equivalent gas adsorption loop. Hence, mercury intrusion and capillary evaporation follow a similar trend in gas adsorption curve. F.Porcheron et al [115] have confirmed this by using statistical mechanics to model the mercury porosimetry. The porosities of the sample are determined from the values of cumulative pore volume at external pore filling and internal pore filling process of the intrusion branch in Fig.2.5. The total porosity is obtained by dividing the volume of mercury intruded (Vt) which corresponds to the difference between the volume obtained at 3 nm and the volume at the end of the first step due to the compaction of the bed by the total volume which is the sum of the volume of intruded mercury

-3 (Vt) and the volume occupied by the solid (1/s) (s=2.2 g.cm for silica):

Vt (2.23) εt = V + 1 t ⁄ρs The external porosity is obtained by dividing the macroporous volume (volume of the second step corresponding to the intrusion of mercury into the empty space of the interparticle pores for spherical particles or the interskeleton pores in the case of monoliths) by the total volume:

Vmacro (2.24) εe = V + 1 t ⁄ρs The particle porosity and skeleton porosity are obtained by dividing the mesoporous volume (volume of the intraparticle filling step) by the volume of the particle:

Vmeso (2.25) εp = V + 1 meso ⁄ρs Data from mercury porosimetry can also be used to determine the pore size, surface area and particle size. The pore size is evaluated by Eq.2.22. Surface area is derived from the intrusion curve by the equation proposed by Rootare and Prenzlow [116] as:

1 V (2.26) A = − ∫ PdV γcosθ 0

______Characterization techniques

The correction of the estimated surface area is influenced by small measurement errors in the high pressure part of the intrusion-extrusion curve. The surface areas calculated by mercury porosimetry are found to be in good agreement with that values determined by gas adsorption measurement [116]. The particle size can be estimated by the value of pore size with the assumption of a detailed packing structure of the powder particles. Obviously, in this case, the correction of this parameter depends on the assumed packing structure. With the sphere particle, the diameter d’p can be achieved from the specific density of the material ρs by this equation: 6 (2.27) d′p = ρs. A

2.3.2. Apparatus The measurement of mercury intrusion/extrusion data is performed on a Quantachrome Poremaster PM 60-GT-12-18 equipped with a vacuum pump and a cold trap assembly. The system is operated with the Poremaster software. This apparatus can reach the maximum pressure of 414MPa and the minimum pressure at the low-pressure stage at 1379Pa. The pore volumes can be measured by the apparatus in the range of about 1000 to 0.0035 m diameters. Fig.2.6 shows the penetrometer used to keep the sample.

Figure 2.6. A view of mercury penetrometer cell The penetrometer is composed of glass sample cell (an insulator) made of a capillary part and a bulb part plugged by a conductor. The capillary part of the cell operates as a reservoir for the mercury. When pressure drives mercury out of the capillary part of the cell into the bulb part, the mercury volume inside the capillary decreases and so is the capacitance between the mercury (which is in contact with the conductive plug) and the second electrode which a metallic cylinder around the capillary. The decrease in capacitance is proportional to the volume of mercury which leaves the cell with each change in pressure. In the low-pressure stage, the penetrometer cell is 46

______Characterization techniques kept in a horizontal position to minimize the effect of gravity. When switching to the high- pressure stage in vertical position, the initial mercury level is adjust to about one cm from the open end of the capillary to have more correct intrusion volume with the vertical position of the cell. In the low pressure range (low pressure stage) mercury is pushed by nitrogen. In the high pressure range (high pressure stage) it is pushed by oil thanks to a high pressure pump.

2.3.3. Method analysis for pore size determination Different methods have been developed to determine the pore size distribution of porous materials. The pore size ranges in which the methods are most applicable are different. The gas adsorption can be used to characterize the pore size range of 0.4nm to 30nm when using nitrogen as an adsorptive [96]. Meanwhile, the porous systems with the size from about 3nm to 400µm [1,2,97,98] can be characterized by mercury porosimetry. The working pore dimension range in the inverse size exclusion chromatography (ISEC) method is 1 – 400 nm [117]. ISEC method is particularly suitable for the evaluation of meso- and macroporosity of swelling polymers that cannot be analyzed by other techniques, but is applicable to any material that can be introduced in a chromatographic column. Some recommendations for selecting a method were proposed by IUPAC [1]. • The complexity of the porous texture of porous materials is such that even on theoretical grounds the concepts which can be used to describe the texture usually require the introduction of simplifying assumptions. • No experimental method provides the absolute value of parameters such as porosity, surface area, pore size, and surface roughness. Each gives a characteristic value depending on the principles involved and the nature of the probe used (atom or molecule, radiation wavelength...). • The selection of a method of characterization must start from the material and from its intended application. •The method chosen must indeed estimate a parameter related as directly as possible to phenomena involved in the application of the porous material. In this respect, it may often be advisable to select a method involving physical phenomena similar or close to those involved during the practical application (i.e. adsorption or capillary condensation method if the porous substance is to be used as a desiccant, or a freezing point depression method if one is interested in the frost resistance of a construction material...) so that determined parameters are appropriate.

______Characterization techniques

• Rather than to "check the validity" of distinct methods, certified reference porous materials are needed to establish how these methods differ and, of course, to calibrate any individual equipment or technique. • As a consequence, one must not look for a "perfect agreement" between parameters provided by different methods. Such an agreement, when it occurs, is not necessarily a proof of the validity of the derived quantities. Instead, one must be aware of the specific, limited, and complementary significance of the information delivered by each method of characterization of a porous solid.

2.4. Determination of tortuosity by conductivity measurements 2.4.1. Theory The electrical measurement is used to determine the tortuosity of materials through measuring the conductivity. Tortuosity at several scales is derived from electrical measurements on both fixed and fluidized beds of porous media. In such type of measurement, the probe is small as compared to the pore size, which allows reaching a measurement specific of the topology of the pore network and not of the pore size. Two types of measurements are done: determination of the tortuosity of spherical particles by the suspension dilution method and determination of the total tortuosity of a column applied here to monoliths. Case of powders In a case of powders, the sample is added by mass increments to a solution and kept fluidized by a magnetic stirrer. The resistance is measured at each step. After each measurement, the stirring is arrested, the powder is left to settle for 10 min and the resistance of the bulk fluid is verified to be stable during the experiment. The total porosity of the suspension is determined after each added increment by:

Vi (2.28) ε = mp Vi + ρs where: Vi is the initial volume of an electrolyte,

mp is the cumulative mass of dried powder added to the solution at each step,

3 ρs is the density of the material, for Silica ρs= 2.2 g/cm [26]. Basing on the ratio of the conductivity of the porous medium filled with an electrolyte to that of the free electrolyte [118], tortuosity is expressed in the relationship with resistivity and conductivity by Carman [3] as:

______Characterization techniques

ε. σ ε. R (2.29) τ = o = eff σeff Ro where Reff andRo are the resistance of the porous particle suspension saturated with the conductive solution and of the free electrolyte, respectively. The electrical resistance measurements were carried out by impedance spectroscopy (from 1 kHz to 1 MHz) using a standard two-electrodes conductivity cell. Impedance spectroscopy based on Nyquist diagram is a very precise tool for measuring electrical properties of materials and fluids. The total tortuosity (τ) modeled by Maxwell equation at infinite dilution is calculated in the relation with internal conductivity, σp, of the the particle by:

σ σ (2.30) 2 + p + (1 − ε ) (1 − p ) σ e σ τ = ε o o σp σp 2 + − n(1 − εe) (1 − ) σo σo where εe is the external porosity of the particle, the value of n is 2 for the porous powder and 1 for the monolith. The minimization of the difference between the experimental total tortuosity and the one modeled with Eq.2.30 could be performed by changing the ratio of particle conductivity to electrolyte conductivity p/o until experimental and theoretical data are concurrent when porosity tends towards 1 at infinite dilution. Then the internal tortuosity of the particle τp can be calculated by the following relationship with the internal porosity derived from ISEC methods:

εpz. σo (2.31) τp = σp

Case of monoliths In the case of monoliths, a conductivity measurement of a blank column with the same length as that of the monolithic column was carried out due to the contribution of the dead volume in the monolithic column. To eliminate the electrical resistance through the end-fittings of the monolithic and blank columns, a conductivity measurement of a zero-volume column was also carried out. The stainless-steel end fittings are used as electrodes. The reference resistance of electrolyte Ro in the monolith was then obtained by the correction to the resistance of the blank column:

______Characterization techniques

So l (2.32) R0 = Ra Slo where Ra is the resistivity of the free electrolyte in the blank column corrected from the influence of the electrical resistance through the end-fitting; S, l and So, lo are the cross-section and the length of the monolith and blank column, respectively. From the resistance of electrolyte and the measured value of resistance of the monolith saturated with the conductive solution, the total tortuosity of the monolith was then obtained by Eq.2.29 [15,16].

2.4.2. Equipment For the dilution method, conductivity measurement is carried out in a double envelop cell connected to a circulating thermostated bath Polystat 24 from Fisher Scientific at 20°C. The porous particles, i.e. fully porous particle and core-shell powders recovered from the chromatographic columns are added with an increment mass in the cell filled with NaCl 1M. A standard two-electrodes conductivity cell connected with a ModuLab electrochemical system (Model 2101A, Solartron Analytical, UK) is put into the cell to measure the electrical resistance of the suspension through the Z-plot software installed in the personal computer. An overview of electrical measurement for a suspension is shown in Fig.2.7. The magnetic stirrer is used to fluidize the porous particles in the solution. The expression for impedance on the Nyquist plot comprises of a real part on the horizontal axis and an imaginary part on the vertical axis. The resistivity is obtained by the x-intercept of the linear section of impedance and horizontal axis of the Nyquist plot. In the case of a monolith, a voltage-controlled AC impedance experiment was performed in a thermostated bath at 20°C by applying an alternative voltage axially across the monolithic column filled with an electrolyte solution and measuring the electrical impedance of the column. The frequency of the AC signal was varied from 1 kHz to 1 MHz using the ModuLab electrochemical system (Model 2101A, Solartron Analytical, UK). From the measured impedance, the column resistance was determined with the ModuLab software.

______Characterization techniques

Figure 2.7. Overview of electrical measurement for porous particle and Nyquist impedance plot

______Characterization of porous materials

3 CHARACTERIZATION OF POROUS MATERIALS

The basic parameters such as mean pore size and pore size distributions (PSD), specific surface area and porosity are used for the characterization of porous materials. Various methods such as nitrogen adsorption, mercury porosimetry and inverse size exclusion chromatography (ISEC) have been used to determine the porous properties of chromatographic stationary phases. The tortuosity of the silica’s (monolithic column or particles) is determined by electrical measurements. The chromatographic experiments were firstly performed on the three commercial columns and the solid materials, packed in these columns, were then recovered to be characterized by standard methods. 3.1. Determination of the porosities and mean pore size 3.1.1. Characterization by ISEC In ISEC method, toluene and a set of polystyrenes with different sizes, as shown in Table 2.2, are eluted (1 µL) through each of the chromatographic columns given in Table 2.1 at a flow rate of 0.5 ml.min-1 in THF (non-adsorbing conditions). As an example, Fig 3.1 shows the chromatograms and relationship of retention time (tr) as a function of molecular diameter (rm) of toluene and the series of polystyrenes P1 to P12 in the case of Chromolith column. a) b)

1.0 3,2 Toluene P1 0.8 P2 3 P3 P4 0.6 P5 2,8 P6 Chromolith column

P7 min , r 0.4 t A/A max P8 2,6 P9 P10 0.2 P11 2,4 P12 0.0 2,2 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 0 10 20 30 40 50 time (min) rm,nm Figure 3.1. a) Chromatograms and b) evolution of retention time to the molecular diameter of all the probes at the flow rate of 0.5ml.min-1 through Chromolith column

______Characterization of porous materials

As seen in the figure, the retention time decreases with an increase in the size of the molecules. The molecules with the size larger than the pore size of the column are excluded from the particle porosity and eluted quickly through the interparticle volume. There are the small peaks between 3 and 3.3 min, which correspond to the peak of THF (the so-called solvent peak) or of residual monomer.

The total porosity to the probe of rm radius, εt[rm], is easily extracted from an inverse size exclusion data by using Eq.2.4 with the correction of dispersion in the extra-column volume. The comparison of total porosity in three commercial columns is then shown in Fig.3.2.

1 Chromolith Si column 0,9 Poroshell 120 column 0,8 Si 100 column

] 0,7

[r t

ε 0,6 0,5

0,4 εe 0,3 0 20 40 60 80 100 (M ) 1/3,g/mol w Figure 3.2. Total porosity of polystyrenes as a function of the cubic root of the molecular weight of the polystyrenes for Chromolith, Poroshell and Si100 columns The molecular weight is expressed in a cubic root form to give a better visualization of the exclusion volume by the molecules excluded from the mesopores when their molecular weight increases. By extrapolation of the exclusion branch to a polymer mass of 0 (the black line in the figure for the case of Si100), the value of external porosity of the column is determined [11,14]. It is easily seen from Fig.3.2 that the total accessible porosity decreases with an increase of the molecular size. The total porosity of monolithic porous silica column is higher than that of Poroshell or Si100 columns, which is mainly due to the fact that the external porosity is higher for the monolithic column (0.71) than for the columns filled with spherical particles (around 0.4). AS expected, the external porosity of Poroshell column is comparable to the value of Si100 column. Using Eq.2.5, the values of accessible particle or skeleton porosities to each polymer are calculated and given in Fig.3.3. The results show that the intraparticle porosity decreases with an increase of the molecular size. There is a difference in the values of the accessible particle porosity

______Characterization of porous materials for each molecule through the different columns. This variation comes from the difference of porous structures in these columns. With Chromolith column, only toluene and the polystyrene series from P1 to P6 can enter the mesopores. In addition, P7 can also enter Poroshell and Si 100 column mesopores. The molecules with a size larger than pore size will be excluded from the mesopores. This is also pointed in Fig.3.3 where the values of accessible particle porosity of larger molecules reach a negligible amount. The particle porosity of Poroshell column seen in Fig.3.3, corresponding to the particle porosity of the smallest probe, toluene (εpz= 0.57) is lower than that of Si100 (εpz= 0.68) and Chromolith columns (εpz= 0.72).

0,8 0,7 0,6

0,5 Chromolith Si column m]

r 0,4 Poroshell 120 column

pz[ ε 0,3 Si 100 column 0,2 0,1 0 0 2 4 6 8 10 rm,nm

Figure 3.3. Particle porosity of polystyrenes as a function of the molecular size rm for Chromolith Si, Poroshell and Si100 columns

The comparisons of experimental and calculated distribution coefficient, kd for Si 100, Chromolith and Poroshell columns are plotted in Fig.3.4.a. The experimental distribution coefficient kd for each column was calculated from Eq.2.2 by applying the values of porosity acquired with ISEC method when eluting toluene and polystyrenes through the columns. By using the function ‘Solver’ in Microsoft Excel, the minimum difference between experimental and theoretical kd by using equations 2.2 and 2.8 was calculated and f(r) is obtained. The value for mean pore radius rp can be accessed by applying Eq.2.10. The kd values from model and experiment are in a good agreement for these silica columns. The porosities and the mean pore size obtained for the three columns are given in Table 3.1.

______Characterization of porous materials

a) b)

1 Si 100-Kd by Si 100- modelling 1,4 Cumulative 2 Si 100-Kd by ISEC volume 1,8 1,2 Chromolith- 0,8 1 Chromolith-Kd by - Cumulative 1,6

modelling volume 1 1 Poroshell- 1,4 -

Chromolith-Kd by .g 1 0,6 ISEC Cumulative 1,2 - 0,8 volume d Poroshell- Kd by k modelling Si 100-dV/dr 1 Poroshell-Kd by 0,6 0,4 ISEC 0,8 Chromolith- 0,6 0,4 dV/dr /ml.nm dV/dr 0,2 Poroshell- 0,4

Cumulative volume /ml.g volume Cumulative 0,2 dV/dr 0,2 0 0 0 0,1 1 10 100 0,1 1 10 100 1000 r , nm r , nm p p Figure 3.4. ISEC for Si 100, Chromolith Si and Poroshell columns -a) Experimental and calculated distribution coefficient and b) Pore size distribution

3.1.2. Characterization by gas adsorption The fully porous particle, monolith and core-shell samples retrieved from the chromatographic columns are analysed by gas adsorption with Nitrogen at 77K. Before measurement, these samples were degassed under vacuum at 1200C for at least 12 h. The surface area from the BET method is calculated by Eq.2.16. The results are listed in Table 3.1. The nitrogen adsorption isotherms of Si 100, Chromolith and Poroshell are plotted in Fig.3.5.a. These adsorption isotherms have curves corresponding to type IV as usually obtained for mesoporous materials. Due to the highest surface area, the adsorbed amount in Si 100 sample is greater than that in the other samples in all the P/Po region. The pore size distribution curves for these samples obtained from the NLDFT method are displayed in Fig.3.5.b. A significant volume of pores is seen in all samples with a mesopore size between 8 and 18nm. Chromolith and Poroshell samples have similar mean pore diameters around 16nm. One can see that Chromolith sample gives a narrowest pore size distribution when compared to the other solids. The particle porosity can be easily calculated by using Eq.2.17 with

______Characterization of porous materials pore volume obtained from this figure. The values of internal porosity, pore size and surface area of these samples are presented in Table 3.1.

a) b)

40 9

35 8

1 -

Si 100 Si 100 1

30 - 7 .g

Chromolith 3 25 6 Chromolith , , mmol.g Poroshell 20 5 Poroshell

uantity 4 15

dV/dln(d), cm dV/dln(d), 3 10 2

Adsorbed Adsorbed q 5 1 0 0 0 0,2 0,4 0,6 0,8 1 0 10 20 30 40 50 Relative pressure P/P0 dp, nm Figure 3.5. Nitrogen adsorption for Si 100, Chromolith and Poroshell –a) Adsorption-desorption isotherms and b) Pore size distribution (NLDFT)

3.1.3. Characterization by Mercury porosimetry The pore size distribution achieved from nitrogen adsorption measurements for Si 100, Chromolith and Poroshell samples displayed the pores in the range of 0.4 to 30nm. Hence, the mercury porosimetry method was performed to characterize the pores of large size. Samples were prepared at room temperature. Si100, Chromolith and Poroshell samples were intruded at low pressure until 344.8kPa, then the intrusion and extrusion in the pressure range of 0.14 to 407MPa have been carried out. The contact angle θ of 1400 was used to calculate the pore size distribution by Eq.2.22. The evolution of Hg porosimetry isotherm and pore size distribution with the pore diameter for Si 100, Chromolith and Poroshell are shown in Fig.3.6. Here the intruded volume/diameter curves are shown on a logarithm scale. As in gas adsorption, the lower values of the cumulative pore volume of Poroshell sample compared with that of Si 100 and Chromolith are also seen from Hg porosimetry in Fig.3.6a. The pore size distribution curves obtained from the intrusion curves are

______Characterization of porous materials given in Fig.3.6b. The mean pore sizes are given in Table 3.1. In the mesoporous domain, the PSD is sharp for core-shell and monolithic silica but it is very broad for the Si100 silica with two peaks at 6.5 and 17nm. The broad PSD for Si 100 sample was also observed in the work of Goworek et al [119]. The manufacturer indicates a pore diameter around 10 nm for the totally porous silica Si100. a) b)

Si 100-Intrusion 9 1 Si100

- 6 Si100-Extrusion 1 Chromolith-Intrusion - 8 Chromolith Chromolith-Extrusion 5 Poroshell Poroshell-Intrusion 7 Poroshell-Extrusion 6 4 5 3 4 3 2 Cumulative volume, mL.g volume, Cumulative 2

1 1 log differential intrusion,mL.g log differential 0 0 1000 10 0,1 0,001 1000 10 0,1 0,001 Pore diameter, µm Pore diameter, µm

The porosities of these materials obtained by applying Eqs.2.23, 2.24, 2.25 with the values of the cumulative pore volume shown in Fig.3.6.a are given in Table 3.1. The external porosities obtained for the spherical particles are around 0.37 as usually found for a dense packing of spheres. It shows that after the compaction step, the compacity in mercury cell and in column are similar. The external porosity of the silica monolith is at 0.63. The particle porosities are comparable for the totally porous silica and the silica monolith around 0.64, whereas slightly lower values are obtained for the core-shell particles (0.53).

3.1.4. Comparison of the porosities and mean pore size obtained by different methods

______Characterization of porous materials

Table 3.1 summarizes the values of porosity and pore size for Si 100, Chromolith Si and Poroshell samples by different methods. The surface area obtained from gas adsorption method is also given in this table.

Table 3.1. Surface area, porosity and pore size obtained by ISEC, Hg porosimetry and N2 adsorption for Si 100, Chromolith Si and Porosell 120 column

Materials ISEC measurements Mercury porosimetry N2 adsorption

a t e pz dp t e pz dp (nm) dmacro pz dp As 2 ) (nm) (μm) (nm) (m /g

Si 100 0.80 0.38 0.68 14.2 0.77 0.36 0.64 6.5 and 2.21 0.73 11.68 457 17

Chromolith 0.92 0.71 0.72 11.7 0.87 0.63 0.65 12.0 1.77 0.68 16.09 306 Si

Poroshell 0.67 0.42 0.57 13.3 0.62 0.37 0.53 11.8 1.05 0.68 16.09 145 120

a from desorption branch The porosity data achieved from the mercury intrusion measurement and nitrogen adsorption are comparable to the results calculated from the ISEC measurements. A small difference between these values can be explained by the limit in the range of pore size in which each method is applied. The porosity in Chromolith column is greater than the one in the other columns. The values of mean pore size for Si 100, Chromolith Si and Poroshell columns obtained by ISEC and shown in Table 3.1 are higher than that obtained by gas adsorption and mercury porosimetry. The pore size measured by ISEC methods depends on the size of the probe. The pore size of the largest molecule in this work is 42.9 nm. For ISEC, only a few polymers enter the mesopore. ISEC method can reach pore sizes smaller than pore size seen by Hg porosimetry. The smallest molecule (toluene) with the size of 0.44nm can explore almost all pores. Meanwhile, Hg porosimetry is well useful for the determination of large pores. With gas adsorption, it can determine the pore size in the range of 0.4nm to 30nm when using nitrogen as an adsorptive [96]. Finally the agreement between methods is good, considered the differences between approaches. It

______Characterization of porous materials enables a safe collection of material structure parameters that can then be used in transport models.

3.2. Determination of tortuosity by electrical measurements Fully porous particle and core-shell powders packed in the chromatographic columns were recovered to determine the tortuosity by conductivity measurements by impedance spectroscopy. The total porosity of the suspension is determined by following the suspension dilution method described in paragraph 2.4.1. The experimental total tortuosity is obtained by applying the relationship of total porosity and the ratio of conductivity of free electrolyte to the conductivity of suspension (Eq.2.29). The total tortuosity is also modeled by the modified Maxwell equation (Eq.2.30) in which the ratio of particle conductivity to electrolyte conductivity (σp/σo) is unknown. By using the function ‘Solver’ in Microsoft Excel to fit the experimental total tortuosity and the one from the modified Maxwell equation, the value of the ratio of particle conductivity to electrolyte conductivity is accessed. Hence, the internal tortuosity of the particle is calculated by multiplication of the internal porosity to a multiplicative inverse of this ratio (Eq.2.31). The obtained values of internal particle tortuosity τp are 1.6 and 1.7 for fully porous particles and core-shell, respectively. These values are reported in Table 3.2. The correlation of experimental and calculated total tortuosity with total porosity for Si 100 and Poroshell samples is presented in Fig.3.7. The total tortuosity decrease with an increase in total porosity. a) b)

Si 100 Maxwell equation Poroshell 120 Maxwell equation 1,007 1,02 1,006 1,016 1,005

1,004 1,012 τ τ 1,003 1,008 1,002 1,004 1,001 1 1 0,999 0,996 0,994 0,996 0,998 1 1,002 0,975 0,98 0,985 0,99 0,995 1 1,005 ε ε Figure 3.7. The relation of experimental total tortuosity (square point) and calculated total tortuosity by equation 2.30 (solid line) for-a) Si 100 and b) Poroshell 120

______Characterization of porous materials

In the case of core-shell particles, the tortuosity of the porous zone (pz) τpz (which is the porous shell around the solid core) is calculated as:

εpz. σo (3.1) τpz = σpz where σpz is the particle conductivity in the porous zone, it is determined by:

ρ3 (3.2) σ = σ (1 + ) pz p 2 Hence, the particle tortuosity of the porous zone is about 2.0 and this value is greater than the particle tortuosity of all core-shell (1.7). The topological parameter, p is then determined by applying the expression proposed by

Weissberg [36] as stated in Eq.1.6 (τp = 1 − plnεpz) in which the internal tortuosity of the shell is used in the case of Poroshell column. The obtained values of p are 1.7 and 2.7 for the fully porous particle and for the porous shell, respectively. In the case of core-shell, the value of p is 1.1 with 3 the internal porosity of the core-shell of 0.51 following the expression εp = εpz. (1 − ρ ). For monolithic sample, the conductivity measurements were performed directly with the chromatographic column. Chromolith column is saturated by a solution of sodium chloride at a concentration of 1M by the Intelligent HPLC Pump PU-980. The electrical measurement was made every 20 minutes until attaining stable value of impedance. A blank column with the same length of the monolith (100mm) and a zero-volume column were also used. An evolution of the obtained resistance values with the length of the bank column and zero-volume column is plotted. The retrieved intercept of this graph is used to correct the measured electrical resistance values for the monolithic and blank columns. Then the electrical resistance of electrolyte was obtained by applying Eq.2.32. Finally, one obtained the total tortuosity of Chromolith Si sample of 1.13 according to Eq.2.29. Applying this total tortuosity into Eqs.2.30, 2.31, one obtains the value of internal tortuosity for the monolith of 1.5. Hence, the value of p (calculated by Weissberg equation) is 1.4. The results for tortuosity and the p values for the three samples by electrical measurements are given in Table 3.2.

______Characterization of porous materials

Table 3.2. Tortuosities obtained by electrical measurements

Columns Total tortuosity τt Intraparticle tortuosity τp p

Si 100 - 1.6 1.7

Poroshell Core-shell - 1.7 1.1

120 Porous Shell 2.0 2.7

Chromolith Si 1.13 1.5 1.4

______Transport properties in multiscale materials in non-adsorbing conditions

4 TRANSPORT PROPERTIES IN MULTISCALE MATERIALS IN NON- ADSORBING CONDITIONS

This section presents the determination of the diffusion coefficients of polystyrenes through silica columns having different morphologies in dynamic and static conditions in non-adsorbing conditions. In non-adsorbing conditions, effective intraparticle diffusion depends on the structural parameters of the porous solids such as accessible porosity, pore size, friction and tortuosity. 4.1. Transport properties in the dynamic condition The diffusion in a porous material is studied in dynamic condition through band broadening by liquid chromatography. Liquid chromatography was performed with toluene and polystyrenes through three chromatographic columns in non-adsorbing condition as described in Section 1.3.2.1. The chromatograms recorded with toluene at 0.5 ml min-1 for Si 100, Chromolith and Poroshell column are shown in Fig.4.1. It is easily seen from the figure that toluene takes a longer time to transport in Si 100 column than in Poroshell and Chromolith column. This is explained by the greater volume of Si 100 column compared to other columns.

1,0

0,8 Si 100 Poroshell 120 Chromolith Si 0,6

A/Amax 0,4

0,2

0,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0 time (min)

Figure 4.1. Chromatograms for toluene at the flow rate of 0.5 mL.min-1 for Si 100, Chromolith and Poroshell columns The peaks obtained for all columns in non-adsorbing conditions have a Gaussian profile. The HETP curves of non-corrected data (following Eq.1.8) and corrected data from the contribution of the extra-column volume (using Eq.1.9) as a function of interstitial velocity for a small molecule (toluene), the intermediate molecules (P3 and P6) and a big molecule (P10) are presented in Fig 4.2 for Si 100 column.

______Transport properties in multiscale materials in non-adsorbing conditions a) b)

Toluene corrected data 230 70 P3 corrected data Toluene non-corrected data 210 60 P3 non-corrected data 190 50

m 170

m μ μ 40 150 30 130

HETP, HETP, 110 HETP, HETP, 20 90 10 70 0 50 0 0,2 0,4 0,6 0,8 0 0,2 0,4 0,6 0,8 Interstitial velocity, cm/s Interstitial velocity, cm/s c) d)

600 250 P6 corrected data P10 corrected data

500 P6 non-corrected data 200 P10 non-corrected data

m μ

400 μ 150

300 100

HETP, HETP, HETP, HETP,

200 50

100 0 0 0,2 0,4 0,6 0,8 Interstitial velocity, cm/s 0 0,2 0,4 0,6 0,8 Interstitial velocity, cm/s Figure 4.2. HETP curves with and without the contribution of extra volume through Si 100 column for a) a small molecule as toluene, b) and c) molecules with intermediate size (P3 and P6), and d) a molecule excluded from the pore of the particle like P10.

As seen obviously from this figure, there are only small differences between corrected and non- corrected HETP values at small flow rates. The effect of the extra column volume on the dispersion along the column is not considerable. However, when increasing the flow rate this effect is important and must be considered, especially in the case of small molecules and molecules excluded from the pores. It should be emphasized, at the low velocity values, that HETP curves for small size molecules decrease with increasing velocity: the transport of these molecules inside the column is mainly under the control of axial diffusion. At high velocities, the corrected HETP values are approximatively constant, which indicate a a process controlled by the eddy diffusion phenomenon: it also shows that the HETP correction due to extravolume is approximately linear with the variation of the mobile phase velocity. This is also the case for the

______Transport properties in multiscale materials in non-adsorbing conditions transport of the molecules excluded from the mesopores (Fg. 4.2 (d)). When the size of the molecule increases, the contribution of longitudinal diffusion decreases at low velocity as observed from the HETP curves. This is due to smaller effective diffusion in the whole column when the size of the polystyrene increases. The shape of HETP curves at high velocity for the intermediate size molecules is steeper than the one for small size molecules or molecules completely excluded from the mesopores. Hence, the transport of these intermediate size molecules is affected by the film mass transfer resistance and the effective intraparticle diffusion coefficient inside the mesopores. One can easily obtain the C term in Van Deemter equation from the slope of these curves HETP=f(u) at high flow rates. The corrected HETP curves of the three commercial columns are presented in Fig 4.3. a) b)

90 300 P4-Si100 80 250 P4-Poroshell 70 P4-Chromolith Toluene-Si100 200

60 m

m Toluene-Poroshell μ μ 50 Toluene-Chromolith 150 40

HETP, HETP, 100 HETP, HETP, 30 20 50 10 0 0 0 0,1 0,2 0,3 0,4 0,5 0 0,1 0,2 0,3 0,4 0,5 Interstitial velocity, cm/s Interstitial velocity, cm/s c) d)

600 70 P6-Si100 P10-Si100 500 P6-Poroshell 60 P10-Poroshell P6-Chromolith P10-Chromolith 50

400

m μ μ 40 300

30 HETP, HETP, HETP, HETP, 200 20

100 10

0 0 0 0,1 0,2 0,3 0,4 0,5 0 0,1 0,2 0,3 0,4 0,5 Interstitial velocity, cm/s Interstitial velocity, cm/s

Figure 4.3. Comparison of corrected HETP data through three columns: Si 100, core-shell and Chromolith with a) toluene, molecules with intermediate size as b) P4 and c) P6 and a big molecule (P10) 64

______Transport properties in multiscale materials in non-adsorbing conditions

In the case of toluene and large molecules as P10, the same phenomenon in Poroshell and Chromolith columns are seen as in Si 100 column. The HETP curves at flow rates above 0.1cm.s-1 has a flat slope. The transport in these columns for small molecules relies on longitudinal diffusion and eddy diffusion. The outstanding influence on the theoretical plate of the eddy diffusions is observed with the molecules excluded from the mesopores. This is in good accordance with the shape of HETP curves in the work of E.Olah et al [12] in which the transport properties of a small probe (Estradiol) through columns with different morphologies were investigated. In the case of intermediate size molecules, the contribution to band broadening of mass transfer between mobile and stationary phase through Poroshell and Chromolith columns (C term) are not as strong as in the fully porous particle column, Si 100. In these former cases, the effect of eddy diffusion is prevalent. Fig 4.3 shows that the values of corrected HETP in Si 100 column for the intermediate size and larger molecules are larger than those obtained with Poroshell and Chromolith columns. It means that Chromolith column, which has the highest porosity, is more efficient and has better resolution than the other columns. This was also evidenced by other authors [10,52]. Figure 4.4 illustrates the evolution of corrected HETP of the intermediate size molecules as a function of velocity through Si 100 column.

600 Si 100 column

500 P3

400 m

μ P5 300

P6 HETP, HETP, 200 P7

100 P8

0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Interstitial velocity, cm/s Figure 4.4. Corrected HETP of molecules having sizes smaller than pore size as a function of interstitial velocity for Si 100 column

______Transport properties in multiscale materials in non-adsorbing conditions

The values of corrected HETP for molecules with intermediate size are directly proportional to the interstitial velocity in a large range of velocity. When increasing the molecular size from P3 to P6, the slopes of the curves become greater. This may be explained by the fact that the probe size tends toward the pore size of the column. The slope goes to the maximum value at P6. With P7, the molecule can still enter the mesopores of Si 100 column, a value close to that of P6 but smaller is obtained. From P8 to P12, molecular samples cannot diffuse inside the mesopore of the column and the slope for these big molecules goes to zero: the transport is no longer under the control of mass transfer between mesopores and macropores. This was also seen in the study of Gritti and Guiochon [51]. From the slopes of the linear branches of corrected HETP data of molecules from P3 to P7 at high velocity, the values of mass transfer coefficient of Van Deemter equation can be obtained. Then by applying this value into Eq.1.16, the effective intraparticle diffusion coefficients of the probes are determined. The ratio of this coefficient to the molecular diffusion coefficient Dm for these intermediate molecules are presented in Fig.4.5. It is clearly seen that the effective intraparticle diffusion coefficient decreases with an increase in molecular size. This decrease may come from (i) the steric hindrance corresponding to a decrease of accessible particle porosity due to the exclusion of the molecule from the pore walls , (ii) friction of the molecule with the pore wall, and (iii) modification of the pore network seen by the probe. This is in a good agreement with other authors [27,120].

0,14

0,12

0,1

] 0,08

r [ 0,06

eff p

D 0,04

0,02

0 0 0,2 0,4 0,6 0,8 1

rm/rp Figure 4.5. Evolution of the ratio of the effective intraparticle diffusion coefficient to the molecular diffusion coefficient of the intermediate molecules through Si 100 column as a function of molecule/pore size ratio

______Transport properties in multiscale materials in non-adsorbing conditions

For Poroshell and Chromolith columns, the profiles of the branches of corrected HETP curves with the intermediate size molecules at high velocity are not linear. These curves are presented in Fig.4.6. The data scattering could be explained by the fact that the peaks for those columns are very sharp at high velocities. a) b)

70 35 Poroshell column Chromolith Column 60 P2 P2 30

50 P3 P3

m m

μ 25 P4 μ P4 40 P5

20 P5 HETP, HETP, 30 HETP, P6 P6 15 20 P7 P7

10 10 0 0,1 0,2 0,3 0,4 0,5 0 0,1 0,2 0,3 0,4 0,5 0,6 Interstitial velocity, cm/s Interstitial velocity, cm/s

Figure 4.6. Experimental HETPs of some polystyrene standards measured on a) Poroshell and b) Chromolith columns

Figure 4.7 shows the variation of peak shape and peak variance for all the chromatographic columns and the zero-volume column in the case of P5 . a) b)

0,005 Si 100 column 1,0 Poroshell column Chromolith column Zero-volume column Chromolith column 0,004 Zero-volume column 0,8

Poroshell column 2 Si 100 column 0,003

o 0,6

C/C 0,4 0,002

0,2 0,001

0,0 variance, min Peak -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0 time (min) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Interstitial velocity, cm/s Figure 4.7. Comparison of a) normalized peaks and b) evolution of peak variance versus velocity, for zero-volume column and studied columns.

______Transport properties in multiscale materials in non-adsorbing conditions

In the case of Fig. 4.7.a, the absorbance values for the chromatographic peak of different columns were divided by the maximum absorbance from each peak. The peak variance of each column is obtained from Eq.1.8. It can be observed for Poroshell and Chromolith columns that the peaks are very sharp and the peak variance is close to the variance of the zero-volume column. Hence, one obtains a strong error on the HETP curves at high velocities for Poroshell and Chromolith columns. The peak parking method presented in the next part will be used to evaluate the effective intraparticle diffusion coefficients in the columns.

4.2. Diffusion in static conditions by peak parking (PP) method 4.2.1. Effective diffusion coefficient and apparent total tortuosity determined by the PP method In this study, we compare the RTW and EMT models used to calculate the effective diffusion coefficient of the different probes in the 3 columns as a function of diffusion coefficients in particles using either a constant or a variable value of particle tortuosity. In the case of variable tortuosity,the Weissberg equation is used. The effective diffusion coefficient is obtained obtained from peak parking method. From the model, which is in the best accordance with experimental data, the effective intraparticle diffusion coefficient is calculated. The experiments were carried out with the molecules that can enter the mesopore, i.e toluene and polystyrenes from P1 to P7, and one molecule totally excluded from mesopores, i.e P8. The PP method was used to measure the effective diffusion coefficient in the column. In the PP experiments, 1 µL of a dilute sample solution was injected at 0.5 mL min-1, respectively at a constant interstitial velocity of 0.18 cm s-1 (for Si 100 column), 0.07 cm s-1 (for Chromolith column) and 0.13 cm s-1 (for Poroshell column). The temperature for all experiments is fixed at 250C. During the elution of samples through columns, the flow rate was stopped when the probe was in the middle of the column during a given time called parking time and noted tp. The samples diffuse freely along the columns during the parking time. After the parking time, the pump is restarted and the chromatogram is recorded by the detector. The peak shapes obtained from the peak parking experiments for toluene and an intermediate size molecule (polymer P5) for the different values of parking time tp are shown in Fig.4.8. The solid lines are the reference peak without time. The y axes correspond to the ratio between the

______Transport properties in multiscale materials in non-adsorbing conditions absorbance of the peak with parking time and the absorbance at the maximum of the peak without parking time. It is obviously seen from Fig 4.8 that the peak width increases and the peak height decreases when the parking time increases. The band broadening with increasing parking time for toluene tends to increase more than the one obtained with P5 which is due to faster diffusion through the porous media for smaller molecules. From the chromatograms of P5 with the three columns, it is observed that the totally porous particle packed column (Si100) has the smallest variation in the increase of band broadening and decrease in peak height with an increase in parking time due to slower diffusion in this material. The same results were obtained with the other studied molecules. This may be due to the structural characteristics of Si 100 different from the other columns.

______Transport properties in multiscale materials in non-adsorbing conditions

1,0 1,0 tp=0h tp=0h 0,8 0,8 tp=19h tp=2h tp=43h tp=5.4h tp=68,1h 0 0,6

tp=14.8h 0,6 0

C/C

0,4 C/C 0,4 0,2

0,2 0,0

-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,0 time (min) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

time (min) a) d)

1,0 1,0

0,8 0,8 tp = 0h tp=0h tp = 3.7h tp=2.8h

0,6 0,6 tp =7.5h 0 o tp=7.7h tp = 25.9h

C/C tp=16.5h C/C 0,4 0,4

0,2 0,2

0,0 0,0 -0,8 -0,4 0,0 0,4 0,8 -0,4 -0,2 0,0 0,2 0,4 time (min) time (min) b) e)

1,0 1,0

0,8 0,8 tp=0h tp=0h tp=4 h tp=1h tp=6 h 0,6 tp=2.4h 0,6 0 tp=18.9 h

C/C 0

0,4 0,4 C/C

0,2 0,2

0,0 0,0 -0,4 -0,2 0,0 0,2 0,4 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 time (min) time (min) c) f) Figure 4.8. Representation of the obtained peak shape by the peak parking method in THF for a low molecular weight molecule (toluene) in a) Si 100, b) Poroshell, c) Chromolith columns and an intermediate size molecule (polymer P5) with d) Si 100, e) Poroshell and f) Chromolith columns

______Transport properties in multiscale materials in non-adsorbing conditions

Three series of peak variances acquired from increasing parking time for molecules being able to enter the mesopore of Si 100, Poroshell and Chromolith columns are shown in Fig.4.9 a) Si 100 column Toluene 300 2 P1 P2 P3 200 P4 P5 P6

100 P7 Peak variance, variance, s Peak

0 0 70000 140000 210000 280000 350000 parking time, s b) Poroshell column 400 Toluene 350 P1 P2

2 300 P3 250 P4 200 P5 150 P6 P7 Peak variance, s variance, Peak 100 P8 50 0 0 30000 60000 90000 120000 150000 180000 parking time, s c) 160 Chromolith Si Toluene P1 140

P2 2 120 P3 100 P4 80 P5 P6 60 P7

40 P8 Peak variance, s variance, Peak 20 0 0 15000 30000 45000 60000 75000 parking time, s

______Transport properties in multiscale materials in non-adsorbing conditions

Figure 4.9. Plots of the peak variances of peak parking experiments as a function of parking time for a) Si 100, b) Poroshell and c) Chromolith columns It should be noticed that all the lines are straight. The linear behavior of the plots shows the

2 eff applicability of Eq.1.20 (Δσz = 2D tp ). Interestingly, when increasing the size of a probe equivalent to a decrease in molecular diffusion, the slope of the plot decreases. Similar results have been obtained by other authors [28,50,65,70]. The relative error on the slope is about 5%, the value estimated by making three times the same experiments in the same conditions. The errors on the interstitial velocity and on the porosities are lower than 1%. The relative error made on the determination of Dm is about 5% for toluene and the polymers P01-P04 (TDA measurements) and less than 0.5% for Dm measured by DLS eff (polymers P05-P12). The relative error made on D /Dm is +/- 10% for toluene and P01-P04 and

+/- 5% for P05-P12 essentially due to the relative errors made on Dm and on the slope. The mean values of the effective diffusion coefficient, Deff, are directly determined for all

2 2 eff 1 ∆σt u 2 columns by using Eq.1.22 (D = 2 ) in which the ratio ∆σt /tp values are the slopes 2 tp (1+k1[rm]) of the curves retrieved from Fig.4.9 and the zone retention factor values for each sample are

(1−εe) calculated from Eq.1.18 in non-adsorbing conditions (k1[rm] = εpz[rm]). The external εe porosity results from ISEC method (see Table 3.1) for each column are used to calculate the zone retention factor. The values of internal porosity for each sample through silica’s columns are taken from ISEC method as shown in Table 4.1 (see also Fig.3.3). Table 4.1. Values of internal porosity used in the calculation of Deff by PP method Sample Si 100 column Poroshell Column Chromolith Column Toluene 0.68 0.57 0.72 P1 0.66 0.54 0.68 P2 0.60 0.50 0.60 P3 0.55 0.45 0.59 P4 0.44 0.35 0.40 P5 0.25 0.21 0.21 P6 0.13 0.09 0.07 P7 0.03 0.01 -0.03 P8 -0.01 -0.05 -0.13

eff The ratio D /Dm as a function of rm/rp with the error bars in Si 100, Poroshell and Chromolith eff columns is presented in Fig.4.10. The ratio D /Dm decreases with an increase in the molecular eff size until rm/rp around 0.3 due to slower diffusion in the particles then the D /Dm increases until a 72

______Transport properties in multiscale materials in non-adsorbing conditions constant value corresponding to diffusion mainly in macropores. It can be observed that this value is slightly the same for the columns made of spherical particles which have almost the same value of the interparticle porosity as shown in Table.3.1. The effective diffusion in Chromolith column is faster than that in the columns packed with spherical particles which could be explained by a higher external porosity leading to higher diffusion in the macroporosity [10]. In fact, the external porosity of Chromolith column is around 0.71 against around 0.4 for Si 100 and Poroshell columns.

1,2

m 0,8 /D

eff 0,6 D

0,4 Chromolith Si column Poroshell 120 column 0,2 Si 100 column 0 0 0,3 0,6 0,9 1,2 1,5 1,8

rm/rp

Figure 4.10. Plots of the ratio of the experimental effective diffusion coefficient and the molecular eff diffusion coefficient (D /Dm) versus the ratio of molecular size and mesopore size (rm/rp) through Si 100, Poroshell and Chromolith columns.

The apparent total tortuosities of the silicas evaluated as the ratio of molecular bulk diffusion coefficients and experimental effective diffusion are plotted in Fig.4.11. The apparent tortuosity is smaller for the monolith than for the spherical particles. The value of apparent total tortuosity for Chromolith column measured by PP method with toluene of 1.14 is close to that obtained by electrical measurements (1.13, see table 3.2). Meanwhile, the total tortuosity for Si 100 and Poroshell columns is around 1.25. The apparent tortuosity which is a combination of the external and intraparticle tortuosities increases as the molecular size increases until rm/rp=0.3 and then decreases as the molecular size increases. For rm/rp larger than one the external tortuosity or the tortuosity in the macropore is obtained. The external tortuosity, i.e. tortuosity of the macropore is 1.02 for the monolith and around 1.3 for the columns packed with spherical particles (core-shell

______Transport properties in multiscale materials in non-adsorbing conditions and totally porous silicas). Using the Weissberg equation (Eq.1.6) with the external tortuosity and external porosity, the p values are t 0.3 and 0.4 for monolith and columns packed with porous spherical particles, respectively. The results are given in Table 4.2. Johnson et al (2017) [121] determined tortuosity in the macropores of columns packed with ceramic and cellulose by X-ray computed tomography systems and values around 1.40 and 1.79 were obtained, respectively. Those values are close to the values obtained in this study by the PP method. Hormann et al (2016) [122] determined the tortuosity of the macropores in silica monoliths by using medial axis analysis and a geodesic distance propagation method after reconstruction of the macropores by confocal laser scanning microscopy. The pore connectivity in the medial axis analysis is determined after compartmentalization of the open macropore space. The geodesic distance is the shortest path between the two positions in the space. The global geometrical tortuosity found by [122] in the macropore is 1.09 for the silica monoliths, close to the value of our Chromolith column (1.02). The low external tortuosity value obtained for monolith reflects an open macropore space that provides a weak obstruction to transport.

2,4 Chromolith Si column Poroshell 120 column 2 Si 100 column

1,6

1,2

0,8 Apparent tortuosity total Apparent 0,4 0 0,3 0,6 0,9 1,2 1,5 1,8

rm/rp

Figure 4.11. Plots of the apparent total tortuosity of Si 100, Poroshell and Chromolith solids

obtained by PP method versus the ratio of molecular size and mesopore size (rm/rp)

4.2.2. Determination of effective intraparticle diffusion coefficient of polystyrenes in non- adsorbing conditions To have a better understanding of the effect of probe size on the diffusion through the columns, the effective intraparticle diffusion coefficient for each probe with size rm through the mesopores

______Transport properties in multiscale materials in non-adsorbing conditions

eff in the columns, Dp [rm], must be identified. It is shown in Section 4.1 that there is a decrease in the effective intraparticle diffusion coefficient of intermediate molecules through the totally porous column when the molecular size tends to the mesopore size. This decrease is caused by steric hindrance, friction, and tortuosity evolution. Suzuki [79] presented that the effective diffusion coefficient in the particle is considered to be proportional to the diffusion in the bulk phase with a proportionality constant called the diffusibility. The diffusibility depends on the structure or configurations of the pore network. For a probe having an infinitely small size when compared to the pore size, the friction can be neglected. In this case, the diffusibility ordinarily comprises two parts: the contribution of internal porosity, εpz, and that of internal tortuosity, τp,. This leads to the equation as [79]:

eff Dp εpz (4.1) = Dm τp A review of the impact of the tortuosity on diffusion in porous media was performed by Shen et al [33]. The tortuosity can be determined by theoretical or empirical methods that generally established a relationship between tortuosity and porosity. In addition to the independence on a porosity parameter like in the theoretical method, the total tortuosity from the empirical method can be controlled by adjustable parameters. In the case of particle tortuosity, when the solute is small as compared to the pore size, the values of τp in Eq.4.1 used by other authors are often fixed to a constant value in a range of 1.4 and 2 [4–8]. Generally, the probes are not infinitely small as compared to pore size so Eq.4.1 has to be modified to take into account the accessible particle porosity εpz[rm], the particle tortuosity, τp[rm] and also friction to calculate the effective eff intraparticle diffusion coefficient of a probe of rm radius Dp [rm]. Due to the contribution of friction between molecules and pore wall, the value of the reference bulk diffusion coefficient, Dm-

, is decreased by a friction factor that depends on the molecule radius rm, (kf[rm]) as follows: (4.2) Dm[rm] = kf [rm]Dm

Then, the effective particle diffusion coefficient of a probe of radius rm is derived from the following relationship [27,56,79,81,84]:

[r ] k [r ]D (4.3) eff εpz m f m m Dpz [rm] = τp[rm] Renkin [120] proposed that the pore hindrance factor corresponds to the diffusion of a sphere in a cylindrical pore. The theory consists of the exclusion effect and the wall drag effect. The exclusion effect is developed from the solute molecule being excluded from the region adjacent to the pore wall. The wall drag effect can be simulated using centerline hydrodynamic [27]. When 75

______Transport properties in multiscale materials in non-adsorbing conditions the molecule is moving in a pore, the frictional resistance within the pores is determined in the present study by the Renkin equation:

3 5 (4.4) kf [rm] = 1 − 2.104 λ + 2.09 λ − 0.956 λ where λ is the ratio of the molecule radius rm to the pore radius rp. The particle tortuosity is calculated by the Weissberg equation (Eq.1.6) [36]. This equation was theoretically established for spheres [36] and experimentally verified for spheres and some other shapes [16,34]. The same equation will be used to calculate the variation of tortuosity with probe size rm in a pore network as a function of accessible porosity of the probe with size rm. This is indeed shown by random walk simulations of the diffusion process through a porous bed of spheres [123]. When the size of the molecule increases, the accessible porosity decreases but the topological properties of the pores are assumed to be not modified. Hence, the particle tortuosity of the accessible porosity seen by a probe of size rm may be reasonably given by Eq.1.7 where

εpz[rm] is measured by ISEC or modeled by the following equation with an assumption that the molecules are under the control of the exclusion effect [27]:

2 εpz[rm] = εpz(1 − λ) (4.5)

As mentioned in Section 1.3.2.2, EMT and RTW models are used to determine the effective intraparticle diffusion coefficient from the effective diffusion coefficient obtained from the PP eff method. Fig.4.12 shows the comparison of D /Dm obtained from the PP experiments and calculated from the RTW and EMT models (see Table 1.1). The external tortuosity in RTW model eff is used as a multiplicative inverse of the ratio D /Dm for the molecules excluded from the mesopores. Ω is calculated with Eq.4.3 by using either a constant value for the internal tortuosity (1.4 or 2) (Figs.4.12.a and 4.12.b) or by using a variable particle tortuosity (Fig.4.12.c). The variable particle tortuosity is given by Eq.1.7 which is derived from the Weissberg equation. The best fit is obtained with the Maxwell model by calculating the particle tortuosity with the Weissberg equation. By using the Solver function in Microsoft Excel, the topological parameter p eff obtained for Si100 column to fit the experimental D /Dm values from the Maxwell model is about eff 1.5. The Landauer model completely fails to account for the ratio of D /Dm of the probes in non- adsorbing conditions. This is due to the fact that the Landauer model totally ignores the spatial arrangement of the different homogeneous phases in the chromatographic columns [124].

______Transport properties in multiscale materials in non-adsorbing conditions a) 1 Model with constant τp=1.4

0,8 Si 100 Maxwell model 0,6 RTW model

Torquato model /Dm

eff Landauer model

D 0,4

0,2

0 0 0,5 1 1,5 rm/rp b) 1 Model with constant τp=2 0,8 Si 100 0,6 Maxwell model

/Dm RTW model eff

D 0,4 Torquato model Landauer model 0,2

0 0 0,5 1 1,5 rm/rp c) 1 Model with a variable τp

0,8

0,6 Si 100

Maxwell model /Dm

eff RTW model

D 0,4 Torquato model Landauer model 0,2

0 0 0,5 1 1,5 rm/rp eff eff Figure 4.12. D /Dm acquired experimentally and D /Dm estimated by RTW or EMT models as a function of molecule/pore size ratio for Si 100 column a) model with τp =1.4, b) model with τp =2

and c) model with a variable τp obtained by Weissberg equation with p=1.5.

______Transport properties in multiscale materials in non-adsorbing conditions

eff For Poroshell column, the agreement between the values of D /Dm provided by the Maxwell model and the PP experiments can be reached with the same p value of 1.5. The plots of the eff D /Dm values from the Maxwell model (Eqs.1.30 and 1.32) and also from the PP experiments as a function of the ratio of rm/rp for silica’s columns are shown in Fig.4.13. The fit is relatively correct for the spherical particles but for the monolithic column, the Maxwell model with the p eff value of 1.5 fails to fit the experimental values for rm/rp larger than 0.7 The values of the D /Dm ratio obtained experimentally for this column are higher than the one given by the Maxwell model.

1,2

0,8

/Dm eff

D 0,6 Si100 Poroshell 0,4 Chromolith Si100-Maxwell model 0,2 Poroshell-Maxwell model Chromolith-Maxwell model 0 0 0,3 0,6 0,9 1,2 1,5 1,8 r /r m p eff eff Figure 4.13. Comparison of the D /Dm achieved experimentally and the D /Dm assessed by the Maxwell model versus the ratio of molecular size and mesopore size of for Si 100, Poroshell and Chromolith columns By using the Maxwell equation given either in Eq.1.30 for Si 100 and Poroshell columns or in eff Eq.1.32 for Chromolith column with the values of D /Dm obtained from the PP experiments, the eff Dpz [rm]/Dm values through the mesopores in particle for Si 100 column, in skeleton with Chromolith column and inside the porous shell of Poroshell column with toluene and a series of polystyrenes from P1 to P7 can be calculated. It is easily seen from Fig.4.14 that the effective intraparticle diffusion coefficient decreases when increasing the size of the molecule. When the eff molecular size approaches the mesoporous size, the value of the ratio Dpz [rm]/Dm for the columns packed with spherical particles tends to zero whereas a value around 0.2 is obtained for the monolithic column. Those values are not shown because the effective diffusion coefficient evaluated from the Maxwell model does not fit the experimental results for rm/rp larger than 0.7, which has been already shown in Fig.4.13. In the case of the smallest molecules, the value of

______Transport properties in multiscale materials in non-adsorbing conditions

eff Dpz [rm]/Dm around 0.46 is obtained both for the monolithic column and the totally porous column. Lowest values are obtained for the core-shell column. For intermediate sized molecules, the intraparticle diffusion for the monolithic column is higher when compared to the value through the columns filled with spherical particles.

eff Comparisons of the values of Dpz [rm]/Dm determined by analysing the experimental height equivalent to a theoretical plate (HETP) data using liquid chromatography under dynamic conditions and the one derived either from the Maxwell model or by Eq.4.3 with a constant or variable value of τp through silica’s columns are shown in Fig.4.14. In the case of Si100 column, the dynamic and static methods give similar results and are in good accordance with the values eff modeled by Eq.4.3. The Dpz [rm]/Dm values from the model at a constant tortuosity of 1.4 for Si 100 column are significantly different from the experimental results. A better fit between the eff values of the ratio Dpz [rm]/Dm from Maxwell model and the one from Eq.4.3 can be achieved with a p value of 1.5 in the Weissberg equation. A good fit is also seen in the case of Poroshell column with the same value of topological parameter p of 1.5.

0,6 Si100 in dynamic condition Si100 in static condition Poroshell 120 in static condition 0,5 Chromolith Si in static condition Si100 with τp=1.4 0,4 Si100 with τp variable, p=1.5 Poroshell 120 with τp variable, p=1.5

m Chromolith Si with τp variable, p=1.5

/ 0,3 ] Si 100 by Tallarek

r [ Poroshell by Tallarek

eff pz

0,2 Chromolith Si by Tallarek

0,1

0 0 0,2 0,4 0,6 0,8 1 rm/rp

Figure 4.14. Plots of the ratio of the effective particle diffusion to the molecular diffusion versus the molecule/pore size ratio

______Transport properties in multiscale materials in non-adsorbing conditions

Tallarek et al [86] carried out the simulation of the diffusion of tracers with different sizes for the case of mesoporous silica monoliths reconstructed by scanning transmission electron eff microscopy tomography of realsamples. The ratio of Dpz [rm]/Dm was called the global diffusive hindrance factor and determined from the local hindrance factor and accessible porosity as a function of the molecule-to-pore radius ratio (λ) as follows:

eff Dpz [rm] ε (4.6) = o (1 − 3.416 λ + 3.338 λ2 − 5.433 λ3 + 24.063 λ4 Dm τ0 − 43.298 λ5 + 33.022 λ6 − 9.282 λ7) in which the value of εo is the particle porosity and τ0 is the particle tortuosity. Hence, the values eff of Dpz [rm]/Dm for Si100, Poroshell and Chromolith columns obtained by the model of Tallarek are also plotted in Fig 4.14 by using the value of particle porosity determined with toluene for εo and the particle tortuosity τ0 from the electrical measurements. The values from Tallarek model for Poroshell and Chromolith column are in good accordance with the one retrieved by the PP method. Meanwhile, Tallarek model fails to fit the value from the PP method with Si 100 column. The diffusion in the mesoporous zone has been determined by Hlushkou et al [125] by reconstructing a macroporous-mesoporous silica monolith. The empty space of the amorphous mesoporous silica from the monolithic skeleton was physically reconstructed by scanning transmission electron microscopy tomography. The effective intraparticle diffusion coefficient in the mesopore space as a function of the size of the tracer diameter was determined by numerical simulation using a random-walk particle-tracking method. They also found a decrease of the effective diffusion by increasing the tracer size and the results are comparable to the results eff obtained here by the PP method. For the smallest tracer, the value of Dpz [rm]/Dm around 0.6 is obtained, this value is close to the values obtained with the smallest probe, i.e. toluene in this eff study. They also found a decrease of Dpz [rm]/Dm to values close to zero for rm/rp close to 0.3. Maier and Schure [13] determined the effective intraparticle diffusion coefficient in wide-pore superficially porous particles using pore-scale simulation. They found higher values for the eff intraparticle diffusion coefficient around 0.6 for Dpz [rm]/Dm when the size of the molecule is negligible as compared to the pore size which could be explained by the fact that their materials have a pore radius around 50 nm and is probably less tortuous than the porous shell silica used in this study. The value of the apparent particle tortuosity can be derived from Eq.4.3 as follows:

εpz[rm] kf[rm] (4.7) τ [r ] = p m Ω 80

______Transport properties in multiscale materials in non-adsorbing conditions

eff in which the value of Ω = Dpz [rm]/Dm derived from Maxwell equation by applying the PP experimental results. Figure 4.15 shows the relationship between the particle tortuosity obtained by Eq.4.7 and the particle tortuosity retrieved from Weissberg model (Eq.1.7) by using the p value obtained by the PP method (p=1.5) versus the internal porosity through silica’s columns. For rm/rp>0.3 the intraparticle tortuosity for spherical particles becomes very high (data not presented) because the particle diffusion becomes close to zero. The apparent total tortuosity is thus dominated by the external tortuosity. This could explain the minimum value observed in the eff D /Dm curves around rm/rp=0.3 (Fig.4.10). For the smallest molecule, i.e. toluene, the particle tortuosity from the PP experiments is around 1.5±0.1 for Si 100 and Chromolith columns. Poroshell column gives a higher value of 1.9. The topological parameter p will be retrieved from these values of intraparticle tortuosity through Weissberg equation (Eq.1.6). The values of particle porosity and topological parameter p for silica’s column are given in Table 4.2. Generally, the tortuosity is taken as a constant value whatever the size of the molecule. From Fig.4.15 we can see that the apparent particle tortuosity depends on the particle porosity. When the particle porosity decreases the apparent particle tortuosity increases. Such behavior has also been observed by Richard and Striegel [126]. They used narrow dispersity polymer standards through chromatographic columns with different particle sizes in the solvent THF. The particle obstruction factor was obtained by performing experiments at several flow rates, measuring the change in peak variance as a function of flow rate. The particle tortuosity which is the inverse of the particle obstruction factor was seen to vary with analyte molar mass. It decreases to a minimum value at the molar mass around 30000 g/mol. With the molar mass larger than 30000 g/mol, the internal tortuosity factor increases with increasing molar mass, i.e with an increase in molecular size.

The relationship of the internal tortuosity (τp) to the internal porosity (εpz) was also expressed (1−m) by the simple form of Archie law [118,127,128] as follows 휏p = εpz whereas m is the cementation exponent. From the values of τp obtained by electrical measurement (1.6 for Si 100,

2.0 for Poroshell and 1.5 for Chromolith samples) and the εpz value by gas adsorption (0.73 for Si 100, 0.68 for both Poroshell and Chromolith columns ), the values of cementation exponent (m) are calculated at 2.2, 2.8 and 2.1 for Si 100, Poroshell and Chromolith columns, respectively. Hence, one obtains the value of τp[rm] for the different molecules through the silica columns with these values of m following the Archie law. These τp[rm] values from Archie law and the ones from the Weissberg model in which the p value are taken at 1.5 for three silica columns are compared with the values of τp[rm] from peak parking method (Eq.4.7) as shown in Fig 4.15. It could be seen that

Archie law only fails in expressing the τp[rm] values for Poroshell column.

______Transport properties in multiscale materials in non-adsorbing conditions

5 Si 100- Eq.4.7 4,5 Poroshell-Eq.4.7

4 Chromolith-Eq.4.7 Weissberg model 3,5 Archie model-Si 100 3 Archie model-Poroshell

Particle tortuosity Particle 2,5 Archie model-Chromolith

1,5

1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 Particle porosity

Figure 4.15. Particle tortuosity obtained from the PP method and by the Weissberg equation as a function of particle porosity

A summary of the results of the tortuosity parameters acquired by the PP measurements and the tortuosity data from the electrical measurements is presented in Table 4.2. As observed in Table 4.2, the values of the particle tortuosity obtained by the conductivity measurements and PP method in the core-shell particles are higher than those in the fully porous particles. The prevalent cause of this phenomenon is pointed out by the structural characteristic of the core shell with the existence of a solid core increasing the degree of obstruction for axial diffusion [73]. It means that the transport of a probe through the porous network in Poroshell is more sinuous than in other columns. Another explanation comes from the lower porosities in Poroshell, resulting in a higher tortuosity [129]. The same obtained values of the external tortuosity for Si 100 column and for Poroshell 120 column is due to the same structure of spheres in columns. There is a good agreement between the total and internal tortuosities from chromatography and from conductivity measurement for all the columns studied. The p values for the columns packed with spherical particles are different between fully porous and core shell samples. This may be due to different pore structure. The different value of p for the particle tortuosity by PP method from

______Transport properties in multiscale materials in non-adsorbing conditions the conductivity measurement with totally porous particles could be due to the two types of pore size distribution system in this sample.

Table 4.2. Estimation of the internal, external and total tortuosities as well as the topological factor (p) by the PP and electrical measurements for Si 100, Poroshell and Chromolith columns

Columns Tortuosity Tortuosity obtained by PP method conductivity

τt τp τt (toluene) External tortuosity Particle tortuosity (toluene)

τp p τext p τp p

Totally porous 1.6 1.7 1.23 1.42 0.4 1.4 0.9 particles

Porous shell - 2.0 2.7 1.29 1.29 0.3 1.9 1.5 particles

Monolith 1.13 1.5 1.4 1.14 1.02 - 1.5 1.4

4.3. Study of eddy diffusion Eddy diffusion (A term) is under the control of three main contributions, i.e. the transchannel (TS), the short-range interchannel (IC), and the transcolumn (TC) eddy dispersion terms [48]. The eddy diffusion term is variable for different molecules with different sizes [88,130,131]. In this section, one presents the value of A term obtained by fitting the HETP data with different plots and the one retrieved by the subtraction of the values of the longitudinal diffusion and mass transfer terms (B, C terms) from the experimental data of HETP. The values of A term for different molecules are also considered. 4.3.1. Van Deemter, Knox and Giddings plots As mentioned previously in part 1.3.3, the value of HETP can be fitted by the expressions of Van Deemter, Knox and Giddings equations. For the sake of comparison, the reduced value of HETP and the reduced velocity will be applied. Under the concept of reduced HETP and velocity, the models depend only on the structural parameters λi and ωi [48]. The reduced value of HETP is obtained by dividing the HETP data by the particle diameter. The reduced velocity is determined

______Transport properties in multiscale materials in non-adsorbing conditions by Eq.1.38 in the relationship with interstitial velocity, particle size and molecular diffusion of the molecules.

When using the Giddings equation (Eq.1.40), the values of the empirical parameters λi, ωi for transchannel and short-range interchannel eddy diffusions are derived from the simulation of the computer-generated packings for each type of a chromatographic column at the different values of external porosity [48,73,88,132]. Tallarek et al [88,132] used a computational approach to reconstruct a packed bed based on the experimental particle size distribution of core-shell and fully porous particles. The fluid flow in the interstitial empty space was simulated by the lattice- Boltzmann method. The convective-diffusive mass transport was simulated using a random-walk particle-tracking technique with the tracers distributed throughout the packing empty space. The eddy dispersion term was predicted. The parameters for the transchannel (λ1, ω1) and the short- range interchannel contribution (λ2, ω2) were found depending on the packing density of the column bed. For the column packed with spherical particles having an external porosity of 0.40, the values of λi, ωi are (0.45; 0.0045) and (0.25; 0.13) for transchannel and short-range interchannel, respectively. In the case of the porous particle column having the external porosity of 0.38 as Si 100 used in this study, the values of these parameters are (0,45;0,0041) for the transchannel contribution and (0.23; 0.13) for the short-range interchannel eddy diffusion. These

(λi, ωi) values for transchannel and short-range interchannel were also used in other studies

[73,130,133,134]. Hence, the transchannel eddy diffusion, hTS, for column filled with the fully porous particles is expressed as [132,135]: 1 (4.8) h = TS 1/0.9 + 1/(0.0041ν) and for the core shell column [73,88,133]: 1 (4.9) h = TS 1/0.9 + 1/(0.0045ν)

The short-range interchannel eddy term, hIC, for columns packed with fully porous particle is obtained by [132,135]: 1 (4.10) h = IC 1/0.46 + 1/(0.13ν) and for the core shell porous columns [73,88,132]: 1 (4.11) h = IC 1/0.5 + 1/(0.13ν) In the case of monolith column, eddy diffusion was studied by morphology reconstruction and simulation of convective-diffusive mass transport by a random-walk particle tracking technique on a high-performance computing platform [74]. The authors considered the reconstructed silica

______Transport properties in multiscale materials in non-adsorbing conditions monolith with the external porosity of 0.704. This value is very close to the macroporosity of Chromolith (0.71) used in this study. The lateral equilibration in the macropore space of the monolith, i.e on the transchannel scale results not only from pure diffusion but also from the convective motion due to the complex morphology of pores. This phenomenon causes an increase in the velocity inequality on the scale of a single pore, resulting in a smaller contribution of the channel to the total eddy diffusion. Hence, the transchannel eddy diffusion,characterized by a relatively high reduced transition velocity, is reduced to a simple velocity proportional term (ωiν) as follows [74]: (4.12) hTS = 0.133ν The contribution of the short-range inter-throughpore were written as [63,74,131]: 1.641ν (4.13) h = IC 1 + 1.154ν The transcolumn eddy diffusion term can be obtained by the substruction of the transchannel, short-range interchannel from the total eddy diffusion. The parameters for the transcolumn (λ3, ω3) will be presented later. The reduced HETP data in dynamic conditions will be fitted by Giddings equation, in which the contributions of transchannel, short-range interchannel eddy diffusion calculated from Eq.4.8 to Eq.4.11 will be applied. Fig.4.16 shows the comparison of reduced experimental HETP and the one by Van Deemter, Knox and Giddings equations (Eq.1.10, Eq.1.38 and Eq.1.40) through Si 100, Poroshell and Chromolith columns for some small molecules and molecules with intermediate size. It can be seen from the Fig.4.16 that the reduced HETP proposed by Knox equation almost fails to fit the reduced experimental HETP. It could be due to the fact that the Knox equation does not concern the different contributions of overall eddy diffusion as in the Giddings equation. A fail in fitting the reduced experimental HETP to the Knox equation was also seen in the study of Kirkup et al [136]. They calculated the height equivalent to a theoretical plate for propylparaben through a packed column. The main source of the difference between HETP obtained from the Knox equation and from the experiments was pointed out that the curvature in the HETP versus velocity u cannot be qualitatively or quantitatively represented by Knox equation. Meanwhile, the reduced HETP predicted by the Van Deemter and Giddings equations fits satisfactorily the HETP curves for all the studied polymers.

______Transport properties in multiscale materials in non-adsorbing conditions

a) b)

40 Toluene 30

35 P2 25 30 Si100 25 Poroshell Si100 Chromolith 20 Poroshell Si 100 (Giddings) Chromolith 20 Poroshell (Giddings) Si 100 (Giddings) Chromolith (Giddings) Poroshell (Giddings) Si 100 (Knox) 15 Chromolith (Giddings)

15 Poroshell (Knox) Si100 (Knox) Reduced HETP Reduced HETP Reduced Chromolith (Knox) Poroshell (Knox) Si 100 (Van Deemter) Chromolith (Knox) 10 Si100 (Van Deemter) Poroshell (Van Deemter) 10 Chromolith (Van Deemter) Poroshell (Van Deemter) 5 Chromolith (Van Deemter)

0 5 0 5 10 15 20 0 20 40 60 Reduced interstitial velocity Reduced interstitial velocity c) d)

65 P5 100 90

55 80 Si100 Poroshell 70 45 Chromolith Si100 (Giddings) 60 Si100 Poroshell (Giddings) Poroshell 35 Chromolith (Giddings) 50 Chromolith Si100 (Knox) Si100 (Giddings) Poroshell (Knox) 40 Poroshell (Giddings)

Chromolith (Knox) Chromolith (Giddings) Reduced HETP Reduced Reduced HETP 25 Si100 (Van Deemter) 30 Si100 (Knox) Poroshell (Van Deemter) Poroshell (Knox) Chromolith (Van Deemter) 20 Chromolith (Knox) 15 Si100 (Van Deemter) 10 Poroshell (Van Deemter) Chromolith (Van Deemter) 5 0 0 50 100 150 200 0 100 200 300 Reduced interstitial velocity Reduced interstitial velocity Figure 4.16. Comparison of reduced experimental HETP and the one obtained by different models through silica columns: Si 100, Poroshell and Chromolith with a) toluene, molecules with small size b) P2 and intermediate size as c) P5 and d) P6

4.3.2. B and C terms

______Transport properties in multiscale materials in non-adsorbing conditions

The reduced longitudinal diffusion coefficient b for any type of column can be directly obtained from Eq.1.25 by peak parking method. A comparison of experimental reduced b value from the PP method and the value from fitting Van Deemter or Giddings equations is shown in Fig 4.17. a) b)

4,5 Si100 100 Si100

4 Poroshell 90 Poroshell chromolith 80 3,5 chromolith 70 3 60 2,5 50 2 40 1,5

b HETP (Giddings) HETP b 30 b HETP (Van Deemter) (Van HETP b 1 20

0,5 10

0 0 0 1 2 3 4 0 1 2 3 4 b term PPM b term PPM

Figure 4.17. Comparison of reduced b term for oluene and polystyrene from P1 to P8 in dynamic and static conditions through silica columns a) data obtained by PP method and Van Deemter equation, b) data evaluated by PP method and Giddings equation

The figure shows that the values of the reduced B obtained by fitting the reduced experimental HETP with Van Deemter equation fit well with the values from PP method for all columns and all polymers when the reduced b values are larger than approximatively 2. For values lower than 2, the dynamic methods cannot be used to determine the b term. When using the Giddings equation, the values are far from the one calculated by the PP method. Due to the limitations of the pump (the lowest possible flow rate is 0.001 ml.min-1), the decrease in the flow rate to measure the b term with good accuracy is not applicable for the intermediately sized molecules having a low eff D /Dm value. The reduced b term from Knox equation (not shown here), the valued fails to fit the result by the PP method, either. The reduced b term for the polystyrenes from P4 to P8 through Chromolith column is higher than in Si 100 and Poroshell columns due to higher effective

______Transport properties in multiscale materials in non-adsorbing conditions diffusion in Chromolith column. For smaller molecules, the effective longitudinal diffusion in Si column is dominant when comparing to the one in other columns. The smaller reduced b term in Poroshell column is due to the presence of the solid core [124]. It causes lower diffusion in the Poroshell column. The C term value is determined through the general rate model in Eq.1.18 in which the ratio of the effective intraparticle diffusion coefficient to the molecular diffusion coefficient is determined by Maxwell model for toluene and polystyrene either from P1 to P4 for Si 100 and Poroshell column or from P1 to P6 for Chromolith column. For the intermediate molecules, i.e. from P5 to P8, the C term values are obtained from the slope of the experimental HETP curves. The results of C term value for toluene and polystyrenes from P1 to P8 at the flow rate of 1.2 ml.min-1 through silica columns are shown in Fig.4.18. Chromolith column has the lowest value of the C term. It suggests that the monolithic column is more efficient than other columns as already evidenced in the literature [10].

0,05 Si 100 at 1.2ml/m

0,04 Poroshell at 1.2ml/m

Chromolith at 1.2ml/m

0,03 C,s 0,02

0,01

0 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 rm/rp Figure 4.18. Evolution of C term for toluene and a series of polystyrene from P1 to P8 as a function of the molecule/ pore size

4.3.3. Eddy diffusion term and polydispersity In this study, polydisperse polystyrenes are used. The question arises that the band variance of the chromatogram could be also due to the dispersity of the sample used. The contribution of the polymer polydispersity HETP term (Hpoly) to the experimental HETP values should be considered. This constant contribution is independent on the velocity and expressed as [51,137,138]: 88

______Transport properties in multiscale materials in non-adsorbing conditions

2 σpoly (4.14) Hpoly = L 2 tr where L is the length of the column, tr is the mean retention time of the sample eluted through the 2 chromatographic column, σpoly is the space variance of the sample due to the dispersity of the polymer used. For toluene and P1, the variance resulting from the polydispersity is strictly zero because they are single molecules. The value of constant Hpoly is significant for the polystyrene standard.

In relation to the reduced experiment HETP value, the reduced hpoly value and the reduced total eddy diffusion is written in the form of Van Deemter equation as: b (4.15) h + h = h − − cv eddy poly v whereas h is the reduced experimental HETP data, b and c are the previously measured reduced longitudinal diffusion and mass transfer terms, respectively. The hpoly value is derived from 2 Eq.4.14 with the peak variance resulting from polydispersity of the sample, σpoly determined by the Knox model [137]: 2 ′2 σpoly = S (P − 1)(1 + α) (4.16) where: - P is the polydispersity index (PDI) given in Table 2.2 for each sample, - α is the correction term, expressed by: 11 137 (4.17) α = (P − 1) + (P − 1)2 4 12 - S’ is the reciprocal of the gradient of the linear section of the calibration curve of lnMw to the mean retention time. An example of the calibration curve for Si 100 with the elution of molecules of intermediate size at 0.5ml.min-1 is shown in Fig. 4.19.

12 11 10 9 8 7 y = -1,7876x + 15,309 ln Mw ln 6 R² = 0,9838 5 4 3 2 2 3 4 5 time, min Figure 4.19. Calibration curve of the molecular weight to the retention time of intermediate samples 89

______Transport properties in multiscale materials in non-adsorbing conditions

The calculation of Hpoly values for intermediate molecules which can enter the mesopores at the flow rate of 0.5 ml.min-1 through Si 100, Poroshell and Chromolith column are performed following Eqs.4.15 and 4.12. The results are shown in Table 4.3.

Table 4.3. Hpoly from Knox model for various molecules in the different chromatographic columns at 0.5ml.min-1 Si 100 Poroshell Chromolith

Hpoly/HETP, Hpoly/HETP, Hpoly, Hpoly/HETP, Polymer PDI Hpoly, μm % Hpoly, μm % μm % P2 1.09 419.7 383 148.4 333 33.3 214 P3 1.05 217.6 162 76.5 180 16.8 80 P4 1.05 264.8 133 87.6 131 18.3 84 P5 1.03 206.1 85 63.0 108 11.8 61 P6 1.03 267.7 110 77.7 187 13.0 83 P7 1.03 369.6 152 91.6 327 14.4 110

It can be easily seen from Table 4.3 that the values of Hpoly are higher than the experimental HETP. It means that the Knox model overestimates the value of plate height due to the polydispersity. This could also be due to the real values of PDI which could be smaller than the

one given by the manufacturer. The overestimation of Hpoly by Knox model was also seen in the work of Heyden et al [138].

Another approach for determination calculation of the Hpoly is using the model which is the sum of n Gaussian functions when assuming that each polystyrene is comprised of some fractions with a different number of units of styrene (see Section 5.2). In chapter 5 in adsorbing conditions it will be shown that the composition of a polystyrene sample of a given mean size, which is in fact a mixture of polystyrene with a different number of units of monomers, can be checked. The separation was possible in adsorbing conditions for P2 and P3 for the three commercial columns studied. For polystyrenes of larger molecular weight, the adsorption was too strong in the experimental conditions studied and the chromatograms can not be recorded in a reasonable time. In adsorbing conditions, each fraction can be separated through silica columns due to different affinities with the silica surface. In fact, adsorption of polystyrenes on the silica surface increases with the length or the number of units p of the polystyrenes. In chapter 5 the percentage of each fraction present in the polystyrenes P2 and P3 are determined (see Fig.5.7). The chromatogram obtained for polystyrene of a given size (P2 or P3) can be fitted with one Gaussian function or by

______Transport properties in multiscale materials in non-adsorbing conditions the sum of n Gaussian functions depending on the numbers of fractions n which are present in P2 or P3. The sum of n Gaussian functions is given by this expression:

n t−t 2 A 2( r,i) (4.18) y = ∑ i e w i=1 w √π/2 where:- y is the absorbance of the molecule, - t is the time,

- Ai is the area of the peak for each fraction (calculated from the percentage of each fraction obtained in adsorbing conditions),

- tr,i is the mean retention time of each fraction, - n is the number of fraction in the polystyrene, - w is the peak width of the chromatogram of each fraction. We assume that all the fractions of the polymer have the same value of peak width of w. The retention time value of each fraction is obtained from the polynomial equation which is the best fit with the curve of mean retention time and the molecular weight of some intermediate molecules that can enter the mesopore of the chromatographic columns in the solvent of THF. Fig 4.20 shows the best fit with some molecules through Si 100 column at 0.5ml.min-1.

5,50

5,00

4,50

y = 2,06E-07x2 - 6,61E-04x + 5,10E+00

, min , r t 4,00 R² = 1,00E+00

3,50

3,00 0 500 1000 1500

Mw, g/mol

Figure 4.20. Relationship of the mean retention time tr and the molecular weight through the Si column at 0.5ml.min-1 with some intermediate molecules In the case of P2, the number of fractions is assumed to be 10. The first fraction has 3 units of styrene, the second fraction has 4 units of styrene and so on. From the chromatogram of P2 with 10 fractions through Si 100 column at 0.5 ml.min-1, the percentage of mass for each fraction to the 91

______Transport properties in multiscale materials in non-adsorbing conditions total mass of all the fractions are obtained (see Section.5.2). The retention time of each fraction in the solvent of THF is calculated by the equation shown in Fig.4.20. By using the Gaussian function to fit the experimental data with Origin software, one obtains the retention time, area and width of the peak. From this area and the percentage of mass for each fraction, the area of each fraction Ai is evaluated. By using the Solver function in Excel to make the fit between the total absorbance of all the fractions calculated by the sum of n Gaussian functions (Eq.4.18) with absorbance of the polymer achieved by one Gaussian function, the width of each fraction is obtained. Fig.4.21 shows the experimental peak of the polymer, the Gaussian peak fitting it, the peak sum of all the fractions and the peaks of some fractions given by Gaussian function (for 3,6, and 9 units) in the case of P2 at 1.2ml.min-1 through Si100 column.

35 P2-Experiment 30 P2-Gauss function 25 All fractions - Gauss function 20 Fraction of 3 units 15 Fraction of 6 units Absorbance, Absorbance, mAu 10

5 Fraction of 9 units

0 1,7 1,9 2,1 2,3 2,5 2,7 2,9 Time, min

Figure 4.21. Chromatograms of P2 and different fractions of P2 having different number of units at 1.2ml.min-1 through Si 100 column Hence, the peak variance due to the polydispersity is calculated by the subtraction of the peak variance of the fractions (w in Eq 4.18) from the peak variance of the polymer. Finally, one obtains the value of Hpoly by applying Eq.4.14. The reduced hpoly value for P2 and P3 through Si 100, Proshell and Chromolith column will be given in Table 4.4.

For the sake of illustration, one uses the overall eddy diffusion (A term) and Hpoly instead of reduced values for packed and monolithic columns in liquid chromatography. A sum of the overall eddy diffusion and polydispersity contribution for toluene and a polystyrenes from P1 to P8 through Si 100, Poroshell and Chromolith columns are plotted in Fig.4.22.

______Transport properties in multiscale materials in non-adsorbing conditions a) b)

250 70 Si 100 column Poroshell column

Toluene 60 Toluene P1 200 P1 P2 P2 50 P3

P3 P4 m

μ 150 P4 40 P5 P5 m

μ P6 P6 , P7

poly 30 100 P7 P8

A+Hpoly, A+Hpoly, P8 A+H 20 50 10

0 0 0 0,2 0,4 0,6 0,8 0 0,1 0,2 0,3 0,4 0,5 Reduced velocity, cm/s Interstitial velocity, cm/s c) d)

35 250 Chromolith Column Si 100 at 1.2ml/m Toluene 30 P1 Poroshell at 1.2ml/m 200 P2 Chromolith at 1.2ml/m P3 25

m m

P5 μ 150 μ

, , P6 20 poly P7 P8

100 A+H

15 A+Hpoly,

10 50

5 0 0 0,2 0,4 0,6 0 0,5 1 1,5 2 Interstitial velocity, cm/s rm/rp

Figure 4.22. Evolution of A +Hpoly term for oluene and polystyrenes from P1 to P8 as a function of velocity (a,b,c) and as a function of the molecule/pore size ratio (d) for the three columns

It may be noted from Fig 4.22 that the total contribution of the overall eddy diffusion and polydispersity is dependent on the studied molecules. The total contribution value is proportional to the molecular size for the samples from toluene to P4 for all the columns. When the molecular 93

______Transport properties in multiscale materials in non-adsorbing conditions size is close or larger than the pore size of column., i.e. for the molecules with the size over than that of P4, the total value is inversely proportional to the size of a molecule. The total value of overall eddy diffusion and polydispersity in Chromolith column are variable to the interstitial velocity. Nevertheless, Si100 column gives a higher total value than Poroshell and Chromolith columns. This phenomenon is also seen in the work of Gritti and Guichon [139], in which the columns packed with shell particles has smaller A term about 34% with toluene than column packed with fully porous particles. The smaller A terms was due to smaller transcolumn velocity bias in poroshell kinetic columns. In the study of Gritti and Guiochon [140], where the eddy diffusion of the monolithic columns are measured for a non-retained (uracil) and a retained (naphthalene) compound, a large difference of eddy diffusion between the two compounds was seen. Generally, the eddy diffusion which is due to the difference of the velocity near the wall and in the center of the column should be the same for different molecular sizes and independent on the stream path of the molecule along the column. The eddy diffusion is expected to depend only on the structure of the porous support material and the interstitiel flow space. Here, one obtains different values of eddy diffusion and polydispersity term for different molecular sizes. This may be due to the different values of polydispersity for different molecules. The transcolumn dispersion has also an influence on eddy diffusion. From Eq.4.8 to Eq.4.13, one sees that the reduced transchannel, short-range interchannel eddy diffusions depends only on the reduced velocity. To have a better understanding of the variation of eddy diffusion on the molecule size, the transcolumn eddy diffusion should be considered. The transcolumn velocity biases originate from the slurry packing process of the column and from the synthesis/preparation of monolith columns [50]. The transcolumn eddy diffusion, hTC, can be estimated by subtracting the transchannel and short-range interchannel terms from the overall eddy diffusion term as [140]: (4.19) hTC = heddy − hTS − hIC Fig 4.23 shows a comparison of the reduced sum value of transcolumn eddy diffusion and polydispersity through the three chromatographic columns.

______Transport properties in multiscale materials in non-adsorbing conditions

45 Si 100 at 1.2ml/m 40 Poroshell at 1.2ml/m 35 Chromolith at 1.2ml/m 30

poly 25 h

TC + TC 20 h 15 10 5 0 0 0,5 1 1,5 2 rm/rp Figure 4.23. A plot of the reduced total contribution of trans column eddy diffusion and polydispersity as a function of the molecule/ pore size by different models for toluene and polystyrenes from P1 to P8

The reduced value in Si 100 column is considerably higher than the one in other columns. Meanwhile, Poroshell column sees the lowest value of the reduced contribution of eddy diffusion and polydispersity. It also can be seen from the figure that the reduced sum values are variable to the molecular size. It may be explained by the polydispersity of the polymer. Horman and Tallarek [141] studied the eddy diffusion of uracil and naphthalene in the different silica monolithic columns. The transcolumn was obtained by the subtraction of the longitudinal diffusion, mass transfer term, transchannel and short-range interchannel eddy diffusions. The values of transcolumn eddy diffusion for uracil in the six analytical silica monoliths with different mesopore sizes were different. Different values were also obtained in the case of naphthalene for these columns. It means that the transcolumn term with the monomer is different.

For toluene and P1, the reduced polydispersity (hpoly) is strictly zero. The molecules having a size larger than the pore size (rm/rp>1) cannot enter the mesopore and the value of hTC+hpoly is constant for each column. For those molecules, the band broadening is mainly the contribution of transcolumn eddy diffusion. By assuming that all the molecules have the same value of transcolumn, one can calculate the polydispersity for the molecules which can enter the mesopore by the subtraction of the transcolumn (hTC for rm/rp>1) from the sum value of transcolumn and polydispersity (hTC+hpoly). A comparison of polymer dispersity by the subtraction and the one calculated by the sum of n Gaussian functions is shown in Table 4.4. 95

______Transport properties in multiscale materials in non-adsorbing conditions

Table 4.4. hpoly for P2 and P3 from experiment and model of the sum of n Gaussian functions

Si 100 column Poroshell column Chromolith column

From From From Sample experiment By model experiment By model experiment By model

P2 15.8 6.0 2.2 1.7 4.1 3.8

P3 23.2 9.8 5.0 5.7 7.0 7.1

As seen from this table the hpoly values for Poroshell and chromolith from the experiment and the model are comparable. A significant difference is observed for Si 100 column. The difference could be explained by the fact that the pore size distribution of the Si100 column is broad as compared to Poroshell and Chromolith columns (see Figs.3.5 and 3.6) which could cause a supplementary spreading of the peak during elution. The mesopore range in Si 100 column is indeed very large and may be there is a flow inside the particles which is not present with the two other samples. This flow is probably affected by the presence of molecules that have a non negligible size as compared to pore size.

4.4. Conclusion In dynamic conditions, at low and moderate velocities the axial dispersion controls band dispersion. The HETP values in Si 100 column for the intermediate size and larger molecules are higher than those in Poroshell and Chromolith columns. The Chromolith column is more efficient and has a better resolution than the other columns. The effective diffusion coefficient in dynamic condition is only obtained for the column packed fully porous particle by the slope of the HETP curves for intermediate molecules. In static conditions, the effective intraparticle diffusion coefficient is calculated from the PP method and by using the Maxwell model. The particle tortuosity is modeled by Weissberg equation. The topological parameters determined by the PP method are in good agreement with the data obtained by electrical measurement except for the Poroshell column. The sum of eddy diffusion and polymer polydispersity depends on the molecular size. It is increasing with molecular size for the smallest molecules and then goes through a maximum. For molecular sizes approaching the pore size of the chromatographic column, the sum value of eddy diffusion and polymer polydispersity decrease with increasing the molecular size. The value for

______Transport properties in multiscale materials in non-adsorbing conditions the column packed with fully porous particles is stable with interstitial velocity and higher than the one for other columns. The eddy diffusion, which depends mainly on the trans column effect, is constant for all molecules. Meanwhile, the polymer polydispersity is variable to different molecules and has an effect on the apparent A term.

______Transport properties in adsorbing conditions

5 TRANSPORT PROPERTIES IN ADSORBING CONDITIONS

5.1. Adsorption isotherm of toluene on silica from heptane Before studying the transport properties of polystyrene in a silica column in adsorbing conditions, the choice of the solvent enabling adsorbing condition must be considered. n-heptane was suggested by some authors [93,142,143] as a solvent when investigating the adsorption of organic chemicals on the silica surface. The measurement of an experimental adsorption isotherm from n-heptane on silica should be performed firstly. Adsorption equilibrium experiments were performed on Si100-12µm powder at 25°C by the solution depletion method. The solutions of toluene in n-heptane with different initial concentrations were prepared in glass vials. Silica powder at around 300mg was added to each vial. All the vials were sealed and kept in a rotating mixer during 24hour at 250C until adsorption equilibrium was reached. Then the liquid and solid phases were separated by centrifugation for 20 minutes at 12000 rpm. The liquid phase was analyzed in order to determine the concentration of toluene in the solution at equilibrium. The absorbance values of initial solutions and solutions obtained after adsorption and centrifugation were measured with UV-visible spectroscopy at 262 nm by using the HPLC device. The calibration curve of the absorbance for pure solutions of toluene in n-heptane as a function of concentration was plotted. From the slope of this calibration curve and the absorbance of the solution after adsorption, the concentration of toluene in the liquid phase at equilibrium was determined. The adsorbed amount of toluene on silica calculated by the following equation: C − C (5.1) q = o e V e m where C0 and Ce (in mol/L) are the concentrations of the solution before adsorption and at equilibrium, respectively. V (in L) is the volume of the solution and m (in g) is the mass of adsorbent. The amount of adsorbed adsorbate at equilibrium was also expressed by the model of Langmuir as follows [144]:

qmKL Ce (5.2) qe = 1 + KL Ce

In which KL is the Langmuir equilibrium constant and qm (mol/g) is the theoretical monolayer saturation capacity.

______Transport properties in adsorbing conditions

Results for equilibrium adsorption experiments of toluene from n-heptane with Si100-12µm are shown in Fig.5.1. The adsorbed amount of toluene increases with the equilibrium concentration and then reaches the maximum value at around 0.9 mmol.g-1. The curve can be fitted with the Langmuir model where the Langmuir equilibrium constant and the theoretical monolayer saturation capacity are obtained at 0.536 and 0.0015 mol/g, respectively. This behavior has also been found in adsorption on silica gel [142].

0,0012 y = 8,21E-04x 0,001 R² = 1

0,0008 Adsorption isotherm

0,0006 Langmuir model

0,0004 Henry constant from isotherm

y = 7,28E-04x Henry constant from Adsorbed mol/g Adsorbed amount, 0,0002 R² = 0,9937 chromatography

0 0 1 2 3 4 5 Equilibrium concentration, mol/l Figure 5.1. Equilibrium adsorption isotherm of toluene in n-heptane with silica

For the low-concentration range, the adsorption isotherm curve is linear. The slope of this linear section is known as the Henry constant, KH [90]. The value of KH from the for Si100-12µm powder is 0.000728 l/g. Normally, the Henry constant is also defined as the ratio of the equilibrium concentrations of the solute in the stationary phase and in the mobile phase, respectively. The equilibrium concentration in the solid phase is considered as the mass of the adsorbate per unit mass of the adsorbent while the unit of solute in the mobile phase has the unit of mass per unit volume. Therefore, the Henry constant has the unit of l/g [53]. In chromatography, the Henry constant is derived from the first absolute moment through the column length, superficial velocity, retention time, porosity and the density of the adsorbent

[53,94]. The relationship between the Henry constant and the equilibrium constant (Ka) is expressed as KH = Ka(1- εt)/(ρs(1- εe)) [53]. The equilibrium constant is determined by Eq.1.19 in

______Transport properties in adsorbing conditions which the retention of solute represents the distribution ratio of the analyte between the stationary and mobile phases in equilibrium and is known as the retention factor (k’) [95]:

tr − to (5.3) k′ = to whereas tr and to (min) are the retention times of a retained and unretained compound, respectively. When performing the elution of toluene in n-heptane through Si100 column (particle size of 5µm) at 0.5ml/min, one obtains the value of k’ of 1.4 and then Ka of 5.6. From this value of

Ka, the Henry constant is calculated at 0.000821 l/g. This value is shown in Fig.5.1. There is a good agreement between the value of the Henry constant obtained from the isotherm of adsorption of toluene in heptane on silica Si100 (particle size of 12 µm) and the one achieved from chromatography with Si100 (5µm). Toluene and P1 were eluted in n-heptane through Chromolith column at the flow rate of 0.5 ml/min. A comparison of the chromatograms of toluene and P1 in THF (non-adsorbing solvent) and n-heptane (adsorbing solvent) with Chromolith column is presented in Fig 5.2.

1.0

0.8 Toluene in THF P1 in THF Toluene in n-Heptane 0.6 P1 in n-Heptane

C/C 0.4

0.2

0.0 2.8 3.2 3.6 4.0 4.4 time (min)

Figure 5.2. Chromatograms of toluene and P1 in THF and n-heptane at 0.5 ml/min through Chromolith column

100

______Transport properties in adsorbing conditions

It can be obviously seen in Fig 5.2 that the values of retention time of toluene in n-heptane are higher than that in THF. The same phenomenon is observed with P1. The values of the retention factor for toluene and P1 are 0.17 and 0.15, respectively. A weak hydrogen bond between a hydrogen atom of silica and more electronegative oxygen of THF causes the non-adsorbing condition when using THF as a solvent. In the case of n-heptane, the aromatic ring of polystyrene and the silica surface are parallel to each other. There is an interaction between the highest occupied molecular orbital of the phenyl group and the lowest unoccupied molecular orbital of silica. For larger polystyrene P2, a stronger adsorption is observed. Fig 5.3 shows the chromatogram of P2 in the solvent of n-heptane through Chromolith column.

1.0

0.8

0.6 P2 in THF P2 in Heptane

C/Co 0.4

0.2

0.0 0 5 10 15 20 25 30 time (min)

Figure 5.3. Chromatograms of P2 in THF and n-heptane at 0.5 ml/min with Chromolith column

As already mentioned before, the number of units of styrene is derived from the molecular weight of polystyrene determined by Eq.2.23. P2 has the molecular weight of 690 g/mol so the number of units is around at 6. A lot of peaks of P2 with the same UV spectrum were obtained through Chromolith column in the solvent heptane. Each peak may correspond to each fraction of P2 with different numbers of units. These peaks are different from the Gaussian shape, they caused difficulty when analyzing the band broadening in studying the transport properties of P2 through 101

______Transport properties in adsorbing conditions the column. The idea is then to use the mixture of THF and n-heptane as a solvent in order to decrease the affinity of polystyrenes with silica surface when studying transport in adsorbing conditions. By changing the percentage in volume of n-heptane in the mixture of n-heptane and THF, one obtains solvents with different compositions. The chromatograms of P2 in the different solvents through Chromolith column are presented in Fig 5.4.

1.0

0.8 P2-80% Heptane- 20% THF P2-85% Heptane- 15% THF P2-97% Heptane-3% THF 0.6

0 P2-100% THF

C/C 0.4

0.2

0.0 2.5 3 4 5 6 7 time (min)

Figure 5.4 Chromatograms of P2 at 0.5ml/min in four different eluent compositions through Chromolith column

For solvents with the percentage of THF higher than 20%, one gets only one peak of P2. The adsorption capacity of polystyrenes increases along with increasing the percentage of n-heptane. For the mixture of n-heptane/THF (85/15, v/v), two peaks of P2 are seen from the chromatogram. In the case of the mixture of THF and n-heptane (3/97, v/v), many peaks of P2 having the Gaussian shape are attained.

The retention factor for P2 in each mixture is calculated by Eq.5.3 in which t0 is the retention of

P2 in the solvent THF and tr is the retention time of P2 in each mixture THF/heptane. For the mixtures THF/heptane with the percentage of n-heptane above 80% in volume, the values of tr are taken from the retention time values of the highest peak. The evolution of retention factor as a function of the percentage of heptane in the mixture THF/heptane is shown in Fig 5.5. It should be noticed that the retention factor of P2 increases with an increase in the percentage of n-heptane. There is a very high variation of retention factor when the amount of n-heptane tends to 100 %

102

______Transport properties in adsorbing conditions volume. Hence, the mixture of n-heptane and THF (97/3, v/v) with a high retention factor but keeping Gaussian shape for the peak of P2 is chosen as a solvent in adsorbing conditions in this study.

1,6 1,4 1,2 1 P2 0,8 0,6

0,4 Retention factor k' 0,2 0 0,4 0,5 0,6 0,7 0,8 0,9 1 Percentage of n-Heptane, % Figure 5.5. The calculated retention factors of P2 for Chromolith column in the different mixtures of n-heptane and THF

5.2. Band broadening of polystyrenes in adsorbing conditions In dynamic conditions, P2 and P3 are eluted through each column shown in Table 2.1 in the mixture of n-heptane and THF (97/3, v/v) at different flow rates (0.5, 1, 1.5 and 2 ml.min-1). The absorbance is measured at 262nm. The examples of band broadening of P2 through Si 100 column and P3 through Poroshell column at the flow rate of 1ml/min are shown in Fig 5.6.

103

______Transport properties in adsorbing conditions a) b)

6 units 1.0 1.0 11 units

0.8 P2-Si 100 column 0.8 P3-Poroshell column

0.6 0.6 0 0

C/C 0.4 C/C 0.4

0.2 0.2

0.0 0.0 3 4 5 6 7 8 9 10 11 3 6 9 12 15 time (min) time (min)

Figure 5.6. Chromatograms obtained for polystyrenes in the mixture of n-heptane and THF at the flow rate of 1ml.min-1 a) P2 with Si 100 column and b) P3 with Poroshell column

Many peaks are obtained for the chromatograms of P2 and P3 in this mixture. These peaks show different heights and widths. For P2, 10 peaks are obtained with a Gaussian profile. The number of peaks for P3 through Poroshell column is 14. As presented previously, P2 has 6 units of styrenes and P3 has 13 units of styrenes. The values of polydispersity index (PDI) given by supplier for P2 and P3 are 1.09 and 1.05, respectively. When PDI is different from 1, the polystyrene is composed of molecules of different molecular weights. In the case of P2, the main fraction of polystyrene has 6 units but there are also smaller and bigger polystyrenes which have different affinities for the silica surface. By assuming the number of units for each peak in Fig 5.6 and calculating the area of each peak, the value of mass fraction of each compound can be calculated from the ratio of peak area on sum of peak areas.. The PDI, a measure of the width of molecular weight distributions, is defined as a ratio of the weight average molecular weight to the number average molecular weight [145] M (5.4) PDI = w Mn with: - Mw is the weight average molecular weight. It is based on the fact that a bigger molecule contains more of the total mass of the polymer sample than the smaller molecules do. It is calculated from the weight fraction distribution of molecules at different size: 104

______Transport properties in adsorbing conditions

n (5.5) Mw = ∑ wiMi i=1 - n is the degree of polymerization,

- wi and Mi are the area fraction and the molecular weight of each fraction, respectively

- Mn is the number average molecular weight. It is just the total weight of all the polymer molecules in a sample, divided by the total number of polymer molecules in a sample. It is calculated by this relationship: n n (5.6) Mn = ∑ NiMi / ∑ Ni i=1 i=1

- Ni is the number of molecules of size i, determined from this equation:

Ni = NAmi/Mi (5.7)

- mi is the total mass for each fraction. It is the product of the area fraction by the mass of sample,

- NA is the Avogadro’s number From the chromatograms shown in Fig 5.6, one can obtain the mass fraction of each peak. The assumption of the number of units for each peak must be done. The first peak has been assumed to correspond to 3 units, 4 units for the second peak, and so on. Then, the highest peaks for P2 and P3 have 6 and 11 units, respectively. By using Eqs.5.4, 5.5, 5.6 and 5.7, the PDI values for P2 and P3 are 1.07 and 1.08, respectively. They are close to the values from the manufacture. The distribution of units in sample P2 calculated from experiments with Si 100 column is plotted in Fig 5.7. It may be noted that the peak with 6 units is the main fraction of P2

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______Transport properties in adsorbing conditions

6 units 7 units

8 units 5 units

9 units

4 units

3 units 11 units 12 units

Figure 5.7 The distribution of each fraction of the sample P2 in the mixture of THF and n-heptane through Si 100 column Before calculating the retention factor of each fraction with different units, the retention time of unretained fraction in non-adsorbing conditions should be considered. The unretain retention time of each fraction with different units in the solvent of THF can be obtained by modeling the relationship between retention time and molecular weight of of polystyrenes in THF through each column as follows:

−b2 tr = b1(Mw) (5.8) where b1 and b2 are the empirical parameters dependent on the chromatographic column type. Fig 5.8 shows the relationship between the retention time and molecular weight of different polystyrenes in THF through Poroshell, Chromolith Si and Si 100 columns at the flow rate of 1ml.min-1. A power equation (R2=0.965) fits the curve of retention time and molecular weight through Poroshell column. A pair of empirical parameters (b1 and b2) are obtained at 2.5517 and 0.071, respectively. By fitting the same type of power equation to the retention time curve, one can obtain the values of these empirical parameters for Si 100 and Chromolith columns.

106

______Transport properties in adsorbing conditions

2,6

2,3 Poroshell 120 column Chromolith Si column 2 Si 100 column 1,7

1,4

Retention time, min Retention y = 2,5517x-0,071 1,1 R² = 0,9659 0,8 0 500000 1000000 1500000 2000000 Mw, g/mol

Figure 5.8. Plots of the retention time of a series of polystyrenes as a function of the molecular weight through the chromatographic columns Finally, the retention factor of each fraction in the mixture of n-heptane and THF can be calculated by applying Eq.5.3 with the retention time of each retained compound achieved by fitting the chromatograms with a Gaussian function and the retention time of each non-retained fraction obtained by Eq.5.8. Evolution of retention factor of P2 and P3 through Si 100, Poroshell and Chromolith columns at the flow rate of 1ml.min-1 is shown in Fig 5.9.

3,5 P2-Si 100 3 P3-Si 100 P2-Poroshell 2,5 P3-Poroshell P2-Chromolith 2

P3-Chromolith k' 1,5

0,5

0 3,0 6,0 9,0 12,0 number of units Figure 5.9. Retention factor versus the number of units for the 3 chromatographic columns in the case of P2 and P3

107

______Transport properties in adsorbing conditions

It can be seen obviously that the retention factor increases with increasing the number of units. This type of behavior has also been observed in the high-performance liquid chromatography analysis of polystyrene oligomer on the bare silica column in the mixture of THF and n-hexane. The mass spectrometry was used to recognize the degree of polymerization for the oligomer peaks in this case [95]. In the case of Si 100 column, the values of the retention time of the fractions of P2 with 7,8,9 and 10 units in this figure are in accordance with that of the fractions of P3 with the same numbers of units. It means that the affinity to the silica surface in this column for the fractions of P2 and P3 with the same numbers of units is comparable. Meanwhile, there is a slight difference in the retention factor for the fraction of P2 and P3 with the same number of units for Poroshell and Chromolith columns. The figure also shows that the values of the retention factor of P2 and P3 in Si 100 column are higher than the one in other columns. It may show different surface chemistry between the materials: in the case of silica it is often related to the surface density of surface groups.

5.3. Diffusion in adsorbing conditions 5.3.1. Determination of effective diffusion coefficients in adsorbing conditions As previously shown in Section 4.2, the effective diffusion coefficient in Chromolith column is higher than in other columns. For the sake of simplicity, Si 100 column is chosen to study the transport properties in adsorbing conditions. The PP method was applied to Si 100 column in order to measure the effective diffusion coefficient, Deff of different fractions of P2 and P3 in the mixture of n-heptane and THF (97/3,v/v). The flow rate of 1 ml.min-1 was used in these experiments. The corresponding PP chromatograms recorded for P2 with different parking times for Si 100 column are shown in Fig 5.10. As expected, like in non-adsorbing conditions, the peak width increases and the peak height decreases when increasing the parking time. For some larger fractions, there is a small shift between the peak with different values of parking time and the reference peak.

108

______Transport properties in adsorbing conditions

1.0 6 units 1.0 6 units

tp = 0 min 0.8 tp = 0 min 0.8 tp = 15 min tp = 15 min tp = 31.5 min tp = 31.5 min 0.6 0.6 tp = 60 min tp = 60 min 0 0 tp = 127 min tp = 127 min tp = 220 min C/C 0.4 tp = 220 min C/C 0.4

0.2 0.2

0.0 0.0 -1 0 1 2 3 4 5 6 7 -0.2 -0.1 0.0 0.1 0.2 time (min) time (min) a) b) Figure 5.10. Chromatograms recorded during the peak parking experiments a) for P2 and b) for the fraction of 6 units of P2 through Si 100 column at the flow rate of 1 ml.min-1

The relationship between the peak variance and the parking time for some fractions of P2 for Si100 column is plotted in Fig 5.11. It is worth mentioning that the behavior of the plot is linear.

0,16

0,14

0,12 P2 5 units

2 P2 6 units 0,1 P2 7 units

P2 8 units 0,08 P2 9 units

0,06 P2 10 units peak variance, cm peak variance,

0,04

0,02

0 0 2000 4000 6000 8000 10000 12000 14000 parking time,s

Figure 5.11. A plot of peak variance as a function of the parking time for P2 in Si 100 column

109

______Transport properties in adsorbing conditions

It should be noticed that the peak variance in this figure is expressed in a scale of length in order to have an easier visualization. The slope of these linear plots decreases with increasing the number of units for each fraction. A comparison of the peak variance of the fraction with the same number of units for P2 and P3 is given in Fig 5.12. An interesting feature to note is that the peak variance values for the fraction with the same number of units are on the same linear line whatever their sample origin, P2 or P3.

0,2 0,18 0,16 0,14

2 0,12 P3 7 units 0,1 P2 7 units 0,08 P3 10 units 0,06 P2 10 units

0,04 peak variance, cm variance, peak 0,02 0 0 5000 10000 15000 20000 25000 30000 parking time, s Figure 5.12. Comparison of the peak variance for the fraction of 7 units and 10 units for P2 and P3 for the Si100 column

From the slope of the curve of the peak variance, the effective diffusion coefficient for P2 and

2 2 eff 1 ∆σt u P3 can be derived by applying Eq.1.22 (D = 2 ) in which the zone retention factor 2 tp (1+k1[rm])

(k1[rm]) for each analyte in the mixture of n-heptane and THF is determined by combining

(1−εe) Eqs.1.18, 1.19 and Eq.5.3 in adsorbing conditions (k1[rm] = (εpz[rm] + (1 − εe ′ εt[rm] k εpz[rm]) )). The ratio of the effective diffusion coefficient to molecular diffusion (1−εt[rm]) eff coefficient, D /Dm as a function of retention factor in Si 100 column is presented in Fig.5.13 with the values of molecular diffusion (Dm) of each fraction of P2 and P3 at different values of Mw are −8 −0.599 evaluated by Eq.2.14 (Dm = 4.021 x 10 (Mw) ). The value of this ratio decreases with an increase of the retention factor. 110

______Transport properties in adsorbing conditions

0,85

0,75

0,65

0,55 P2 m

/D P3

eff 0,45 D 0,35

0,25

0,15 0,0 0,5 1,0 1,5 2,0 2,5 3,0 k'

Figure 5.13. Ratio between experimental effective diffusion coefficient and molecular diffusion coefficient as a function of retention factor for Si 100 column

5.3.2. Determination of surface diffusion

eff As mention already in Section 1.3.4, the intraparticle diffusion (Da ) consists of two parallel contributions: pore diffusion of the molecules in the pores filled with the mobile phase (with a eff effective intraparticle diffusion coefficient (Dp )) and surface diffusion occurring when analyte is adsorbed to the surface of the pores (with a surface diffusion coefficient Ds). The value of the surface diffusion coefficient of each fraction is derived from Eq.1.41 as follows:

eff eff Da − Dp (5.9) Ds = (1 − εpz)Ka

The adsorption equilibrium constant Ka is calculated by Eq.1.19, in which the retention factor values for each fraction of P2 and P3 are already determined in Section 5.2.

eff The effective intraparticle diffusion coefficient values (Dp ) for a series of polystyrenes are already determined under non-adsorbing conditions (k’=0) with variable particle tortuosity (Eq.4.3 eff and Eq.1.7) in Section 4.2.2. The relationship of Dp and Mw for these molecules through Si100 eff column is shown in Fig. 5.14. There is a decrease of Dp with increasing the value of Mw. A

eff -1,47 power equation (Dp = 2E-06Mw ) gives a good fit for this curve. With a given value of Mw for

111

______Transport properties in adsorbing conditions

eff each fraction of P2 and P3, the values of Dp for all fraction of P2 and P3 are obtained by this power equation.

2,5E-09

2E-09

2 1,5E-09

eff p 1E-09

D y = 2E-06x-1,47 5E-10 R² = 0,9316

0 0 5000 10000 15000 20000 25000

Mw,g/mol Figure 5.14. Evolution of the effective intraparticle diffusion coefficient to the molecular weight of a series of polystyrenes in non-adsorbing conditions through Si100 column

The molecular weight, Dm for each fraction of P2 and P3 are determined by Eq.2.14 using

eff molecular weight dependent on the number of units. The results of the ratio Dp /Dm for different

eff fractions of P2 and P3 are plotted in Fig.5.15. The value of this ratio Dp /Dm decreases with an increase in the number of units.

0,25

0,20 P2

0,15

/ P3

eff p

D 0,10

0,05

0,00 0,0 5,0 10,0 15,0

number of units Figure 5.15. Relationship between the ratio of the pore diffusion coefficient to the molecular diffusion coefficient with the number of units for P2 and P3

112

______Transport properties in adsorbing conditions

As mentioned previously, the effective intraparticle diffusion coefficient for spherical particles can be derived from the Maxwell model. The Maxwell model given in Eq.1.30 may be also written:

Deff ε (1 + k [r ]) − 1 (5.10) 1 e 1 m D β = m Deff 1 − εe εe (1 + k1[rm]) + 2 Dm Whereas the ratio of the effective diffusion coefficient to the molecular diffusion in adsorbing eff conditions D /Dm is obtained by PP method. Hence, the ratio of the effective intraparticle diffusion coefficient to the molecular diffusion eff coefficient of each fraction of P2 and P3 through Si 100 column in adsorbing conditions, Da /Dm is obtained by the relationship:

Deff 1 + 2β (5.11) a = Dm 1 − β

Therefore, the value of the surface diffusion (Ds) can be calculated by subtracting the contribution of the pore diffusion from the intraparticle diffusion following Eq.5.9. An example

eff eff of D /Dm , Da /Dm, Ds/Dm results for different fractions of P2 are presented in Table 5.1.

Table 5.1. Calculated results for some fractions with different numbers of units for P2 in adsorbing conditions through Si100 column

Number of eff eff eff units k' Dm D /Dm Dp /Dm Da /Dm Ds/Dm 4 0.46 1.01E-09 0.79 0.23 0.80 0.99

5 0.63 8.93E-10 0.58 0.20 0.57 0.49

6 0.83 8.09E-10 0.45 0.17 0.44 0.27

7 1.08 7.43E-10 0.38 0.15 0.40 0.19

8 1.39 6.9E-10 0.35 0.13 0.45 0.18

9 1.78 6.45E-10 0.28 0.12 0.38 0.12

10 2.28 6.08E-10 0.24 0.11 0.37 0.10

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______Transport properties in adsorbing conditions

eff The Da /Dm results and the evolution of the ratio of surface diffusion and molecular diffusion with retention factor for each fraction of P2 and P3 through Si 100 column are also shown in Fig 5.16.

0,9 P2-Ds/Dm 0,8

0,7 P3-Ds/Dm m

0,6 P2-Da,eff/Dm D/D

0,5 P3-Da,eff/Dm 0,4 0,3 0,2 0,1 0 0,0 1,0 2,0 3,0 4,0

k' Figure 5.16. Ratio of the intraparticle diffusion and surface diffusion coefficients to the molecular diffusion coefficient as a function of retention factor for P2 and P3

eff As can be seen obviously from Fig 5.16 the values of the ratio Da /Dm for all the fractions with the same number of units for P2 and P3 are different. This difference may be explained by the influence of the mutual relation of different fractions in the polystyrene in the process of migration through the pores of the particles. The figure also shows that the surface diffusion decreases with an increase in the retention factor. The extrapolated intercept of the curve of the ratio Ds/Dm versus k’ at the origin is probably close to unity. The surface diffusion of a fraction with the small retention factor (weak retained conditions) is predicted to be the same value of molecular diffusion. It suggests that there is a close relationship between surface diffusion in the stationary phase and molecular diffusion in the bulk mobile phase. This behaviour has been also observed in the work of Miyabe and Guichon [82]. They studied the adsorption properties of alkylbenzene and p-alkylphenol homologues in the Methanol/water mixture. Eq.5.9 was also used eff eff to evaluate the value of Ds from Da and pore diffusion, Dp . The Ds/Dm value for some alkylbenzenes was seen to increase with decreasing the adsorption equilibrium constant, Ka. It was

114

______Transport properties in adsorbing conditions also shown that retention strength affects the migration of the sample molecules by surface diffusion. The attractive interaction between the sample molecules and the stationary phase surface caused the restriction of the mass transfer of the sample molecules in the stationary phase by surface diffusion. The ratio Ds/Dm tended toward a value close to unity at K = 0. It indicated that Ds and Dm have the same order of magnitude when the adsorption energy is negligibly small. It is also seen from Fig.5.15 that the surface diffusion has a significant contribution to intraparticle diffusion in porous adsorbents, especially for the small fraction of polystyrene.

5.4. Conclusion The mixture of n-heptane and THF (97/3, v/v) was chosen as a solvent to study the transport properties of polystyrenes through silica columns in adsorbing conditions. One obtained the chromatograms of polystyrenes in this mixture with several peaks which corresponds the polydispersity of polystyrenes. The main fraction has the same number of units of styrene as the polystyrene sample. The retention factors of the fractions of different polystyrenes with the same number of units are comparable. The adsorption increases with the number of units and also with the molecular weight of the polystyrenes. This in agreement with expected influence of chain length on adsorption properties of polymers. The surface diffusion phenomena were investigated by regarding the influence of the polystyrenes interaction with silica surface. The relationship between surface diffusion and molecular diffusions was pointed out. The surface diffusion value decreases with an increase in the retention factor of the sample compounds. The surface diffusion of a weak retained fraction approaches to the value of molecular diffusion. The surface diffusion is prevalent in the overall intraparticle diffusion.

115

______Conclusions and perspectives

6 CONCLUSIONS AND PERSPECTIVES

The objective of this thesis was to have a better understanding of the effect of porous structure parameters on the transport properties through multiscale porous media. The diffusion model at molecule scale expressing the relationship between the transport properties, i.e. effective internal diffusion coefficient, and the structural parameters such as porosity, tortuosity is studied for porous silica material in which the internal tortuosity based on the Weissberg equation is determined by liquid chromatography and verified by electrical measurement. Other types of porous media as core-shell and monolith are investigated via the mass transfer contribution. The longitudinal diffusion and eddy diffusion through porous materials is described. The influence of structural parameter on the transport properties is also extended to adsorbing conditions. Chromatographic columns containing silica having different morphologies (fully porous particles, core-shell and monolith) were characterized by Inverse Size Exclusion Chromatography in non-adsorbing conditions by injecting polystyrenes of different sizes. The porosities (total, external and particle) and the pore size distribution were obtained. The results obtained were comparable to the values obtained by classical methods such as gas adsorption and mercury porosimetry. The tortuosity was determined by electrical measurements and compared to the values obtained by liquid chromatography by using the peak parking (PP) method. Similar results were obtained for the apparent total tortuosity of the monolithic column with the smallest probe and for the particle tortuosity of the totally porous particles. Different particle tortuosity results are obtained for the core-shell column. The advantage of the PP method is that the apparent total tortuosity and the particle tortuosity could be determined as a function of the size of the molecule. The effective diffusion coefficient for different molecules of toluene and polystyrenes were determined by the peak parking method in non-adsorbing conditions. Chromolith column has higher effective diffusion coefficients than other columns as a result of higher porosities and lower apparent total tortuosity. Different equations are used to model the ratio of the effective diffusion coefficient to the molecular diffusion coefficient. The Maxwell equation is seen as the best fit for the experimental effective diffusion coefficient with a variable particle tortuosity given by Weissberg equation. The effective intraparticle diffusion coefficient in the mesopore of the skeleton for Chromolith column is higher than the one in the mesopore of both porous particles for Si 100 column and core-shell particles with Poroshell column.

116

______Conclusions and perspectives

HETP data for the intermediate size and large molecules in Chromolith column are lower than the one in Poroshell and Si 100 column. It suggests that the monolithic column is more efficient and has a better resolution than the other columns. The sum of eddy diffusion and polydispersity contribution depends on the molecular weight of the polystyrenes for all chromatographic column. For the small molecules, the combined contribution of eddy diffusion and polydispersity increases with increasing the molecular size. It decreases with an increase in the molecular size for the molecules having the size close to the mesopore size of the stationary phase. The polydispersity is the main source of this difference. For Si100 the broad pore size distribution leads to an increase of the spreading of the peak due to various retention times depending on the size of the molecule and the various sizes of the pores. Different from the study in non-adsorbing conditions, the chromatograms of polystyrenes in the mixture of n-heptane and THF show several peaks, which is due to the polydispersity of the polystyrenes evidenced by the adsorption versus chain length. The ratio of the surface diffusion coefficient to the molecular diffusion increases with decreasing retention factor for n-heptane/THF mixture. From the relationship between the surface diffusion and the retention, the surface diffusion tends towards the molecular diffusion when the retention factor tends zero, i.e. under non-adsorbing conditions. This document describes a small part of the knowledge of the effect of pore structure on the transport in porous media. Some matter is still being questionable. The behavior of the surface diffusion in the regions of very low values of retention factor and at very large values of retention factor should be considered further. Moreover, due to the limitation of determination of the average radial dispersion coefficient, the influence of the transcolumn eddy diffusion on the effective diffusion should be also investigated.

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