Effect of tortuosity on diffusion of polystyrenes through chromatographic columns filled with fully porous and porous –shell particles and monoliths

Microporous and Mesoporous Materials xxx (xxxx) xxx

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Effect of tortuosity on diffusion of polystyrenes through chromatographic columns filled with fully porous and porous –shell particles and monoliths Khac Long Nguyen a, b, V�eronique Wernert a, *, Andr�e Morgado Lopes a, c, Loïc Sorbier c, Renaud Denoyel a a b c

Aix-Marseille Universit�e, CNRS, MADIREL, UMR 7246, Centre Saint-J�er^ ome, F-13397, Marseille cedex 20, France Hanoi University of Mining and Geology, 18 Vien Street, Bac Tu Liem, Hanoi, Viet Nam IFP Energies nouvelles, Rond-point de l’�echangeur de Solaize, BP 3, 69360, Solaize, France

A R T I C L E I N F O

A B S T R A C T

Keywords: Peak parking Effective diffusion coefficient Intraparticle diffusion coefficient Tortuosity Maxwell equation

The tortuosity parameter, essential in the prediction of molecular transport properties, is determined for several materials with multiscale porosities: interparticular/interskeleton macropores and intraparticular/intraskeleton mesopores for beds of spherical porous particles and monoliths. Electrical measurements (impedance spectros­ copy) and peak parking experiments are used. The former measures the electrical resistance between two electrodes surrounding the porous material impregnated with a concentrated electrolyte, while the latter uses a set of polystyrene molecules in non-adsorbing conditions within a chromatographic setup. Tortuosity mea­ surements as well as characterization via mercury porosimetry, nitrogen adsorption and inverse size exclusion chromatography (ISEC) are performed on four materials: fully porous silica and alumina particles, core-shell silica particles and silica monoliths. The tortuosity determined by electrical measurements is in agreement with the value determined by peak parking with the smallest probe (toluene). For molecules which size is not negligible as compared to pore size, the apparent particle tortuosity is determined from the intraparticle diffusion coefficient obtained by the Maxwell model estimating the hindrance factor via the Renkin correlation. The apparent particle tortuosity is estimated from the Weissberg equation τp[rm] ¼ l-pln(εp[rm]), where εp[rm] is the particle porosity accessible to a molecule of size rm and p a parameter depending on the material topology. A model with just one adjustable parameter p can thus estimate the intraparticle diffusion coefficient in nonadsorbing conditions.

1. Introduction A better understanding of transport through porous materials has become very important for the effective design, preparation and pro­ duction of columns in the separation and catalysis fields. For example, silica columns with different morphologies are used for the separation of molecules: totally porous particles, core-shell particles or monoliths are available. γ-aluminas are used as catalyst supports in refinery processes such as petroleum hydrotreating. The mass transfer properties are very important for the activity of the catalyst [1]: the diffusion of the mole­ cules and especially heavy liquid petroleum fractions in the pore network of alumina is the limiting step in the processes and it is often necessary to adapt the pore organization to increase the efficiency. The tortuosity of a porous material is a useful parameter to evaluate a-priori the influence of pore morphology on the easiness of transport. It can be

measured by diffusion or electrical experiments. One of the ways to measure the effective diffusion coefficient (Deff ) in a column is the analysis of band broadening at very slow elution speed or at zero flow rate by keeping the eluent for a given time in a column, which is depicted as arrested elution or peak parking (PP) method initially re­ ported by Knox [2–4]. The peak parking method was invented by Gid­ dings and Knox during a taxi ride in 1962 [5]. In this method, the molecule is arrested in the column far enough from the extremities by stopping the pump during a given time called the parking time (tp ) where the molecule can freely diffuse through the porous media. The pump is then restarted and the dispersion of the molecule during this time can be evaluated from the peak spreading increase as compared with the one without parking time. The effective diffusion coefficient (Deff ) is obtained from the Einstein’s diffusion equation by measuring the variance of the peak in unit length (Δσ 2z ) for a given parking time tp

* Corresponding author. E-mail address: [email protected] (V. Wernert). https://doi.org/10.1016/j.micromeso.2019.109776 Received 19 August 2018; Received in revised form 25 September 2019; Accepted 30 September 2019 Available online 1 October 2019 1387-1811/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Khac Long Nguyen, Microporous and Mesoporous Materials, https://doi.org/10.1016/j.micromeso.2019.109776

K.L. Nguyen et al.

Microporous and Mesoporous Materials xxx (xxxx) xxx

[4,6]:

measured by NMR [16,17] or by electrical conductivity [13,18–20] using probes having a size negligible as compared to pore size. The intraparticle diffusion or tortuosity could also be estimated by physical reconstruction-pore scale simulations [21,22] or by the combination of PFG NMR and Quasi-Elastic Neutron Scattering (QENS) diffusion mea­ surements with molecular dynamics and Monte Carlo simulations [23]. For other sizes, it is reasonable to speak of “apparent tortuosity”. The second point concerns the intraparticle porosity which is used in the calculation of the intraparticle diffusion coefficient. In the PP method the molecule is already in the porosity when the pump is stopped, and the diffusion is thus only hindered by tortuosity and friction with the pore walls and not by accessibility. Equation (2) is thus questionable. In order to clarify those points, the effective diffusion of polystyrenes of different sizes has been determined in non-adsorbing conditions through commercial columns filled with silica and alumina by using the PP method. For silica, the columns are composed of (i) totally porous par­ ticles, (ii) core-shell particles or (iii) monoliths. For alumina only totally porous particles are used. Three tortuosities are considered: (i) the total, (ii) the external, corresponding to macropores, and (iii) the internal, corresponding to mesopores. They are determined from the effective diffusion coefficients obtained for toluene and a series of polystyrenes by the PP method and compared to the values obtained by electrical mea­ surements in order to validate the model used in the calculation of the intraparticle diffusion coefficient by EMT models. The effect of the size of the molecule on the total and intraparticle apparent tortuosities is determined by PP method. In a first part the used theories are described, then the experimental methods are presented followed by the experimental results.

(1)

2 z

Δσ ¼ 2Deff tp

To determine the intraparticle diffusion coefficient from the global effective diffusion coefficient obtained from equation (1) it is possible to use models such as Effective Medium Theory (EMT) models. EMT ex­ pressions can be found in the literature about the effective electrical and thermal conductivities of packings of spheres and monoliths. The pre­ dictions of the EMT models are better than those obtained with the residence time weighted (RTW) expression because this latter model is based on the additivity of the mass fluxes in each phase. Among EMT models, there are implicit and explicit models. The Landauer-Davis model considers the solid skeleton as a set of microscopic in­ homogeneities dispersed randomly in the bulk mobile phase, however, this implicit model fails to describe diffusion results [7]. Among the explicit models, the Maxwell model is the simplest and is sufficient to describe the effective diffusion in non-adsorbing [7] and adsorbing conditions [8,9], despite the fact that this model is valid only for diluted phases [10]. Models like the Torquato model [44] is a higher order variant of the Maxwell model. When the Maxwell model describes correctly the experimental results it is not necessary to use higher order models as presented in Ref. [7]. The Maxwell equation is widely accepted in the field of chemical engineering to represent the diffusion in packed bed columns. The Maxwell model can be extended to the core-shell particles [8,9,11] and monoliths [8,9,12]. It has also been used to determine the particle tortuosity of spherical particles by elec­ trical measurements using the suspension dilution method for which the model is rigorously applicable [13]. In this study, the Maxwell equation has been used to determine the intraparticle diffusion coefficient (Deff p ) from a peak parking measurement and the particle tortuosity by elec­ trical measurements using the suspension dilution method. In non-adsorbing conditions, the intraparticle diffusion coefficient of molecules, Deff p , whose size is not negligible as compared to pore size, depends on the accessible porosity (εp ½rm �), the friction between the molecule and the pore wall (kf ½rm �), which has to be taken into account when the size of the molecule rm increases, and the tortuosity (τp ) or the obstruction factor (γp ¼ 1=τp ): � Deff εp ½rm � kf ½rm � p ½rm � ¼ (2) Dm τp

2. Theory The Maxwell model is used in this study to determine the intra­ particle diffusion coefficient of polystyrenes of different sizes, starting from total diffusion coefficient in the column. The Maxwell model is also used for the determination of the tortuosity by electrical measurements when the suspension dilution method is used. 2.1. Maxwell model for spherical particles 2.1.1. Fully porous particles The Maxwell expression for the effective electrical conductivity (σeff ) of a suspension of particles is written as [13,18].

The friction coefficient can be calculated by using the Renkin equation given by Ref. [14]: kf ½rm � ¼ 1

2:104λ þ 2:09λ3

0:95λ5

σ eff 1 þ 2βð1 εe Þ εt ¼ ¼ σ m 1 βð1 εe Þ τt

(3)

where λ is the ratio between tracer size rm and pore size rp . For conductivity experiments kf ½rm � is close to 1 because the elec­ trolyte is very small as compared to pore size for systems studied here and this term is generally not considered for the calculation of tortuosity by electrical measurements. The intraparticle porosity accessible to a molecule of size rm ðεp ½rm �Þ may be measured by Inverse Size Exclusion Chromatography (ISEC) by using molecules of different sizes in nonadsorbing conditions or may be estimated by considering a spherical molecule in a cylindrical pore and by knowing the particle porosity (εp Þ [14]:

εp ½rm � ¼ εp ð1

λÞ2

(5)

where σ m is the conductivity of the medium surrounding the particles, εe and εt are the external and total porosities and τt is the total tortuosity. The parameter β depends only on the particle conductivity σp : σp

β ¼ σσmp

1

σm þ 2

¼

εp τp εp τp þ

1 2

(6)

where εp is the particle porosity and τp the particle tortuosity. The Maxwell model is also used to determine the effective diffusion by replacing the electrical conductivity by diffusion coefficient [8]. The equivalence between conductivity and diffusion is [8]:

(4)

σ eff � εt ½rm �Deff

where εp is the particle porosity which can be measured experimentally by nitrogen adsorption, for example. In the model given in eq. (2) two points deserve further scrutiny. The first is about the value of the tortuosity, which is often considered as a constant property of the porous network with values ranging between 1.4 and 2 whatever the size of the molecule. Models with variable tor­ tuosity should also be considered [15]. The value of tortuosity is mainly

(7)

and thus Deff 1 1 þ 2βð1 εe Þ 1 ¼ : ¼ Dm εt ½rm � 1 βð1 εe Þ τt

(8)

where εt ½rm � is the total porosity accessible to the diffusing molecule of size rm. When the size of the molecule is not negligible as compared to 2

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Microporous and Mesoporous Materials xxx (xxxx) xxx

pore size, the tortuosity τt in equation (8) is the total apparent tortuosity of the column. The parameter β is: β¼

Ω 1 Ωþ2

values by the PP method. The value of β can be calculated for spheres and cylinders by using equations (8) and (15) respectively: 2 30 1

(9)

β¼

where Ω¼

εp :Deff p ½rm � Dm

¼

εp ½rm �kf ½rm � τp

β¼

(11)

1 3

(14)

3.3. Columns

2.2. Maxwell model for monoliths

The columns were composed of fully porous particles made of silica (Lichrospher Si100, Merck) or alumina (Chromegasphere Alumina, ES Industries), core-shell silica particles (Poroshell 120, Agilent) and monolithic silica (Chromolith, Merck). The main characteristics of the columns are given in Table 1.

The Maxwell model has also been established for cylinder packings [8,12] and it is assumed this is a good representation of the skeleton of monoliths. The expression is: Deff 1 1 þ βð1 ¼ : Dm εt ½rm � 1 βð1

εe Þ 1 ¼ εe Þ τt

(15)

3.4. Taylor dispersion analysis (TDA) and Dynamic Light Scattering (DLS) measurements

and β¼

Ω 1 Ωþ1

(20)

The experiments were made by using the 1200 HPLC system (Agilent Technologies), including a quaternary gradient pump with a multi-diode array UV-VIS detector, an automatic sample injector with a 100 μL loop, an autosampler and a thermoregulated column compartment. The in­ jection volume was set at 1 μL and all experiments were conducted at 298K. The system is controlled by the Chemstation software.

(13)

2

σpz

! cylinders

3.2. HPLC system

Equation (13) is obtained from the EMT models and a similar equation may be applied to conductivity: 1 þ ρ2

1 þ β εp :Dp ½rm � εp ½rm �kf ½rm � ¼ ¼ 1 β Dm τp

Tetrahydrofuran (THF) used as a mobile phase was purchased from Carlo Erba Reagents (SDS). Toluene was purchased from Aldrich. Twelve polystyrene standards with molecular weights Mw ranging be­ tween 162 and 1,850,000 g mol 1 were provided from Polymer Stan­ dards Service (Mainz, Germany). Samples of toluene and polystyrenes were dissolved in the mobile phase (THF) at a concentration of 1 gL 1. Detection of the polymers was done at 262 nm by UV-VIS spectroscopy.

and Deff pz ½rm � is [8]:

σp ¼

(18)

3.1. Chemicals

To take into account the effect of solid core it is also important to distinguish between Dpz and Dp . Dpz is the diffusion in the meso-porous zone (entire particle in the case of fully porous particles, shell layer in case of a core-shell particle). For spheres, the relation between Deff p ½rm �

1þ

17 B C 7 BcylindersC A þ15 @

3. Experimental

dcore dp ’ ).

Deff pz ½rm �

1

For core-shell columns Deff pz ½rm � is calculated by using equations (12) and (13).

where ρ is the ratio between the core diameter (dcore ) and the particle

ρ3

6ε 6 εe 4ε

eff

Ω¼

2.1.2. Core-shell particles The effect of the solid core has to be taken into account in the models. The porosity of the porous zone εpz and the apparent particle porosity εp are related by Ref. [8]: � εp ¼ 1 ρ3 εpz (12)

1

1 1

30 Deff t½rm � Dm Deff t½rm � Dm

(17)

By knowing β, the value of Ω can be calculated by using equations (9) and (16): ! eff 1 þ 2β εp :Dp ½rm � εp ½rm �kf ½rm � Ω¼ spheres (19) ¼ ¼ Dm 1 β τp

In the PP method the intraparticle diffusion coefficient is only hin­ dered by the friction of the molecule with the pore walls and by the particle tortuosity. The intraparticle diffusion coefficient is not hindered by particle porosity because the molecule is already present in the particle porosity when the PP experiment starts. εp :Deff p ½rm � is the intra­ particle diffusion coefficient taking into account the particle accessibility.

Deff p ½rm � ¼

D

6 εt½rm � Deffm 1 7 B C 6 7 BspheresC A εe 4εt ½rm � DDeff þ 2 5 @ m 2

Ω is the ratio of to the molecular diffusion coefficient Dm. The relationship between particle conductivity and intraparticle diffusion being:

diameter (dp ’) (ρ ¼

1

(10)

εp :Deff p ½rm �

σ p � εp Dp

1

The molecular diffusion coefficients of toluene and of the smallest polymers (P01–P03) were determined by TDA measurements. The TDA experiments were done with a stainless-steel tube (0.876 mm i.d., length 1.20 m) three times at three different flow rates. The molecular diffusion coefficient Dm is calculated with

(16)

Ω is calculated by equation (10). eff 2.3. Determination of Deff p ½rm � and Dpz ½rm � from the effective diffusion coefficients (Deff) obtained by the PP method

Dm ¼

The values of Deff p ½rm � ​ could be obtained by measuring the Deff

R2c ðtR 24ðσ2R

ti Þ

σ2i Þ

(21)

were Rc is the capillary internal radius, tR is the mean retention time and 3

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Table 1 Geometrical characteristics of the considered particle types, including the column’s dimension (length x internal diameter (I.D.)), particle diameter d’p, the ratio ρ between the diameter of the core dc and the diameter of the particle and the mean pore diameter (dp) (data taken from manufacturers documentation).

Totally porous particles Core-shell particles Monoliths

Samples

Suppliers

Support

Column’s dimension length (mm) * I.D. (mm)

Particle diameter d’p (μm)

ρ ¼ dcore/ d’p

Pore size dp (nm)

Lichrospher Si 100 Chromegasphere Alumina Poroshell 120 Chromolith Si

Merck ESI

silica alumina

250x4 150x4.6

5 5

0 0

10 13

Agilent Merck

silica silica

150x4.6 100x4.6

4 skeleton size: 1 μm

0.625 0

12 13

where Vc is the geometrical volume of the column. The external porosities (εe) were derived by extrapolating to zero hydrodynamic molecular radius the branches of ISEC plots corre­ sponding to the accessible porosity of excluded polymers versus the cube root of the molecular weight [24]. This corresponds to the contribution of the excluded volume of the finite-size tracers in the interparticle void space. The accessible particle porosity εpz[rm] to a probe of radius rm is estimated by the following equation [25]:

σ 2R is the variance of the peak with capillary and ti and σ2i correspond to

the retention time and the variance obtained without capillary. This equation is valid as long as two conditions are fulfilled: the dimen­ sionless residence time should be higher than 1.4 and the Peclet number should be higher than 70. Those conditions are verified here. The molecular diffusion coefficients of the polymers having a size larger than 1 nm (P04–P12) were obtained previously in THF by Dy­ namic Light Scattering (DLS) experiments [15]. The hydrodynamic radius of the polymers (rm) is then calculated by using the Stokes-Einstein equation. The size (rm), molecular weight, index of polydispersity (PDI) and molecular diffusion coefficient (Dm) of toluene and of the polystyrenes from P01 to P12 are given in Table 2.

εp ½rm � ¼

The porous properties and the mean pore size of the solids were determined by ISEC measurements using toluene and the polystyrene standards in THF (non-adsorbing conditions) at a flow rate of 0.5 ml min 1. The chromatograms were fitted with a Gaussian function in order to evaluate the mean retention time (tR). The total accessible volume Vt[rm] for a molecule of a given size is calculated from the mean retention time of this molecule multiplied by the flow rate. Preliminary experiments are done without column in order to evaluate the volume of the capillaries (Vi). The value of εt[rm] is then derived from: Vt ½rm � Vi Vc

(22)

Kd ¼ Table 2 Molecular weights (Mw), polydispersity (PDI), bulk diffusion coefficient Dm ob­ tained by TDA for the smallest polymers (Toluene, P01, P02 and P03) and by DLS for the other polymers (P04 to P12) of the solutes used in ISEC and hy­ drodynamic radii rm calculated with Stokes Einstein equation. Polymer code

Molecular weight M(1) w (g mol 1)

toluene P01 P02 P03 P04 P05 P06 P07

92 162 690 1380 3250 8900 19100 33500

P08 P09 P10

PDI(1)

Molecular diffusion coefficient Dm (TDA and DLS measurements) (m2 s 1)

Probe radius rm (nm)

1.00 1.09 1.05 1.05 1.03 1.03 1.03

(2.35 � 0.11).10 9 (1.85 � 0.11).10 9 (7.09 � 0.20).10 10 (5.16 � 0.21).10 10 (3.21 � 0.28).10 10 (2) (2.036 � 0.005).10 10 (2) (1.327 � 0.006).10 10 (2) (8.520 � 0.005).10 11 (2)

0.205 � 0.009 0.26 � 0.02 0.68 � 0.02 0.93 � 0.04 1.50 � 0.13 (2) 2.36 � 0.01 (2) 3.62 � 0.02 (2) 5.633 � 0.004

96000 243000

1.04 1.03

(5.083 � 0.008).10 (3.194 � 0.009).10

11 (2)

9.44 � 0.02 (2) 15.02 � 0.04

546000

1.02

(2.101 � 0.009).10

11 (2) 11 (2)

P11

827000

1.08

(1.650 � 0.008).10

P12

1850000

1.05

(1.12 � 0.01).10

11 (2)

εt ½rm � εe εe Þð1 ρ3 Þ

(23)

where εe is the external porosity and ρ is the ratio of the solid core to the shell particle diameter. ρ ¼ 0 for totally porous particles and monolith columns and ρ ¼ 0.625 for the core-shell column used in this study. The determination of the pore size distribution (PSD) of mesopore zone is described in detail in Ref. [26]. Briefly the pore size distribution (PSD) is calculated by fitting the experimental distribution coefficient Kd with the theoretical one. Kd represents the extent of permeation into the pore volume of the solid phase. The theoretical Kd represents the ratio of accessible pore volume to total pore volume. The fractional accessible volume of a sphere in a cylinder is (1-rm/r)2 because the probe molecule has a finite size rm and a part of the pore adjacent to the wall is inac­ cessible to the center of the probe. Although this partitioning model is idealized, it provides a reasonable measurement of the pore size. The experimental Kd is calculated by:

3.5. ISEC experiments

εt ½rm � ¼

ð1

εt ½rm � εe εt εe

(24)

where εt is the total porosity obtained with the smallest molecule, here toluene. Kd varies between 0 for a totally excluded molecule to 1 for molecule which have access to the total pore volume (here toluene). The theoretical Kd is given by the following equation where f(r) is the poresize distribution function and f(r)d(r) represents the pore volume which has a radius between r and r þ dr [26,27]: R∞ f ðrÞ½1 rm =r�2 dr Kd ¼ r m R ∞ (25) f ðrÞdr 0 The Solver program in Excel was used to fit the model with the experimental Kd values to obtain f(r). The mean pore size is determined by using the following equation: R∞ rf ðrÞdr r ¼ R0∞ (26) f ðrÞdr 0

(2)

All the results are given in Table 3.

(2)

22.84 � 0.10

3.6. Characterization of the solids by mercury porosimetry and nitrogen adsorption

(2)

11 (2)

29.09 � 0.15 (2)

After the experiments, the materials were retrieved from the columns to be characterized by mercury intrusion porosimetry and nitrogen adsorption. Mercury porosimetry experiments were carried out with the

42.90 � 0.40 (2)

(1) Given by supplier (2) Wernert et al. [15]. 4

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Microporous and Mesoporous Materials xxx (xxxx) xxx

Table 3 Structural properties of the columns including the total porosity εt, the external porosity εe, the porosity of the porous zone εpz and the pore diameter (dp) obtained by ISEC, mercury intrusion porosimetry and N2 adsorption. The macropore diameter is obtained by Hg porosimetry and the specific surface area as by N2 adsorption. Materials Lichrospher Poroshell Chromolith Chromegasphere Alumina a

ISEC

Hg porosimetry

N2 adsorption (NLDFT)

εt

εe

εpz

dp (nm)

εt

εe

εpz

dp (nm)

dmacro (μm)

εpz

dpa (nm)

as (m2 g-1)

0.80 0.67 0.92 0.73

0.38 0.42 0.71 0.44

0.68 0.57 0.72 0.51

14.2 13.3 11.7 14.8

0.77 0.62 0.87 0.71

0.36 0.37 0.63 0.38

0.64 0.53 0.65 0.52

6.5 and 17 11.8 12.0 14.9

2.21 1.05 1.77 1.54

0.73 0.68 0.68 0.60

11.68 16.09 16.09 16.69

426 136 276 111

from desorption branch.

Poremaster apparatus from Quantachrome. Intrusion and extrusion are carried out after the sample was evacuated under vacuum. The cumu­ lative intruded volume is measured as a function of intrusion pressure. This latter is transformed in pore size by applying the Laplace Washburn equation with a contact angle of 140� . Nitrogen sorption isotherms at 77 K were determined with an ASAP2010 apparatus from Micromeritics. The solid was outgassed overnight before analysis at a temperature of 120 � C and a pressure below 1 Pa. The pore size distributions were calculated by density functional theory (DFT) [28,29] using the software of Quantachrome ASiQwin. Calculation was made by non-local density functional theory (NLDFT) on the desorption branch assuming cylin­ drical pores, silica surface and using equilibrium transition as model. The structural properties of the materials are given in Table 3.

column, a conductivity measurement of a zero-volume column was also taken into account. The stainless-steel end fittings are used as electrodes. Given the inverse relationship between resistivity and conductivity, the total tortuosity of the monolith was obtained from Refs. [13,19]:

τt ¼ εt

3.8. Peak parking (PP) experiments

3.7.1. Spherical particles The tortuosity of the particles was determined by conductivity measurements as described by Refs. [13,20]. The particles were fluid­ ized in a thermostated cell at 20 � C in a sodium chloride solution at 1 mol L 1, which a high enough so that tortuosity is independent of salt concentration [20]. Impedance spectroscopy (from 1 kHz to 1 MHz) using a standard two-electrodes conductivity cell integrated in the flu­ idized bed was used to measure the electrical resistance. The amplitude of the signal was fixed to 30 mV. A ModuLab® electrochemical system (Model 2101A, Solartron Analytical, UK) controlled by a personal computer using ModuLab software was used. The porosity of the suspension was varied by adding powders by 0.1 g increments to 30 cm3 of electrolyte. At each step, the cumulative volume Vp of particles added to the solution was calculated by dividing the cumulative weight of dried material, mp (in g) by the density ρ of the material: mp

ρs

The peak parking (PP) method was used to measure the apparent effective diffusion coefficient of molecules through porous media. In the PP experiments, 1 μL of a dilute sample solution was injected at 0.5 mL min 1. The columns were eluted during the time necessary for the sample to arrive about half length of the column. Then the flow was stopped, and the molecule left to diffuse freely during a given time called the parking time tp. The flow was returned to the same flow rate and the peak variance of the solute band, σ2t , was measured. The variance of the peak is measured by fitting the chromatograms with a Gauss function. To correct for the other sources of band broadening (injector, connecting tubing) the peak variance values obtained without parking was sub­ tracted from the peak variance values obtained with parking. The variance was then plotted versus parking time and the slope is used to calculate the effective diffusion by using the following equation: � � 1 Δσ2t εe 2 2 Deff ¼ u (30) 2 tp εt ½rm �

(27)

The total porosity εt was then deduced by

εt ¼

Vi Vi þ Vp

(29)

whereas εt is the total porosity of the monolith, σeff is the effective conductivity of the electrolyte filled monolithic column and σ m is the conductivity of the NaCl solution. The total porosity could be obtained by ISEC with a small molecule (toluene in this study), by mercury porosimetry or by nitrogen adsorption using equation (23) where εp is obtained from N2 isotherms and εe from either ISEC or mercury porosimetry.

3.7. Tortuosity determination

Vp ¼

σm ​ σeff ​

where the porosities are obtained by ISEC and u is the interstitial linear velocity.

(28)

4. Results and discussion

where Vi is the initial volume of the electrolyte.

4.1. Characterization of the pore structure by mercury porosimetry, gas adsorption and ISEC

3.7.2. Monoliths A voltage-controlled AC impedance experiment was performed in a thermostated bath at 20 � C by applying an alternating voltage axially across the monolithic column filled with electrolyte solution (NaCl, 1 mol L 1) and measuring the electrical impedance of the column. The frequency of the AC signal was varied from 1 kHz to 1 MHz using a ModuLab electrochemical system (Model 2101A, Solartron Analytical, UK). From the measured impedance, the column resistance was deter­ mined with the ModuLab software. Due to the contribution of the dead volume in the monolithic col­ umn, a conductivity measurement of a blank column with the same length as that of the monolithic column was carried out. To eliminate the electrical resistance through the end-fitting of the monolithic and blank

The samples were analyzed by mercury intrusion extrusion poros­ imetry. The curves are given in Fig. 1a for all samples. The first step is due to the compaction of the bed, the second step corresponds to the filling of the macropores and the third step to the filling of the meso­ pores. In the case of spherical particles, the macropores correspond to the interparticle pores, whereas for monoliths it is the interskeleton pores. The porosities are determined from these curves: 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 by the total volume which is the sum of the volume of intruded mercury (Vt) and the volume 5

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Microporous and Mesoporous Materials xxx (xxxx) xxx

A

B

C

D

E

F

Fig. 1. a) Mercury intrusion-extrusion curves of porous and porous-shell particles and monoliths b) Pore size distribution obtained from Hg c) Nitrogen adsorptiondesorption isotherms at 77 K of porous and porous-shell particles and monoliths d) Pore size distribution determined from the desorption curve by applying NLDFT method e) Partition coefficient Kd obtained from ISEC f) Pore size distribution obtained from ISEC.

occupied by the solid (1/ρs) (ρs ¼ 2.2 g cm alumina): Vt εt ¼ Vt þ 1=ρ

3

for silica and 3.6 g cm

3

for

εe ¼ (31)

Vmacro Vt þ 1=ρ

(32) s

The particle porosity and skeleton porosity are obtained by dividing the mesoporous volume (volume of the third step) by the volume of the particle:

s

The macroporous porosity is obtained by dividing the macroporous volume (volume of the second step) by the total volume:

εp ¼

Vmeso Vmeso þ 1=

ρs

6

(33)

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Microporous and Mesoporous Materials xxx (xxxx) xxx

The porosities are given in Table 3. The external porosities obtained for the spherical particles are around 0.4 as usually found for a dense packing of spheres. It shows that after the compaction step, the com­ pacity in the mercury cell and in column are similar. The external porosity of the silica monolith is around 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 par­ ticles (0.53). The pore size distribution curves obtained from the intru­ sion curves are given in Fig. 1b. The mean pore sizes are given in Table 3. In the mesoporous domain, the PSD is sharp for the core-shell silica, monolithic silica and alumina but it is very broad in the case of Si100 silica with two peaks at 6.5 and 17 nm. The manufacturer indicates a pore diameter around 10 nm for Si100. The samples were also charac­ terized by nitrogen adsorption. Nitrogen adsorption-desorption isotherms are presented in Fig. 1c. The isotherms are of type IV as usually obtained for mesoporous sam­ ples. The pore size distribution obtained from the desorption branches are given in Fig. 1d. Again, the PSD obtained for Si100 is broader than for the other solids. The mean pore sizes obtained by gas adsorption are close to that obtained by mercury intrusion. The PSD is also derived from the distribution coefficient Kd obtained by ISEC. The Kd curves are presented in Fig. 1e and the PSD in Fig. 1f. There are less points on the PSD compared to the two other methods because only 13 probes are used and only some of them enter inside mesopores. The mean pore sizes are given in Table 3: they can be considered in good agreement with nitrogen adsorption and mercury porosimetry given the differences between methods. The total, external and particle porosities are also determined by ISEC. The total and particle porosities determined for each polymer are plotted as a function of the ratio between the size of the molecule (rm) and the mean pore size (rp) in Fig. 2a–b respectively. The total and particle porosities decrease as the molecular size increases. The total porosity is larger for the monolithic column than for the columns packed with the particles which is mainly due to the fact that the external porosity is higher for the monolithic column (0.7) than for the columns filled with spherical particles (0.4). The external porosities obtained by ISEC are comparable to the values obtained by Hg porosimetry (Table 3). The particle porosity obtained with the smallest molecule (i.e. toluene) is slightly higher for the monolithic column than for the other columns. The particle porosity values found by mercury intrusion are comparable or slightly lower than the values obtained by gas adsorption and ISEC. The difference could be explained by the presence of smaller pores (inferior to 3 nm) which cannot be filled by mercury at the highest intrusion pressure applied, around 400 MPa. The particle porosity accessible to a molecule of size rm (εp ½rm �Þ can be calculated by using equation (4) assuming a spherical molecule in a cylindrical pore by using the particle porosity (εp ) obtained by nitrogen adsorption. The plot of

εp ½rm � εp

A

¼ ð1

Si100 for rm/rp lower than 0.3 and then the difference between model and experimental results is about 25%. For core-shell and alumina col­ umns the model overestimates the accessible porosity and for rm/rp lower than 0.3 the difference is about 10%. For the monolithic column the values of

εp ½rm � εp are larger than 1 because the particle porosity obtained

from N2 isotherms is smaller than the value obtained with toluene by ISEC. The model underestimates the particle porosity by a factor of 10–20%. To conclude the assumption of a spherical molecule in a cy­ lindrical pore is too simple to describe with enough accuracy the accessible particle porosity of large polystyrenes in mesoporous silicas. The equation proposed by Reich et al., 2018 [21] � ε ½r � ( pεpm ¼ 1 2:2λ þ1:245λ2 from physical reconstruction of a monolith and random walk simulation does not better fit the experimental values for rm/rp lower than 0.3 (difference between the two models is lower than 8%) as seen in Fig. 2c. For rm/rp larger than 0.3 the fit with Reich et al. model is better but the difference between model and experimental results remains relatively high about 10–25%. Reich et al. (2019) [22] proposed another equation for core-shell silica particles 2

3

εp ½rm � εp

¼ 1

5:491λ þ 10:133λ 6:261λ . This equation underestimates the porosity accessible to a molecule of size rm even for the core-shell particles used in this study (Fig. 2c). The difference between the experimental results and the equations proposed by Reich et al. (2019) could be explained by the fact that the morphologies of the core-shell particles are different. They use core-shell particles from Advanced Materials Technology. In the reconstruction of the core-shell particles, Reich et al. (2019) noticed the presence of small pores that establish connections with larger ones. The small connecting pores block the connection when the size of the molecule increases and explain the steep decrease of the accessible porosity. For the modeling of the particle diffusion coefficients it is thus preferable to use directly the accessible porosity determined by ISEC. 4.2. Determination of particle tortuosity by conductivity measurements When using the suspension dilution method, the total tortuosity of the suspension τt is calculated from the effective conductivity of the suspension σeff and from the conductivity of the electrolyte σm by applying equation (5). The Maxwell equation (eq. (5)) applied to data at infinite dilution of particles is used to determine the particle conductivity (σp) and then the intraparticle tortuosity (τp). A rearrangement of the Maxwell equation by introducing eq. (6) gives: � � σ σ 2 þ σmp þ ð1 εe Þ 1 σmp � � (34) τt ¼ εt σ σ 2 þ σmp 2ð1 εe Þ 1 σmp

λÞ2 is given in Fig. 2c. It can be seen that the fit is good for

C

B

Fig. 2. ISEC measurements of the a) total porosities and b) particle (in the porous zone for the porous-shell column) porosities obtained for each polymer c) particle porosity normalized by particle porosity obtained from nitrogen adsorption, comparison with the Renkin model and the equations proposed by Reich et al. (2018) and Reich et al. (2019). 7

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Microporous and Mesoporous Materials xxx (xxxx) xxx

The tortuosity of the suspension is measured as a function of the porosity and modeled by equation (34) by varying the ratio σp/σ m until experimental and theoretical data are concurrent, tending towards 1 at infinite dilution. The results obtained for porous silica and alumina and core-shell particles are given in Fig. 3. The tortuosity of the particles τp can be calculated by:

τp ¼

εp σ m σp

σ p εp 1 þ β ¼ ¼ σ m τp 1 β

εp being the particle porosity obtained from nitrogen adsorption, εe is

obtained from mercury porosimetry. The intraparticle tortuosities are 1.56 and 1.62 for totally porous silica and alumina particles respectively by using the particle porosity obtained from nitrogen adsorption. The intraparticle tortuosity found for the core-shell column is higher with a value around 2.0. The p values obtained from the Weissberg equation (eq. (36)) are 1.8 and 1.2 for totally porous silica and alumina particles respectively and 2.7 for the core-shell silica. The intraskeleton tortuosity calculated for the mono­ lithic column is 1.17 with a p value about 0.43 by using the external porosity obtained by mercury porosimetry and the particle porosity derived from nitrogen adsorption. The results are given in Table 4. In Table 4 the calculation of the tortuosities by conductivity have been made by using the external porosity obtained by mercury porosimetry and the particle porosity obtained by nitrogen adsorption. The value of tortuosities depends on the value taken for the porosities which could be obtained either by ISEC (εt ; εp , εe ), mercury porosimetry (εt ; εp , εe ), or nitrogen adsorption (εp ). For example, the total tortuosity of the monolithic column varies between 1.06 and 1.13 with a mean value of 1.09 � 0.03 depending on the total porosity used. The total porosity could be obtained by ISEC, by mercury porosimetry or by nitrogen adsorption by using the external porosity obtained by ISEC or by mer­ cury porosimetry. The intraskeleton tortuosity of the monolithic column varies between 1.11 and 1.45 with a mean value of 1.3 � 0.2 depending on the porosities used. The effect of the porosities used on the estimation of parameter p is more important because the Weissberg equation is a logarithmic function. The p values varied between 0.27 and 1.37 for the monolithic column with a mean value of 0.8 � 0.5 depending on the porosities used for the calculation. The values of 1.45 for the intra­ skeleton tortuosity and 1.37 for p are obtained by using the porosities obtained by ISEC. To conclude, the p value depends strongly on the value of the porosities used for the calculation whereas the total and intraparticle tortuosities varies slightly with the porosities used. Kolitchev et al. [1] obtain particle tortuosities around 2 and 3 for columns filled with different boehmite γ-aluminas whatever the size of the molecule. They determine the intraparticle diffusion coefficient and thus the particle tortuosity from the slope of the HETP curves. They obtain higher values since they do not consider neither the friction be­ tween the molecules and the pore walls nor the accessible porosity even with the largest molecule, which is squalane (C30H62), an aliphatic compound. The alumina support they used are polydisperse due to inter and intra-aggregates porosities. Alumina nanocrystals are combined to form aggregates creating an intra-aggregate porosity. The alumina ag­ gregates are then compressed together to obtain the catalyst support creating an inter-aggregate porosity. The inter-aggregate porosity is supposed to be the limiting porous network for mass transfer. The par­ ticle tortuosity found in Ref. [1] in the inter-aggregates domain is in the range 1.6–1.9, values close to the value obtained in this study.

(35)

where εp is the particle porosity. The particle porosity is obtained from nitrogen adsorption. For the core-shell particles equation (14) must be taken into account in order to calculate the tortuosity of the porous zone. The particle tortuosity obtained by impedance spectroscopy are re­ ported in Table 4. The particle tortuosity can be also calculated by the following equation which is theoretically found for freely overlapping spheres (Weissberg) [30] and experimentally verified for nonoverlap­ ping spheres and some other shapes [31]: � (36) τp ¼ 1 p ln εp The p value depends on the pore topology with proposed values of p ¼ 0.41 [31] or p ¼ 0.49 [32] for spheres, p ¼ 0.53 for cubes and p ¼ 0.86-3.2 for plate [31]. This equation has been validated by both experiments [13,20,33] and simulations [34,35]. The justification of such approach is to consider that porous materials are often built by aggregation of small objects. The p value obtained by electrical mea­ surements are given in Table 4. The total tortuosity of the monolith measured by electrical mea­ surements is 1.13. This value could be compared to the value obtained with the PP method with the smallest molecule i.e. toluene which is 1.14. This will be discussed in the next part. The intraskeleton tortuosity is calculated by applying the Maxwell model for monoliths. The Maxwell equation given in equation (15) applied to conductivity is:

σ eff 1 þ βð1 ¼ σm 1 βð1

εe Þ εe Þ

(37)

The parameter β is thus: β¼

1 1

σeff σm εe σσeffm

1 þ1

(39)

(38)

and

4.3. Determination of effective diffusion coefficients Deff and of total and external tortuosities by PP method Examples of peak parking bands obtained with the columns for different parking times are presented in Fig. 4a–b for small and large molecules respectively. The peaks are symmetrical and broaden with increasing parking time which is due to diffusion. Similar results were obtained for all columns and polystyrenes studied. The curves with solid lines are the reference peak with no parking time. The variance of the peak is evaluated for each parking time to determine the effective diffusion coefficient. Firstly, a linear baseline correction of the whole chromatogram is performed to correct for the baseline drift of the signal. Secondly, the chromatogram is fitted with a Gauss function. The R2 are

Fig. 3. Total tortuosities of silicas and alumina as a function of the total porosity determined by electrical measurements. Fit with the Maxwell model. Internal porosities obtained from nitrogen adsorption. 8

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Microporous and Mesoporous Materials xxx (xxxx) xxx

Table 4 Tortuosities obtained by electrical measurements: total tortuosity (τt) for monolith, intraparticle or intraskeleton tortuosity (τpz) and p values. Tortuosities obtained by the PP method: total tortuosity (τt) obtained with toluene, external tortuosity (τext) obtained for molecules having a size larger than the pore size, and the intraparticle or intraskeleton tortuosity (τpz) obtained with toluene. τpz is calculated from the Maxwell model and the p values are calculated from the Weissberg equation (eq. (36)). The p value is also calculated by modelling the experimental results (Ω) (Fig. 6) with eq. (44). columns

tortuosity (conductivity)

τt

Totally porous particles core-shell particles Monoliths

silica alumina silica silica

– – – 1.13

particle or skeleton tortuosity

τpz

p

1.56 1.62 2.03 1.17

1.8 1.2 2.7 0.43

Tortuosity obtained by PP method

τt (toluene)

1.23 1.37 1.30 1.14

External tortuosity

particle or skeleton tortuosity (toluene)

p value (modelling of Ω with eq. (44))

τext

p

τpz

p

p

1.42 1.45 1.38 1.02

0.4 0.5 0.4 –

1.35 1.90 1.87 1.47

0.9 1.4 1.5 1.4

1.5 1.3 1.5 1.4

A

B

C

D

Fig. 4. a) Examples of peak shape for the peak parking method for (a) Toluene and (b) a intermediate size molecule (polymer P5) for the porous-shell silica c) Peak variance as a function of the parking time given as example for the porous-shell column (similar curves are obtained for the three other columns) d) Plots of the ratio of the experimental effective diffusivity and the molecular diffusivity (Deff/Dm) versus the ratio of molecular size and mesopore size (rm/rp). Fit with the Maxwell model using the Weissberg equation.

higher than 0.99 in all cases. The variance of the peak was calculated from the Gauss fit. The peak variance is plotted as a function of the parking time as shown in Fig. 4c. It should be noticed that all the curves are straight lines in agreement with equation (1). The slope of the line decreases as the molecular size increases which is due to slower diffusion when the size of the molecule increases. Applying equation (30), the value of effective diffusion coefficient, Deff, for each probe is determined. A comparison of the ratios of Deff/Dm as a function of rm/rp in the four columns is presented in Fig. 4d. The

relative error made on these coefficient Deff/Dm is �10% for toluene and the polymers P01–P04 and � 5% for the polymers P05–P12. The relative error is mainly due to the relative error made on the measurements of Dm (�5% for TDA measurements and less than 0.5% for DLS) and on the relative error made on the determination of the slope of the experi­ mental plot of the peak variance versus the peak parking time (�5%). The ratio Deff/Dm decreases as the molecular size increases until rm/rp around 0.3 due to slower diffusion in the particles then the Deff/Dm in­ creases until a constant value which corresponds to diffusion mainly in 9

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Microporous and Mesoporous Materials xxx (xxxx) xxx

macropores. It can be observed that this value is almost the same for the three columns made of spherical particles: the intersphere porosity is the same and the size of the polymers is small as compare to macropore size. The effective diffusion in the silica monolith column is faster than that in column packed with spherical particles which could be explained by a higher external porosity leading to higher diffusion in the macroporosity [36]. In fact, the external porosity of the monolithic column is around 0.7 against 0.4 for the columns packed with spherical particles. The apparent total tortuosities of the silicas and alumina obtained from equation (8) are plotted in Fig. 5. It appears that the apparent tortuosity is smaller for the monolith than for the spherical particles. The apparent tortuosity, which is a combination of the external and intraparticle or intraskeleton tortuosities, increases as the molecular size increases until rm/rp ¼ 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 is 1.02 for the monolith column indicating that the transport in this column is very efficient in the macropores. The external tortuosity is around 1.4 for the columns packed with spherical particles (core-shell and totally porous silica and alumina). The results are given in Table 4. The p values are then calculated by the Weissberg equation using the external porosity ob­ tained from ISEC. For packing of spheres p ¼ 0.4-0.5 is in agreement with discussion above and literature [31]. For a packing of spheres tortuosity values ranging between 1.3 and 1.6 are usually obtained [34]. The tortuosity in the macropores can be calculated from the Maxwell model by applying Dp/Dm ¼ 0. This gives:

τext ¼

εe

3 2

by PP method. Hormann et al. (2016) [38] determined the tortuosity of the macropores in silica monoliths by using medial axis analysis (MAA) and a geodesic distance propagation method after reconstruction of the macropores by confocal laser scanning microscopy. The global geometrical tortuosity found by Ref. [38] in the macropore is 1.09 for the silica monoliths (propagation method) and the branch tortuosity is 1.18 (MAA). The branch tortuosity is larger because the medial axis path is longer than the geodesic distance. The diffusive tortuosity obtained by random walk is about 1.47. This value is larger than the geometrical tortuosity due to deviation from the ideal route by Brownian motion and the effect of constriction in the diffusive path. The low tortuosity value obtained for monolith reflects an open macropore space that provides little obstruction to percolation. The Maxwell model overestimated the external tortuosity in the case of monoliths but calculated values are in agreement with the experimental values in the case of a packing of spherical particles. The Deff/Dm and the apparent tortuosity values can be calculated as reported in Figs. 4d and 5 by using the Maxwell equa­ tion (equations (8) and (15)). This will be presented in the next part. 4.4. Determination of the intraparticle diffusion coefficient and intraparticle tortuosity by the PP method The Ω ¼

εe

Dm

values in the porous zone are calculated from the

experimental Deff/Dm values and by using the Maxwell model to separate

the external contribution from the internal contribution. The Ω ¼ εp :Deff pz ½rm �

(40)

Dm

values are reported as a function of the ratio between the size of

the molecule and the pore size (λ ¼ rm/rp) for the four columns in Fig. 6 a-d. The intraparticle diffusion coefficient decreases as the size of the molecule increases with values close to zero for λ around 0.3. In the case of the smallest molecules, comparable diffusion coefficients are obtained for the monolithic column and the totally porous silica particles col­ umns. Lower values are obtained for the core-shell particle and totally porous alumina particle columns. For intermediate sized molecules the intraparticle diffusion is slightly higher for the monolithic column compared to the columns filled with spherical particles. The diffusion in the mesoporous zone has been determined by Hlushkou et al. (2017) [39] by reconstructing a macroporous-mesoporous silica monolith. The void space of the amorphous mesoporous silica from the monolithic skeleton was physically reconstructed by scanning transmission electron microscopy (STEM) tomography. The effective particle diffusion coef­ ficient 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 obtained here by the PP method. For the smallest tracer they obtain values of Ω around 0.6, values which are close to the values obtained with the smallest probe i.e. toluene in this study. They also found a decrease of Ω to values close to zero for λ close to 0.3. Maier and Schure (2018) [40] determined the intraparticle effective diffusion in wide-pore superficially porous particles using pore-scale simulation. They found higher values of intraparticle diffusion coefficient around 0.6 for Ω when the size of the molecule is negligible as compared to the pore size which could be explained by the fact that their material has a pore radius around 50 nm and is probably less tortuous than the core-shell silica used in this study. In the work of Maier and Schure the model particles were generated by considering a solid spherical core surrounded by a porous shell comprising a random packing of spherical sol particles. The Poroshell 120 particles used in this study are prepared through a one-step coating process called the “coacervation method”. The core-shell particles were then strengthened by sintering at higher temperature. The intraparticle tortuosity is lower for the simulated bed compared to the synthesized Poroshell particles which could be explained by the fact that the synthesized particles are sintered. The intraparticle diffusion coefficient calculated from the PP exper­

for spheres, and:

τext ¼ 2

εp :Deff pz ½rm �

(41)

for cylinders. By using these equations the external tortuosities are around 1.3 for all the columns. This value is close to the values obtained for columns packed with spherical particles but for the monolithic column the external tortuosity obtained by PP method is smaller about 1.02. Johnson et al. (2017) [37] determined the tortuosity in the macropores of columns packed with ceramic and cellulose by X-ray computed to­ mography systems and values around 1.40 and 1.79 were obtained respectively. Those values are close to the values obtained in this study

Fig. 5. Apparent tortuosity of the solids obtained by PP method as a function of the ratio between the size of the molecule and the pore size (rm/rp). Fit with the Maxwell model using the Weissberg equation for the calculation of the intra­ particle/intraskeleton tortuosity. 10

K.L. Nguyen et al.

Microporous and Mesoporous Materials xxx (xxxx) xxx

A

B

C

D

Fig. 6. Plots of the ratio of the intraparticle diffusion coefficient and the molecular diffusivity (epz.Dpz/Dm) versus the ratio of molecular size and mesopore size (rm/ rp) for a) the porous silica column, b) the core-shell column, c) the monolithic silica column and d) the alumina column. Fit with the model proposed for the calculation of the intraparticle coefficient by using the particle porosity obtained by ISEC, a constant value for the intraparticle/intraskeleton tortuosity (values obtained by conductivity), a variable value for the intraparticle/intraskeleton tortuosity by using the Weissberg equation, p being obtained from electrical mea­ surements or by fitting the experimental results with eq. (44). The results are compared to the equations proposed by Reich et al. (2018) (eq. (45)) for monolithic silica and fully porous silica and alumina and by Reich et al. (2019) (eq. (46)) for core-shell silica particles.

iments and the Maxwell model are compared to models (see Fig. 6 a-d). The model usually used to estimate the intraparticle diffusion coefficient is given by: Ω¼

εpz :Deff pz ½rm � Dm

¼

εpz ½rm �kf ½rm � εpz ð1 ¼ τpz

λÞ2 ð1

2:104λ þ 2:09λ3

τpz

accessible porosity decreases but the molecule sees the same particle morphology leading to a constant p value in the Weissberg equation whatever the size of the molecule. The Weissberg equation is thus used and extended to the accessible intraparticle porosity [7]: � τpz ½rm � ¼ 1 p ln εpz ½rm � (43)

0:95λ5 Þ (42)

Equation (42) becomes thus:

where λ is the molecule to pore size ratio, the term depending on λ is the Renkin equation, εpz is the particle porosity in the porous zone (for totally porous particles εp ¼ εpz) and τpz is the particle tortuosity in the porous zone (τp ¼ τpz for totally porous particles). As seen in Fig. 2c the fit of the experimental particle porosity ðεpz ½rm �Þ with the model

Ω¼

εpz :Deff pz ½rm � Dm

¼

εpz ½rm �kf ½rm � εpz ½rm �ð1 2:104λ þ 2:09λ3 � ¼ τpz ½rm � 1 pln εpz ½rm �

0:95λ5 Þ

(44)

In this equation there is just one adjustable parameter p. The experimental values of Ω are fitted with equation (44) by using the porosity obtained by ISEC and the p values obtained by electrical mea­ surements. The fit is good for totally porous particles by taking a p value of 1.8 for silica and 1.2 for alumina. For the core-shell and the mono­ lithic column the fit is not so good by using the p value obtained by

λÞ2 is not so good because the assumption of a spherical molecule in a cylindrical pore is too simple as explained in part 4.1. Thus, for the calculation, the particle porosity accessible to a molecule of a given size obtained by ISEC ðεpz ½rm �Þ is used. The particle tortuosity is obtained by electrical measurements. As seen in Fig. 6 a-d the fit is good for coreshell columns but for totally porous particles (silica and alumina) and monolith columns the model with constant tortuosity overestimates the experimental values. The particle tortuosity is also evaluated by using the Weissberg equation. The Weissberg equation assumes that the material can be seen as an assembly of objects. When the size of the molecule increases the

εpz ð1

electrical measurements. The Ω ¼

εpz :Deff pz ½rm � Dm

values for the core-shell and

the monolithic columns are fitted with equation (44) by using the solver of Excel to obtain the p value. The p value obtained with the solver is 1.5 for the core-shell column and 1.4 for the monolithic column. This value is close to the p value obtained for totally porous silica particles. The value of p of 1.4 has been obtained from electrical measurements for the monolithic column by using the porosities obtained by ISEC as presented

11

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Microporous and Mesoporous Materials xxx (xxxx) xxx

in part 4.2. By using the solver, the p values obtained are 1.5 and 1.3 for totally porous silica and alumina particles respectively, those values are close to the values obtained by electrical measurements. The p values obtained with the solver are given in Table 4, the mean p value is 1.4 � 0.1 for all the columns studied. Other equations based on numerical simulation using physical reconstruction of the materials was proposed by Reich et al. [21,22]. They use pore-scale simulations of the diffusion of tracers of different sizes by reconstructing physically the macro-mesoporous silica by using electron tomography. They proposed the following equation which could be applied for mesoporous silica (monoliths or fully porous par­ ticles) obtained through sol-gel processing [21]:

Ω¼

εpz :Deff p ½rm � Dm

¼

εpz ð1

3:416λ þ 3:338λ2

5:433λ3 þ 24:063λ4

the particle tortuosity obtained by PP method will be presented. The model presented in Fig. 5 to fit the experimental apparent total tortu­ osity is the inverse of the Deff/Dm values obtained from the Maxwell model presented in Fig. 4d. There is a good agreement between the Maxwell model and the experimental values. 4.5. Determination of the intraparticle tortuosity by the PP method In the PP method the molecule is already present in the intraparticle porosity so the intraparticle porosity has not to be considered in the calculation of the intraparticle diffusion coefficient. The same reason is invoked to extract tortuosity from NMR measurements [41]. Thus, the

43:298λ5 þ 33:022λ6

εpz :Deff p ½rm � Dm

¼

(45)

τpz

intraparticle tortuosity should be obtained from:

where εpz and τpz are the particle porosity and the intraparticle tortuosity obtained for small tracers in the porous zone. For core-shell silica particles the equation proposed by Reich et al. [22] is:

Ω¼

9:282λ7 Þ

εpz ð1

6:835λ

22:9761λ2 þ 412:272λ3

τpz ½rm � ¼

1946:54λ4 þ 4794:21λ5

τpz

kf ½rm �Dm Dpz ½rm �

6714:55λ6 þ 5084:37λ7

(47)

1623:08λ8 Þ

(46)

By using equation (47) the intraparticle tortuosity obtained for toluene is 1.4 and 1.9 for totally porous silica and alumina particles respectively and 1.9 for core-shell particles. Those values are close to the electrical measurements where 1.6 for silica and alumina porous parti­ cles and 2.0 for core-shell particles are obtained. The intraparticle tor­ tuosity in the mesoporous zone is reported in Fig. 7 as a function of intraparticle porosity for the four columns. For rm/rp > 0.3 the tortuosity of spherical particles becomes very high (data not presented) because the particle diffusion becomes close to zero. The total tortuosity is thus dominated by the external tortuosity. This could explain the minimum

Those equations are applied here in Fig. 6 with εpz obtained from nitrogen adsorption and the intraparticle tortuosity obtained from electrical measurements (values given in Table 4). For alumina equation (45) was used. Fig. 6a-d reveal differences between the experimental results and the expressions proposed by Reich et al. (2018, 2019). This is probably due to differences in the morphology of the materials used in their work and in the present paper: they are not elaborated by the same procedures. The two different approaches are complementary and give different results for different materials and, ideally, should provide close results for the same materials as it is observed for example with the monolith from Merck. For core-shell particles (see Fig. 6b), the steep decrease of intraparticle diffusion with λ predicted by eq. (46) could be explained by the fact they use core-shell particles having unusual small pores connected to larger ones and which are blocked when the size of the molecule increases. Equation (46) is only valid for core-shell parti­ cles having the same morphology which is not the case in this study. To conclude the model which could be used for all the columns to model the intraparticle diffusion coefficient is the model with the intraparticle tortuosity calculated with the Weissberg equation with one adjustable parameter p and by using the porosities obtained by ISEC. The Deff/Dm values obtained by the PP method (Fig. 4 d) can be modeled with the Maxwell equation by estimating the intraparticle diffusion coefficient with equation (44). The intraparticle diffusion co­ efficient is calculated by using the accessible particle porosity obtained by ISEC for each polymer, the friction coefficient calculated by the Renkin equation and the intraparticle tortuosity calculated with the Weissberg equation by taking the p values obtained by taking into ac­ count all the polymers (see values Table 4). A good agreement is found for the four columns. For the monolithic column there is a good agree­ ment until rm/rp ¼ 0.6 whereas for higher values the external tortuosity is overestimated by the Maxwell model. In the next part the evolution of the intraparticle diffusion coefficient with the size of the molecule and

Fig. 7. Intraparticle tortuosity obtained from the PP method as a function of the particle porosity obtained by ISEC. Comparison with the Weissberg model with p ¼ 1.4. 12

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observed in the Deff/Dm curves around rm/rp ¼ 0.3. Often the tortuosity is taken as a constant value whatever the size of the molecule. From Fig. 7 we can see that the apparent particle tortuosity depends on accessible porosity. When the size of the molecule increases, the accessible porosity decreases and the intraparticle tortuosity increases. Such behavior has also been observed by Richard and Striegel (2010) [42]. Beckert et al. (2010) [43] measured the molecular self-diffusion of polystyrenes so­ lutions in porous acrylate-based monoliths by pulsed field gradient (PFG) nuclear magnetic resonance (NMR). They found a constant tor­ tuosity around 1.5 whatever the size of the molecule, which is explained by the fact that the polystyrenes cannot enter in the mesoporosity and thus only the tortuosity in the macropores is measured in their study. The Weissberg equation with a mean value of 1.4 for p is plotted in Fig. 7 and can correctly fit the experimental results obtained by the PP method.

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5. Conclusion Transport of toluene and polystyrene compounds in non-adsorbing conditions were investigated by Peak Parking experiments on columns made of porous materials with various morphologies: fully porous silica and alumina particles, core-shell porous silica and monolithic silica. A good agreement is found between the pore size distribution and acces­ sible porosities obtained from the retention times, nitrogen desorption and mercury intrusion. The total tortuosity obtained for monolith are comparable to the values obtained by electrical measurements. The intraparticle tortuosities obtained by the PP method with the smallest probe, i.e. toluene, for spherical particles is also in agreement with electrical measurements provided the friction coefficient is taken into account. The molecule is already present in the porosity during the PP method, therefore, the accessible porosity has not to be considered in the calculation of the intraparticle diffusion coefficient. Using such data processing, transport of large molecules in column of rather different morphologies can be satisfactory predicted. The effect of the size of the molecule on the effective diffusion and on the apparent and intraparticle tortuosity is thus shown. The apparent tortuosity increases with probe size until rm/rp ¼ 0.3 and then decreases to a constant value which is the external tortuosity or the tortuosity in the macropores. The apparent tortuosity is a combination of internal and external tortuosities. This point will be studied in the future. In non-adsorbing conditions, the intraparticle diffusion coefficient can be satisfactorily estimated by calculating the intraparticle tortuosity (τp[rm]) with the Weissberg equation (τp[rm] ¼ l-pln(εp[rm])) where εp[rm] is the accessible porosity to a molecule of size rm. The Weissberg equation has just one adjustable parameter p which depends on the material topology. This is a good approximation if the exact morphology of the material is not known, since this latter knowledge needs other approaches such as the physical reconstruction of the material as proposed in the papers by Reich et al. (2018, 2019) [21,22]. Despite the good agreement between data and modeling, one must keep in mind two important points. Firstly, the Weissberg equation is adapted to solids made of an assembly of similar particles whereas most materials presented here have a more complex structure. Secondly, the hypothesis that when the size of the probe in­ creases the topology is maintained is another approximation which is probably only valid for a reduce range of probe sizes. Indeed, simula­ tions show that the topology, evaluated in term of branch and node populations, is actually modified when probe size increases [21,22]. Nevertheless, because of the reasonable agreement between data and modeling, applying the empirical Weissberg type equation could be a way to assess if the pore structure has the one expected from the syn­ thesis protocol when it uses assembly of particles with a defined shape. The higher is p, the higher is the difference between particle assembly morphology and the real morphology that depends not only on aggre­ gation, notably in sol-gel processes, but also on other phenomena such as sintering.

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