Transport under confinement: Hindrance factors for diffusion in core-shell and fully porous particles with different mesopore space morphologies

Accepted Manuscript Transport under confinement: Hindrance factors for diffusion in core-shell and fully porous particles with different mesopore space morphologies Stefan-Johannes Reich, Artur Svidrytski, Alexandra Höltzel, Wu Wang, Christian Kübel, Dzmitry Hlushkou, Ulrich Tallarek PII:

S1387-1811(19)30123-4

DOI:

https://doi.org/10.1016/j.micromeso.2019.02.036

Reference:

MICMAT 9349

To appear in:

Microporous and Mesoporous Materials

Received Date: 26 January 2019 Accepted Date: 26 February 2019

Please cite this article as: S.-J. Reich, A. Svidrytski, A. Höltzel, W. Wang, C. Kübel, D. Hlushkou, U. Tallarek, Transport under confinement: Hindrance factors for diffusion in core-shell and fully porous particles with different mesopore space morphologies, Microporous and Mesoporous Materials (2019), doi: https://doi.org/10.1016/j.micromeso.2019.02.036. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Transport under confinement: Hindrance factors for diffusion

2

in core–shell and fully porous particles with different mesopore

3

space morphologies

RI PT

1

4

Stefan-Johannes Reicha, Artur Svidrytskia, Alexandra Höltzela, Wu Wangb, Christian

6

Kübelb, Dzmitry Hlushkoua, Ulrich Tallareka,*

8

a

9

35032 Marburg, Germany

M AN U

7

SC

5

Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse 4,

10

b

11

Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

TE D

Institute of Nanotechnology and Karlsruhe Nano Micro Facility, Karlsruhe Institute of

12

____________________

14

* Corresponding author.

15

Phone: +49-(0)6421-28-25727; E-mail: [email protected] (U.T.)

AC C

EP

13

1

ACCEPTED MANUSCRIPT

Abstract. We quantify confinement effects on hindered transport in mesoporous silica

17

particles through using physical reconstructions of their mesopore space morphology

18

obtained by electron tomography as geometric models in direct diffusion simulations for

19

passive, finite-size tracers. Studied are fully porous particles with mean mesopore sizes of

20

dmeso = 16.0 and 23.9 nm, prepared by classical sol–gel processing, and solid core–porous

21

shell particles (dmeso = 9.4 and 16.8 nm) originating from a layer-by-layer assembly of sol

22

particles around a solid, impermeable core followed by thermal consolidation of the porous

23

shell. Because shell thickness and core size are independently adjustable, core–shell

24

particles allow to decouple the intraparticle diffusion distance in a fixed-bed reactor or

25

chromatographic column from the external surface area of the particles and the hydraulic

26

permeability of the bed, impossible with fully porous particles. Effective diffusivity,

27

accessible porosity, and pore network connectivity recorded in the four reconstructions as a

28

function of λ, the ratio of tracer to mean mesopore size, demonstrate an unfavorable shell

29

morphology for the core–shell particles that opposes their design advantage over fully

30

porous particles. The reconstructions reveal that core–shell particles contain an increased

31

number of narrow and constricted as well as closed pores. These structural features reflect

32

compacted and sintered packings and are most likely formed during shell consolidation.

33

The presented expressions for hindered diffusion and accessible porosity can be used to

34

optimize mesopore space morphologies in sensitive applications, e.g., catalysis under

35

confinement and controlled drug release.

36

Keywords: Brownian dynamics; Diffusion coefficient; Porosity; Closed pores; Tortuosity;

37

Pore network; Sol–gel; Layer-by-layer assembly; Electron tomography.

AC C

EP

TE D

M AN U

SC

RI PT

16

2

ACCEPTED MANUSCRIPT

38

1. Introduction Today mesoporous silica particles formed from amorphous SiO2 find numerous

40

applications as catalysts and catalyst supports, packing materials for liquid and ion-

41

exchange chromatography columns, and oral carriers of drugs, macromolecules, and cells

42

[1–9]. The various applications place different demands on particle properties, such as their

43

chemical purity, alkali metal content, mechanical strength and size, mesopore

44

characteristics and chemical surface modification, but all applications require a good

45

understanding and description of diffusive molecular transport inside the porous structures.

46

This is particularly important for multifunctional catalysts, which are designed to achieve

47

improved catalytic activity or enantioselectivity [10–13], because molecular transport under

48

confinement triggers synergystic effects between different surface functionalities

49

immobilized on the same mesopore surface.

SC

M AN U

Molecular transport inside the mesoporous silica particles depends on the geometry,

TE D

50

RI PT

39

topology, and surface modification of the pore space, on the composition and properties of

52

the solvating liquid, and on the size, shape, and charge of the diffusing solute. The sum of

53

all influences is reflected in the effective (i.e., the long-time asymptotic) diffusion

54

coefficient Deff. Depending on the conditions, Deff may represent several simultaneously

55

occuring phenomena, which are furthermore interrelated in a complex manner, such as

56

hindered diffusion in the pores and sorption and reaction at the surface [14]. The

57

interrelationships can be elucidated by a combination of sophisticated experimental

58

approaches. Pulsed field gradient nuclear magnetic resonance (PFG-NMR), for example,

59

has been intensively used to monitor effective diffusion coefficients in porous particles

60

applied in separation and catalysis [15–23]. These ensemble-tracking data can be

AC C

EP

51

3

ACCEPTED MANUSCRIPT

complemented by single-molecule tracking information from spectroscopic techniques that

62

allow to monitor the trajectories of single molecules and localize their position of sorption

63

and reaction [24–27]. It is even possible to track diffusion in a pore while imaging the pore

64

itself and thus quantify diffusion at the single-pore level [28]. By contrast, IR microimaging

65

has been implemented to determine effectiveness factors in catalyst particles in a single

66

measurement by recording the evolution of reactant and product concentration profiles,

67

which emerge from the direct interplay of diffusion and reaction [29]. The smart

68

combination of these complementary experimental approaches has moreover helped to

69

verify the validity of the ergodic theorem [30] or the applicability of Fick’s laws behind the

70

diffusion of guest molecules in porous materials [31].

SC

M AN U

71

RI PT

61

A particular aspect of the coupled transport scheme of diffusion–sorption–reaction in porous particles is the hindrance to diffusion caused by the steric and hydrodynamic

73

interactions between finite-size solutes and the confinement [32]. For passive, point-like

74

tracers that neither interact with each other nor specifically with the surface (no sorption or

75

reaction), the effective diffusive transport on a global scale (through a particle, subscript H)

76

as well as on a local scale (inside a particle, subscript K) is described by the following

77

equation [17]

EP

AC C

78

TE D

72

,

=

,

=

(1)

79

where ε0 and τ0 denote porosity and tortuosity, respectively, underlying the transport of the

80

point-like tracers (subscript 0) into and through the particles, and the diffusivity Dm

81

represents unhindered diffusion in the corresponding bulk liquid. A key objective of

82

research on hindered transport is to obtain an expression that allows to predict effective

4

ACCEPTED MANUSCRIPT

diffusion coefficients as a function of λ = dtracer/dmeso, the ratio between tracer size and

84

mean mesopore size [33]. Such an expression precisely captures the influence of solute size

85

on hindered diffusion and thus allows to calculate how much the associated intraparticle

86

mass transfer resistance contributes to the overall separation efficiency of chromatographic

87

particles with a given intraparticle morphology (ε0, τ0, dmeso) and mean size (dp) [34], or to

88

the Thiele modulus and, thus, the effectiveness factor of catalyst particles (also

89

characterized by ε0, τ0, dmeso, and dp) in a fixed-bed reactor [35].

SC

We recently introduced a three-step methodology comprising (i) physical reconstruction

M AN U

90

RI PT

83

of the mesopore space by electron tomography, (ii) morphological analysis of the

92

reconstructed mesopore space, and (iii) direct (pore-scale) numerical simulations of tracer

93

diffusion in the reconstructed mesopore space under systematic variation of the

94

fundamental parameter λ to derive expressions describing hindered diffusion in

95

mesoporous silica materials [36]. For our first investigated material, a silica monolith with

96

hierarchical pore space (i.e., with interskeleton macropores and intraskeleton mesopores),

97

we obtained an expression for Deff,K(λ) that decreased more strongly with λ than predicted

98

by the equations of Renkin [32] and Dechadilok and Deen [37]. Their equations were

99

derived for hindered diffusion of finite-size spherical particles in a single, uniform,

AC C

EP

TE D

91

100

cylindrical pore by resolving the problem of enhanced drag due to the hydrodynamic

101

Stokes-friction effect (i.e., of increased molecular friction in constrained space compared to

102

an unbounded solution). Therefore, we were not surprised that a random network of

103

heterogeneous pores with varying shape and size, such as present in the mesoporous

104

skeleton of silica monoliths, closed off at a lower λ (due to bottle-necking at smaller pore

5

ACCEPTED MANUSCRIPT

openings) than a single, straight cylinder. We next investigated a trio of commercial silica

106

monoliths from a synthesis procedure that promised a systematic variation of the mesopore

107

size while keeping all other parameters constant [38]. The samples (with dmeso = 12.3, 21.3,

108

and 25.7 nm) were indeed found to have the same general mesopore shape and topology at

109

varied mean mesopore size and porosity, which offered us a consistent set of model

110

structures for testing the scalability of hindered diffusion with respect to dmeso and λ =

111

dtracer/dmeso. This work revealed that the pore networks evolving in a given reconstruction

112

with increasing λ are not self-similar, prohibiting a prediction of their evolution by a

113

generic diffusive tortuosity‒porosity relationship. The data sets obtained for the local

114

diffusive hindrance factor Kd(λ) = Deff,K(λ)/Dm and accessible porosity ε(λ) from the three

115

reconstructed mesopore spaces collapsed over the entire range of λ-values, so that the

116

overall hindrance factor H(λ) for these silica samples could be accurately reproduced with

117

the following equation (for 0 ≤ λ ≤ 0.9) [38]

TE D

M AN U

SC

RI PT

105

λ =

=

λ ε λ

1 − 3.416λ + 3.338λ − 5.433λ" + 24.063λ% − 43.298λ' + 33.022λ( −

9.282λ)

EP

119

=

λ

(2)

AC C

118

,

120

This equation allows precise predictions about hindered diffusion using accessible values

121

for the relevant solute and material parameters λ = dtracer/dmeso, Dm, ε0, and τ0. Interestingly,

122

we found that the expression derived for Kd(λ) in that work [38] also describes hindered

123

diffusion data reported earlier for porous glasses reasonably well [39,40]. The hindrance to

124

diffusion for the three mesoporous silica samples, expressed by Eq. (2), was gentler than

6

ACCEPTED MANUSCRIPT

observed for the homemade silica monolith studied in our first paper [36]. This observation

126

underlines that the morphological and thus the transport properties of mesoporous silica

127

materials depend sensitively on subtle details of their preparation, for example, the thermal

128

treatment steps used to shape the final morphology of the intraskeleton mesopores [38]. In

129

our most recent work we investigated wide-pore ordered mesoporous silicas, specifically

130

SBA-15 and KIT-6 samples (with dmeso = 9.4 nm), which have a primary, ordered mesopore

131

system surrounded by amorphous silica walls that contain random mesopores of

132

unspecified size [41]. Unexpectedly, the primary, ordered mesopore system was found to

133

cause severe transport limitations. Our reconstruction‒simulation approach demonstrated

134

that a modest amount of structural imperfections in the primary mesopore system had

135

drastic consequences for the transport properties of these materials. Constrictions (SBA-15

136

and KIT-6) and dead-ends (SBA-15) caused diffusive transport to come to a standstill at λ

137

= 0.29, which corresponded to a relatively modest tracer size of dtracer = 2.76 nm.

138

Surprisingly, that study suggested that predicting the extent of hindered diffusion in ordered

139

mesoporous silicas is more complex than in the previously investigated random

140

mesoporous silicas.

SC

M AN U

TE D

EP

The insight gained from the study of random and ordered mesoporous silicas stimulates

AC C

141

RI PT

125

142

further research with the established three-step methodology (physical reconstruction–

143

morphological analysis–pore-scale simulations) to derive global hindrance factor

144

expressions for other types of mesoporous silica materials. Apart from the quantitative

145

prediction of hindered diffusion critical to a variety of applications, this insight is key to

146

identifing potential modifications during material preparation to reduce diffusive mass

7

ACCEPTED MANUSCRIPT

transfer resistance and possibly improve the activity and selectivity of a process. In the

148

present work, we investigate µm-sized core–shell silica particles, which have gained

149

enormous popularity as packing material for liquid chromatography columns since their

150

introduction in 2006 [42,43]. Their basic architecture (Fig. 1) consists of a solid

151

(impermeable) silica core surrounded by a mesoporous silica shell, which originates from a

152

layer-by-layer assembly of nanoparticles onto the core [44]. This unique assembly

153

technique is also critical to the engineering of nanolayered particles for biomedical

154

applications [45,46] or the production of hollow microstructures employed as catalysts

155

[47]. The layer-by-layer engineering can be described as covering a target surface with

156

alternating layers of oppositely charged building units. The core–shell particles studied in

157

this work are, for example, made according to the following procedure [44]: First, an

158

organic polyelectrolyte is bound to the solid core (cf. Fig. 1). The coated core particles are

159

then immersed in a dispersion of nanoparticles with opposite charge as the polyelectrolyte.

160

The first two steps are repeated until the desired shell thickness is reached. The particles are

161

then heated to remove the organic polyelectrolyte and fuse the mesoporous shell onto the

162

solid core. Choosing the size and density of the nanoparticles as well as the number of

163

layers deposited onto the core enables a certain control over shell properties such as

164

thickness, porosity, and pore size. The final morphological properties of the mesoporous

165

shell (and hence the performance of the core–shell particles in their targeted applications)

166

are, however, also influenced by the final arrangement of the sol particles after the layer-

167

by-layer assembly and by the consolidation processes occurring during the fusion step,

168

about which little is known.

AC C

EP

TE D

M AN U

SC

RI PT

147

8

ACCEPTED MANUSCRIPT

The rationale behind the architecture of core‒shell particles is to allow an independent

170

adjustment of the intraparticle diffusion distance and interparticle macropore size in packed

171

beds by an independent adjustment of shell thickness and core size. This decoupling of

172

intraparticle diffusion distance and interparticle macropore size is impossible with fully

173

porous particles, whose size determines both parameters. In this regard, packings of core–

174

shell silica particles challenge macro‒mesoporous silica monoliths (where intraskeleton

175

diffusion distance and interskeleton macropore size can also be varied independently),

176

enabling the fundamental decoupling of the internal diffusion distance within a fixed-bed

177

reactor or chromatographic column from the external surface area and hydraulic

178

permeability [48]. The core–shell particles reflect the so-called egg-shell particles

179

employed in catalysis [49,50], which become useful when the reaction is very fast and

180

intraparticle mass and heat transfer limitations impact activity, selectivity, and lifetime of

181

the catalyst. To understand how the preparation of the mesoporous shell in this alternative

182

particle design impacts the morphology and transport properties of the shell with respect to

183

the conventional fully porous silica particles and the mesoporous skeleton of hierarchical

184

silica monoliths based on classical sol–gel chemistry [51–54], we investigate two

185

commercially available core–shell silica materials with different dmeso and compare their

186

morphological properties and hindered diffusion characteristics with two fully mesoporous

187

silica materials prepared by classical sol–gel processing.

SC

M AN U

TE D

EP

AC C

188

RI PT

169

189

2. Experimental

190

2.1. Mesoporous silica particles

9

ACCEPTED MANUSCRIPT

191

Fused-Core® particles with an overall particle diameter of 2.7 µm, a porous-shell thickness of 0.5 µm (cf. Fig. 1), and nominal mesopore sizes of 9 and 16 nm were

193

purchased from Advanced Materials Technology (Wilmington, DE) and are further denoted

194

as samples CS9 and CS16, respectively. Fully porous Nucleosil® particles with an overall

195

particle diameter of 5 µm and nominal mesopore sizes of 10 and 30 nm came from

196

Macherey-Nagel (Düren, Germany) and are further denoted as samples FP10 and FP30,

197

respectively. All particles are bare-silica particles (i.e., with an unmodified SiO2 surface)

198

and exhibit the strength and stability required for fixed-bed operations in separation and

199

catalysis at pressures of up to at least 400 bar.

200 201 202

2.2. Nitrogen physisorption

M AN U

SC

RI PT

192

Nitrogen adsorption‒desorption isotherms were acquired at 77 K on a Thermo Scientific Surfer gas adsorption porosimeter (Thermo Fisher Scientific, Waltham, MA). Samples

204

were evacuated for 10 h at 250 °C prior to analysis. Specific surface area SBET and pore

205

volume Vpore were calculated from the nitrogen adsorption isotherms recorded up to

206

pressures of p/p0 = 0.98. Pore size distributions were derived from the adsorption branches

207

of the isotherms by applying the non-local density functional theory model with a

208

cylindrical pore geometry [55].

210 211

EP

AC C

209

TE D

203

2.3. Electron tomography To create mesoporous silica crumbs suitable for scanning transmission electron

212

microscopy (STEM) tomography, the samples were ground in a mortar. The crumbs were

213

subsequently dusted over a carbon-coated 100 × 400 mesh Cu grid (Quantifoil Micro

10

ACCEPTED MANUSCRIPT

Tools, Jena, Germany), on which Au fiducial markers (6.5 nm diameter) were deposited

215

from an aqueous suspension (CMC, University Medical Center, Utrecht, The Netherlands).

216

Electron tomography was performed using a Fischione 2020 tomography holder on an

217

image-corrected Titan 80–300 TEM (FEI, Hillsboro, OR), operated at an acceleration

218

voltage of 300 kV in STEM mode with a nominal beam diameter of 0.27 nm. STEM

219

images were collected with a high-angle annular dark-field (HAADF) detector in 2°-steps

220

over a tilt range of around ±74°. Alignment of the tilt-series images was carried out in

221

IMOD 4.7 [56] with Au fiducial markers, yielding an average residual alignment error of

222

about 0.8‒1.2 pixels for all samples.

SC

M AN U

223

RI PT

214

Subsequent 3D reconstruction was performed using the SIRT algorithm [57], implemented in the Xplore3D software package version 3.1 (FEI), with 25 iterations for

225

samples CS9 and CS16 and 50 iterations for samples FP10 and FP30. Images were

226

denoised using the nonlinear anisotropic diffusion filter implemented in IMOD. Final

227

stacks of aligned, reconstructed, denoised, and segmented images had the following

228

dimensions (x × y × z): 146.7 × 183.0 × 103.5 nm3 with 0.463 nm3 voxels (CS9); 133.4 ×

229

156.4 × 101.6 nm3 with 0.463 nm3 voxels (CS16); 109.0 × 202.8 × 92.5 nm3 with 0.463 nm3

230

voxels (FP10); and 714.1 × 740.0 × 445.8 nm3 with 1.853 nm3 voxels (FP30). The

231

reconstructions cover volumes of ~3 nL (CS9), ~2 nL (CS16, FP10), and ~235 nL (FP30).

233 234 235

EP

AC C

232

TE D

224

2.4. Morphological analysis Morphological analysis of the reconstructions comprised chord length distribution (CLD) analysis, which is a complementary approach to the bulk characterization of the four

11

ACCEPTED MANUSCRIPT

236

samples by nitrogen physisorption analysis, and medial-axis analysis, which was used to

237

characterize the evolving pore networks in a reconstruction with increasing λ [38]. For each reconstruction, up to 107 chords were collected and displayed in a histogram

239

(the CLD). From seed points randomly generated in the void space of a reconstruction, 32

240

angularly equispaced vectors were projected radially outwards until they hit the solid phase

241

(SiO2); a chord length is the sum of the absolute lengths of any two opposing pairs of

242

vectors. Chords that projected out of the image were discarded. The CLD was then fitted

243

with a k-gamma function - . 01 .23

* +, = /

-

4.

SC

0

exp 8−9 41 :

M AN U

244

RI PT

238

(3)

where lc denotes the chord length, Γ is the gamma function, µ is the first statistical moment

246

of the distribution (mean chord length), and k is a second-moment parameter defined by the

247

mean chord length and the standard deviation σ as k = (µ 2/σ2). The values for µ and k

248

obtained from the k-gamma fit to the CLDs are quantitative measures for the average pore

249

size and the homogeneity of the pore volume distribution, respectively [58].

TE D

245

Medial-axis analysis was employed by using an iterative-thinning algorithm, available as

251

ImageJ plug-in bundle [59] (Skeletonize3D and AnalyzeSkeleton), to reduce the accessible

252

void space of the reconstructions for each λ-value to a medial axis of one-voxel thickness

253

under conservation of its topological properties. The average pore connectivity Z of the

254

resulting branch–node network is the average number of medial-axis branches that meet at

255

a junction and was calculated as Z = 3nt/nj + 4nq/nj + 5nx/nj (with nx/nj = 1 – nt/nj – nq/nj),

256

where nj is the total number of junctions, nt the number of triple-point junctions (connecting

257

three branches), nq the number of quadruple-point junctions (connecting four branches),

AC C

EP

250

12

ACCEPTED MANUSCRIPT

and nx the number of higher-order junctions (connecting more than four branches) [60].

259

Therefore, nt/nj, nq/nj, and nx/nj give the fraction of nodes in the network that connect three,

260

four, or more than four branches, respectively.

RI PT

258

261

263

2.5. Diffusion simulations

Details of this approach to the direct (pore-scale) simulation of hindered diffusion in

SC

262

physical reconstructions have been presented in our previous papers [36,38], which is why

265

we describe it only briefly here. Initially, a large number of N = 106 passive tracers, which

266

neither interact with each other nor specifically with the surface, were distributed randomly

267

and uniformly in the void space of a reconstruction. During each time step δt of the

268

simulation, the displacement of every tracer due to random diffusive motion was calculated

269

from a Gaussian distribution with zero mean and standard deviation (2Dmδt)1/2 along each

270

Cartesian coordinate. Passive interaction of the tracers with the pore walls was handled

271

through a multiple-rejection boundary condition at the surface, that is, when a tracer hit the

272

impermeable wall during an iteration, that displacement was rejected and recalculated until

273

the tracer position was in the void space. All tracer positions were recorded after each time

274

step to calculate a time-dependent diffusion coefficient

TE D

EP

AC C

275

M AN U

264

; =

< >

(= >?

∑= DF
(4)

276

where ΔCD ; ≡ CD ; – CD 0 denotes the displacement of the ith tracer after time t. The

277

normalized local effective diffusion coefficients Deff,K(λ)/Dm in the four reconstructions for

278

each λ-value were determined from the asymptotes of the normalized transient diffusion

279

curves D(t)/Dm observed in the long-time limit.

13

ACCEPTED MANUSCRIPT

Finite-size tracers (λ > 0) have only restricted access to the void space of a reconstruction

281

due to their steric interaction with the impermeable pore walls. The void space accessible to

282

the center of a finite-size tracer is identical to the void space that is accessible to a point-

283

like tracer when the pore diameter is reduced by the tracer size [61]. Therefore, a reduction

284

of the accessible pore space for finite-size tracers can be accounted for by eroding the pore

285

space accessible to point-like tracers with a structuring element of size dtracer. We used this

286

mathematical morphology operation to generate the accessible pore space in a

287

reconstruction for the following dtracer-values: 7 values increased in 0.92-nm steps from

288

0.92 nm to 6.44 nm for sample CS9; 11 values (0.92-nm steps) from 0.92 nm to 10.12 nm

289

for sample CS16; 15 values (0.92-nm steps) from 0.92 nm to 13.8 nm for sample FP10; and

290

10 values (3.7-nm steps) from 3.7 nm to 37.0 nm for sample FP30. The accessible porosity

291

at a given value of λ, ε(λ), was extracted in a straightforward manner as the void volume

292

fraction of an eroded pore space at this λ.

TE D

M AN U

SC

RI PT

280

The program realization of the simulations was implemented as parallel code in C

294

language using the Message Passing Interface (MPI) standard on a supercomputing

295

platform (GWDG, Göttingen, Germany). All numerical codes and their description can be

296

found in the Supporting Information of [36].

AC C

297

EP

293

298

3. Results and discussion

299

3.1. Nitrogen physisorption analysis

300 301

The nitrogen sorption isotherms and pore size distributions obtained for the four silica samples are summarized in Fig. 2. All isotherms exhibit hysteresis for p/p0 = 0.6‒0.95,

14

ACCEPTED MANUSCRIPT

which is of type H2 [62]. The isotherms are similar to those obtained for the three silica

303

monolith samples studied earlier [38]. The reconstruction of sample Si26 from that work

304

(dmeso = 25.7 nm) was, in turn, used as a model in mean field density functional theory

305

simulations of adsorption and desorption [63]. The simulated isotherms demonstrated good

306

qualitative agreement with experimental nitrogen sorption isotherms at 77 K, with both,

307

experiment and theory, showing isotherms characterized by type H2 hysteresis. For the

308

hysteresis region, the simulated isotherms reproduced the classical scenarios of delayed

309

condensation (adsorption branch) and pore blocking (desorption branch). The pore size

310

distributions in Fig. 2 indicate the absence of micropores from all four samples (which was

311

corroborated by the analysis of cumulative pore volumes). All pore size distributions are

312

positively skewed. The fully porous silica particles (FP10 and FP30) exhibit a pronouned

313

tail towards larger pore diameters.

SC

M AN U

The values of dmeso, SBET, and Vpore derived for the investigated silica samples are

TE D

314

RI PT

302

summarized in Table 1. For the core–shell particles, we first of all notice the good

316

agreement between the determined mean pore sizes (dmeso = 9.4 and 16.4 nm) and the

317

nominal pore sizes of samples CS9 and CS16, respectively. In addition, the values

318

determined for the specific surface area (SBET = 109.5 and 86.1 m2g–1) and the mean pore

319

volume (Vpore = 0.25 and 0.32 cm3g–1) of samples CS9 and CS16, respectively, agree well

320

with values of SBET = 135 and 80 m2g–1 and Vpore = 0.26 and 0.29 cm3g–1 reported by

321

Schuster et al. [64]. In contrast, the mean pore sizes determined for the fully porous

322

particles (dmeso = 16.0 and 23.9 nm) deviate substantially from the nominal pore sizes of

323

FP10 and FP30, respectively, but the values determined for the specific surface area (SBET =

AC C

EP

315

15

ACCEPTED MANUSCRIPT

324

419.3 and 101.6 m2g–1) are in good-to-fair agreement with manufacturer-provided values of

325

SBET = 350 and 100 m2g–1 for FP10 and FP30, respectively.

327

3.2. Physical reconstructions and CLD analysis

RI PT

326

Fig. 3 shows 3D sections and 2D images from two of the four reconstructions obtained

329

by STEM tomography. These visualizations confirm that the investigated silica materials

330

possess a microscopically disordered mesopore space. Larger heterogeneities (on a scale of

331

several pores) cannot be identified by visual inspection. The void volume fractions (or

332

accessible porosities for point-like tracers, ε0) of the reconstructions are summarized in

333

Table 1. Values of ε0 = 0.29 (CS9) and ε0 = 0.37 (CS16) determined for the core‒shell

334

particles show good agreement with reported shell porosities of 0.31 [65] and 0.36 [2],

335

respectively. The shell porosities were derived from the total and interparticle porosities of

336

columns packed with these particles, determined by pycnometry and inverse size-exclusion

337

chromatography, respectively. Independently stated ε0-values for the fully porous particles

338

could not be retrieved in the literature.

M AN U

TE D

EP

339

SC

328

It is noteworthy that we found a significantly larger amount of closed pores in the reconstructions of the core–shell particles than in those of the fully porous particles. This is

341

illustrated in Fig. 4, which highlights closed pores in the reconstructions of samples CS9

342

and FP30. The fraction of closed pores, determined as the number of void voxels belonging

343

to closed pores divided by the total number of void voxels in a reconstruction, is overall

344

quite low: 1.73% and 1.07% in samples CS9 and CS16, respectively, and only 0.01% and

345

0.13% in samples FP10 and FP30. Closed pores cannot be accessed from the remaining

AC C

340

16

ACCEPTED MANUSCRIPT

mesopore network, but during the initial tracer distribution in the void space prior to a

347

diffusion simulation, tracers could land in closed pores and remain trapped for the duration

348

of the simulation. Although given the low fraction of closed pores in the samples this

349

scenario is unlikely to change the calculated diffusion coefficient, the placement of tracers

350

in closed pores was avoided by exchanging the void voxels belonging to closed pores with

351

solid voxels (thus filling the closed pores) prior to simulating diffusion in a reconstruction.

352

The ε0-values in Table 1 are corrected for the contribution of closed pores.

SC

RI PT

346

CLDs for the void space of the reconstructions (Fig. 5) were used to complement the

354

geometrical properties obtained through nitrogen physisorption analysis of the samples.

355

The CLD is abstract but accurate, since it neither requires, nor assumes a pre-defined

356

geometrical shape of the pores, such as the cylinder geometry that was used to derive the

357

pore size distributions from the nitrogen sorption isotherms (Fig. 2). The chord lengths in

358

the CLDs were normalized by the dmeso-value derived from nitrogen physisorption analysis.

359

A fit of Eq. (3) to the CLDs delivered the mean chord length µ and the homogeneity factor

360

k, which are summarized in Table 1. Fig. 5 shows that the CLDs collapse nearly perfectly

361

into separate groups for core–shell and fully porous particles, which emphasizes that both

362

particle types exhibit distinct morphologies. For a given particle type, the geometrical

363

properties among the two samples (chord length characteristics and dmeso) are linearly

364

scaled. This allows to test the scalability of hindered diffusion for each particle type with

365

respect to dmeso and λ = dtracer/dmeso.

366 367

AC C

EP

TE D

M AN U

353

Fig. 5 also reveals that the pore space properties of the fully porous particles can be linearly scaled to those of the mesoporous skeleton of the three commercial silica monolith

17

ACCEPTED MANUSCRIPT

samples investigated earlier, as confirmed by the (representatively shown) CLD for sample

369

Si26 from that work [38]. This is not unexpected, as the mesopore space morphology of all

370

these materials (FP10 and FP30; Si12, Si21, and Si26) is basically formed through classical

371

sol–gel processing. The similarity in geometrical properties between the fully porous silica

372

particles and the silica monolith samples is reflected in the ratio between µ and dmeso

373

derived from CLD and nitrogen physisorption analysis, respectively (Table 1). Samples

374

FP10 and FP30 show values of µ/dmeso = 1.9 and 2.3, respectively, which are close to the

375

value of µ/dmeso ~ 2.2 derived for samples Si12, Si21, and Si26 [38]. In contrast, samples

376

CS9 and CS16, whose mesopore space morphology is not based on sol‒gel processing,

377

have a ratio of µ/dmeso = 1.0. This indicates a fundamentally different pore space

378

morphology for the studied core–shell particles, a point to which we will return below.

M AN U

SC

RI PT

368

379

381

3.3. Hindered diffusion and accessible porosity

TE D

380

Effective diffusion coefficients for the finite-size tracers and the accessible porosity in a reconstruction were determined as described previously [36,38]: After the transient

383

diffusion coefficient D(t), monitored via Eq. (4), reached its long-time limit, which

384

occurred when the tracers had sufficiently sampled the available pore space, the effective

385

diffusion coefficient Deff,K(λ), characterizing transport locally (within a reconstruction),

386

was extracted from the asymptote of that D(t)-curve. The accessible porosity, ε(λ), was

387

available as the corresponding void volume fraction sampled by the tracers.

AC C

EP

382

388

Fig. 6 summarizes these data, that is, Deff,K(λ) and ε(λ), normalized by their respective

389

values for point tracers (λ = 0), for all four reconstructions. As observed for the void space

18

ACCEPTED MANUSCRIPT

CLDs (Fig. 5), these data also collapse neatly into separate groups for core–shell and fully

391

porous particles. In turn, they can be well characterized with the following equations (grey

392

lines in Fig. 6).

393

396

, ,

G

GFH

G

GFH

= 1 − 1.344λ − 40.489λ + 209.827λ" − 392.515λ% + 259.236λ' (5)

= 1 − 5.491λ + 10.133λ − 6.261λ"

397

399 400 401 402

(6)

(ii) Fully porous particles (samples FP10 and FP30): , ,

G

GFH

G

GFH

= 1 − 1.216λ − 0.582λ − 5.199λ" + 13.350λ% − 7.455λ'

= 1 − 2.200λ + 1.245λ

(7) (8)

TE D

398

SC

395

(i) Core–shell particles (samples CS9 and CS16):

M AN U

394

RI PT

390

Importantly, Eqs. (7) and (8) represent the best fits to the hindered diffusion and accessible porosity data of the three commercial silica monolith samples (Si12, Si21, and

404

Si26) [38]. This shows that hindered transport in all investigated silica samples that are

405

based on classical sol–gel processing and whose similarity in geometrical properties was

406

apparent from the CLD analysis (Fig. 5) is uniformly expressed with Eqs. (7) and (8). In

407

contrast, Eqs. (5) and (6) reflect that the pore space in the core–shell particles closes off at a

408

much lower λ than the pore space in the sol‒gel silica samples. At λ = 0.4, diffusive

409

transport in the core–shell particles has almost come to a standstill, whereas the fully

410

porous particles retain nearly 40% of their Deff,K-value at λ = 0 (Fig. 6A). Regarding this

AC C

EP

403

19

ACCEPTED MANUSCRIPT

morphology–transport behavior, the core–shell particles are closer to the ordered

412

mesoporous silicas SBA-15 and KIT-6, for which diffusive transport was found to come to

413

a standstill at λ = 0.29 [41].

416 417

factors Kd(λ) and H(λ) as λ = λ =

,

,

G

G

= =

, ,

G

(9)

GFH

SC

415

From Eqs. (5)–(8), we can now calculate the corresponding local and global hindrance

λ ε λ

(10)

M AN U

414

RI PT

411

418

The required values of τ0 = Dm/Deff,K(λ = 0) and ε(λ = 0) ≡ ε0 are given in Table 1. The

419

H(λ)-expression for the fully porous particles equals that for the three silica monolith

420

samples, already been provided as Eq. (2). For the core–shell particles we have

422 423

λ =

1 − 6.835λ − 22.9761λ + 412.272λ" − 1946.54λ% + 4794.21λ' −

6714.55λ( + 5084.37λ) − 1623.08λJ

TE D

421

(11)

Eqs. (2) and (11) are quantitative expressions that allow to calculate the extent of hindered diffusion through the fully porous and the core–shell particles, respectively, from

425

known values for ε0, τ0, and λ = dtracer/dmeso.Values for theses parameters can be accessed

426

via independent experiments, for example, dmeso and ε0 from nitrogen physisorption

427

analysis and τ0 through PFG-NMR [16,17,20,21] or conductivity measurements [66].

428

Importantly, Eq. (11) for the core–shell particles describes a significantly stronger

429

attenuation of diffusive mobility with increasing λ than predicted by the Renkin equation

430

(restricted to λ ≤ 0.4) [32]

431

AC C

EP

424

λ = 1−λ

1 − 2.104λ + 2.09λ" − 0.95λ'

20

(12)

ACCEPTED MANUSCRIPT

The Renkin equation, Eq. (12), estimates hindrance factors of H(λ = 0.2) = 0.38 and H(λ =

433

0.4) = 0.10, but Eq. (11) predicts much lower hindrance factors of about H(λ = 0.2) = 0.005

434

and H(λ = 0.4) = 0.00005 for the core–shell particles, that is, negligible transport through

435

the porous shell.

436

RI PT

432

These results demonstrate that the Renkin equation yields unrealistic predictions for

certain materials, because it substantially underestimates the hindrance to diffusion caused

438

by the mesopore space morphologies of the core–shell particles studied in this work and the

439

ordered mesoporous silicas (SBA-15, KIT-6) studied in previous work [41]. Replacing Eq.

440

(12) with Eq. (11) can be expected to improve the analysis of chromatographic data and a

441

quantitative discussion of individual mass transfer mechanisms in columns packed with

442

these particles [67]. This is especially important for the separation and purification of larger

443

analytes such as peptides and proteins, for which wide-pore core–shell materials are used

444

with the intention to reduce the intraparticle mass transfer resistance. In the design of the

445

core–shell materials, two parameters are usually adjusted: the thickness and the mean pore

446

size of the shell [42–44,64]. The former parameter determines the diffusion length on the

447

particle scale, the latter is tied to the diffusive hindrance on the pore scale. As we have seen

448

in this and previous work [36,41], however, the mean pore size must be complemented by

449

pore network effects, that is, the concrete pore space morphology, to predict the hindrance

450

to diffusion in the pore network accurately over a wide range of λ-values. In general,

451

thinner shells promote shorter intraparticle residence times. But Fig. 6 reveals that this main

452

advantage of the core–shell particles over fully porous particles (shell thickness vs particle

453

radius as diffusion length on the particle scale) is opposed by an unfavorable shell

AC C

EP

TE D

M AN U

SC

437

21

ACCEPTED MANUSCRIPT

morphology that quite significantly pushes hindrance effects on intraparticle diffusive

455

transport, Eq. (2) vs Eq. (11). It would thus be desirable to combine the core–shell particle

456

design with the mesopore space morphology of the fully porous particles.

RI PT

454

Another morphological parameter worth improving is the shell porosity. Pore-scale

458

simulations using computer-generated, superficially porous particles made of a solid,

459

spherical core and a porous shell (comprising a random and very loose arrangement of

460

spherical sol particles) with a much higher shell porosity (~0.58) and mean pore size (~100

461

nm) than in this work (cf. Table 1) have documented diffusive hindrance behavior closer to

462

that of the fully porous particles in Fig. 6 [68]. It is, however, questionable if these model

463

particles adequately represent the shell morphology as reconstructed by electron

464

tomography, particularly as the shell in the model particles had not experienced further

465

consolidation like sintering.

M AN U

Eqs. (2) and (11) will also improve to predict and analyze data should these silica

TE D

466

SC

457

materials (after appropriate surface modification) be used as catalyst supports in

468

continuous-flow microreactors [69]. In particular, Eqs. (2) and (11) provide specific

469

information about diffusive hindrance for each molecular species involved in the reaction

470

and thereby help to distinguish between external mass transfer resistance, the presence of

471

surface barriers [70], and diffusion only in the mesopore system of the particles. This is

472

important when effective diffusivities are calculated based on the Thiele concept applied to

473

the catalytic conversion [35]. The Thiele concept assumes that the deviation of the initial

474

from the intrinsic reaction rate is only due to a slower rate of mass transfer through the pore

475

system with respect to reaction at the catalytically active sites. PFG-NMR measurements

476

also help to make this distinction [71], but Eqs. (2) and (11) are immediately applicable to a

AC C

EP

467

22

ACCEPTED MANUSCRIPT

wide range of molecular species. This furthermore allows to tune the selectivity via subtle

478

differences in the diffusive hindrance or to predict restricted access to and partial

479

entrapment in the particles (discussion below).

480

482

3.4. Evolving pore networks and connectivity analysis

Complementary to the study of hindered diffusion and accessible porosity for the finite-

SC

481

RI PT

477

size tracers in the four reconstructions (Fig. 6), the pore network connectivities and their

484

evolution with increasing λ were analyzed. Table 2 summarizes the connectivities of the

485

topological skeleton of all reconstructions. The fractions of triple-point, quadruple-point,

486

and higher-order junctions (85–88%, 10–12%, and 2–3%, respectively) and the resulting

487

average pore connectivity (~3.15) in the reconstructions of the fully porous particles very

488

closely resemble the respective values derived for the three silica monolith samples [38],

489

underlining a similarity also in topological properties between these sol–gel silicas

490

(samples FP10, FP30, Si12, Si21, and Si26). The core–shell particles contain somewhat

491

larger fractions of quadruple-point and higher-order junctions (14–15% and 4–5%,

492

respectively) than the sol–gel silicas, which results in a slightly raised average pore

493

connectivity (~3.25).

TE D

EP

AC C

494

M AN U

483

The evolution of the pore networks with increasing λ is illustrated in Fig. 7 through the

495

decline of triple-point and quadruple-point junctions with respect to the initial networks

496

spanned by the point-like tracers in the reconstructions. We immediately notice the already

497

familiar collapse of the data into two groups (cf. Figs. 5 and 6), one for the core–shell

498

particles and one for the sol–gel silicas. Fig. 7 reveals that the degradation of the pore

23

ACCEPTED MANUSCRIPT

network with increasing λ (Fig. 6) is decidedly more progressive in the core–shell particles.

500

At λ > 0.4, their pore space is hardly connected anymore, which explains why diffusive

501

mobility and pore space accessibility also approach zero (Fig. 6). To illustrate this pore

502

network degradation with increasing λ, Fig. 8 compares selected pore networks of samples

503

CS16 and FP30. Whereas the network of sample FP30 is hardly thinned-out at λ = 0.4, the

504

network of sample CS16, already visibly reduced at λ = 0.21, consists of mainly isolated

505

network fractions (islands) at λ = 0.37. In view of Fig. 6, this means that the decrease of

506

Deff,K(λ) and ε(λ) is accompanied by drastic topological changes as entire branches close

507

off and disappear from the active pore network evolving with λ.

M AN U

SC

RI PT

499

Because these islands can neither be entered nor left, their presence implies that products

509

of a catalytic reaction (from smaller reactants) may be enriched or even trapped in the pore

510

space. Vice versa, sufficiently large reactants will only get limited access to the

511

catalytically active surface. Similarly, solutes or particles that need to be immobilized on

512

the surface, such as enzymes, metal clusters, or nanoparticles, may utilize only part of the

513

internal surface of a material. In the extreme case, only the external surface of a material is

514

active and intended confinement effects [11,12] vanish. Together with our previous work

515

on ordered mesoporous silicas [41] the results of this study show that more attention should

516

be paid to the actual mesopore morphology of silica materials instead of relying on generic

517

material properties. It would be worthwhile to review the available multifaceted preparation

518

and functionalization strategies as to whether they produce materials with the desired

519

diffusive transport characteristics [72–75]. Knowing the detailed diffusive hindrance

520

behavior of the catalyst support will help to understand the complex interplay of

AC C

EP

TE D

508

24

ACCEPTED MANUSCRIPT

521

cooperative catalysis, confinement, and transport and thereby allow for a better tuning of

522

efficiency, yield, and selectivity.

524 525

3.5. Special morphological features of the reconstructions

RI PT

523

We now return to the properties of the core–shell particles that are responsible for the rapid decay in diffusive mobility and accessible porosity as well as the associated pore

527

network degradation with increasing λ (Figs. 6–8). As noted in the Introduction, these

528

particles are prepared by fusing a porous shell of sol particles onto a solid core. The sol

529

particles are initially immobilized in a layer-by-layer assembly whereby an organic

530

polyelectrolyte holds neighboring sol layers together [44]; the final thermal treatment

531

removes the polyelectrolyte and lets the sol particles consolidate into a porous shell

532

structure. Little is known about the impact of this consolidation step on the final mesopore

533

space morphology and thus the transport properties of the shell [42]. The consolidation step

534

is important to improve integrity and strength of the core‒shell particles (without particle

535

melting or a reduction in surface area or pore volume) so that they withstand surface

536

modification and column packing procedures [76].

EP

TE D

M AN U

SC

526

In the reconstructions of the core–shell particles, we noticed the presence of many

538

unusually small pores that establish connections between larger ones. Fig. 9 depicts this

539

detail of the pore space for sample CS9 (pore space rendered in blue, small pores indicated

540

by orange circles). This feature is absent from the pore space of the fully porous particles,

541

as shown in Fig. 9 for sample FP30 (pore space rendered in red). The small connecting

542

pores, found in both core–shell particle samples, are prone to bottle-necking with increasing

AC C

537

25

ACCEPTED MANUSCRIPT

λ and responsible for the observed, relatively quick shut-down of the pore network with

544

respect to mobility, porosity, and connectivity (cf. Figs. 6–8). The many small pores (and

545

actually also the closed pores highlighted in Fig. 4) observed in the reconstructions of the

546

core‒shell particles likely result from the thermal consolidation step as they reflect

547

structural features known from compacted and sintered sphere packings, where sphere

548

overlap and the formation of narrow, constricted, or even closed pores are typical,

549

depending on the extent of sintering [77,78]. Sample CS16 has a much higher tortuosity (τ0

550

= 2.84, Table 1) than random packings of hard, non-overlapping spheres (τ0 = 1.48) at the

551

same porosity (ε0 ~ 0.37) [58], and the tortuosity of sample CS9 (τ0 = 4.28) indicates an

552

even higher degree of consolidation at ε0 = 0.29. Increasing the shell porosity while

553

maintaining the mechanical stability of core–shell particles would therefore be a

554

straightforward improvement of their mass transfer properties.

555

557

4. Conclusions

The physical reconstruction of the mesopore space of core–shell and fully porous silica

EP

556

TE D

M AN U

SC

RI PT

543

particles has revealed subtle morphological details related to the different preparation

559

protocols of the two particle types. The conventional, fully porous particles, whose

560

preparation is based on classical sol–gel processing, share the same general mesopore shape

561

and topology at varied mean mesopore size and porosity as three previously studied, also

562

sol–gel derived, silica monolith samples [38]. The porous shell of the core–shell particles,

563

in contrast, forms through consolidation of individually assembled layers of sol particles

564

and thus reflects structural features of compacted and sintered packings, such as narrow and

AC C

558

26

ACCEPTED MANUSCRIPT

highly constricted as well as closed pores. These features, in turn, determine the diffusive

566

transport behavior of the core–shell particles and mark the key difference to the dynamics

567

in the sol–gel silicas. Quantitative expressions for local and global diffusive hindrance

568

factors as well as accessible porosity were derived from diffusion simulations in the

569

reconstructions. Hindered diffusion in the fully porous particles is described by the same

570

equation, Eq. (2), as in the three silica monolith samples [38]. Due to enhanced bottle-

571

necking in the mesoporous shell, the extent of hindered diffusion in the core–shell particles

572

(Eq. (11)) is underestimated by the Renkin equation, Eq. (12). Diffusive mobility,

573

accessible porosity, and the connectivity of evolving pore networks decline more strongly

574

with increasing ratio of tracer to mean mesopore size than in the sol–gel silicas. Slowed-

575

down diffusion is a clear disadvantage when fast transport to and from the active surface

576

sites is required. To realize the benefits of the core–shell particle design, which centers on

577

the adjustment of the diffusion length in the mesopore network through the shell thickness,

578

the shell morphology must be improved. As previously recommended for ordered

579

mesoporous silicas with bottle-necking issues [41], widening narrow and constricted pores

580

towards the targeted mean mesopore size (thereby narrowing the entire pore size

581

distribution) would bring the hindered transport characteristics of the core–shell particles

582

closer to that of the sol–gel silicas. The employed three-step methodology (physical

583

reconstruction–morphological analysis–pore-scale simulations) will be instrumental to

584

refine the morphology–transport relationships of these and other (ordered or random)

585

mesoporous materials and to guide the preparation of materials for sensitive applications,

586

such as catalysis under confinement and controlled drug release, where small

587

morphological adjustments may have a substantial impact.

AC C

EP

TE D

M AN U

SC

RI PT

565

27

ACCEPTED MANUSCRIPT

588 Acknowledgements

590

This work was supported by the Deutsche Forschungsgemeinschaft DFG (Bonn, Germany)

591

under grant TA 268/9–1 and by the Karlsruhe Nano Micro Facility (KNMF) at the

592

Karlsruhe Institute of Technology (Karlsruhe, Germany) under the KNMF long-term user

593

proposal 2017-019-020749. Wu Wang acknowledges his PhD funding by the Chinese

594

Scholarship Council (CSC). We thank Richard Kohns and Professor Dirk Enke (Institute of

595

Chemical Technology, Universität Leipzig, Leipzig, Germany) for the nitrogen

596

physisorption measurements.

597

M AN U

SC

RI PT

589

References

599

[1]

K.K. Unger, R. Skudas, M.M. Schulte, J. Chromatogr. A 1184 (2008) 393–415.

600

[2]

F. Gritti, G. Guiochon, J. Chromatogr. A 1228 (2012) 2–19.

601

[3]

D.I. Fried, F.J. Brieler, M. Fröba, ChemCatChem 5 (2013) 862–884.

602

[4]

S. Lwin, I.E. Wachs, ACS Catal. 4 (2014) 2505–2520.

603

[5]

L. Vilcocq, P.C. Castilho, F. Carvalheiro, L.C. Duarte, ChemSusChem 7 (2014)

605

[6]

606

R. Munirathinam, J. Huskens, W. Verboom, Adv. Synth. Catal. 357 (2015) 1093–

1123.

607

[7]

608

[8]

609

EP

1010–1019.

AC C

604

TE D

598

J. Florek, R. Caillard, F. Kleitz, Nanoscale 9 (2017) 15252–15277.

D.M. Schlipf, S. Zhou, M.A. Khan, S.E. Rankin, B.L. Knutson, Adv. Mater. Interfaces 4 (2017) 1601103.

28

ACCEPTED MANUSCRIPT

[9]

R. Diab, N. Canilho, I.A. Pavel, F.B. Haffner, M. Girardon, A. Pasc, Adv. Colloid

611 612

Interface Sci. 249 (2017) 346–362. [10]

613

M. Iwamoto, Y. Tanaka, N. Sawamura, S. Namba, J. Am. Chem. Soc. 125 (2003) 13032–13033.

RI PT

610

[11]

F. Goettmann, C. Sanchez, J. Mater. Chem. 17 (2007) 24–30.

615

[12]

C. Yu, J. He, Chem. Commun. 48 (2012) 4933–4940.

616

[13]

M.N. Pahalagedara, L.R. Pahalagedara, C.-H. Kuo, S. Dharmarathna, S.L. Suib, Langmuir 30 (2014) 8228–8237.

M AN U

617

SC

614

618

[14]

B. Coasne, New J. Chem. 40 (2016) 4078–4094.

619

[15]

J.L. Coffman, E.N. Lightfoot, T.W. Root, J. Phys. Chem. B 101 (1997) 2218–2223.

620

[16]

U. Tallarek, F.J. Vergeldt, H. Van As, J. Phys. Chem. B 103 (1999) 7654–7664.

621

[17]

J. Kärger, D.M. Ruthven, T.N. Theodorou, Diffusion in Nanoporous Materials,

[18]

624 625

C. D'Agostino, J. Mitchell, L.F. Gladden, M.D. Mantle, J. Phys. Chem. C 116 (2012) 8975–8982.

[19]

A. Galarneau, F. Guenneau, A. Gedeon, D. Mereib, J. Rodriguez, F. Fajula, B. Coasne, J. Phys. Chem. C 120 (2016) 1562–1569.

AC C

626

EP

623

Wiley-VCH, Weinheim, 2012.

TE D

622

627

[20]

F. Elwinger, P. Pourmand, I. Furó, J. Phys. Chem. C 121 (2017) 13757–13764.

628

[21]

T. J. Rottreau, C.M.A. Parlett, A.F. Lee, R. Evans, J. Phys. Chem. C 121 (2017)

629 630

16250–16256.

[22]

631 632

C. D'Agostino, M.D. Mantle, L.F. Gladden, Microporous Mesoporous Mater. 269 (2018) 88–92.

[23]

J. Kärger, D. Freude, J. Haase, Processes 6 (2018) 147.

29

ACCEPTED MANUSCRIPT

[24]

J. Michaelis, C. Bräuchle, Chem Soc. Rev. 39 (2010) 4731–4740.

634

[25]

B. Rühle, M. Davies, T. Bein, C. Bräuchle, Z. Naturforsch. B 68 (2013) 423–444.

635

[26]

D.A. Higgins, S.C. Park, K.-H. Tran-Ba, T. Ito, Annu. Rev. Anal. Chem. 8 (2015)

636

RI PT

633

193–216. [27]

L.D.C. Bishop, C.F. Landes, Acc. Chem. Res. 51 (2018) 2247–2254.

638

[28]

L. Kisley, R. Brunetti, L.J. Tauzin, B. Shuang, X.Y. Yi, A.W. Kirkeminde, D.A.

[29]

641 642

Weitkamp, J. Kärger, Angew. Chem. Int. Ed. 54 (2015) 5060–5064. [30]

643 644

T. Titze, C. Chmelik, J. Kullmann, L. Prager, E. Miersemann, R. Gläser, D. Enke, J.

F. Feil, S. Naumov, J. Michaelis, R. Valiullin, D. Enke, J. Kärger, C. Bräuchle, Angew. Chem. Int. Ed. 51 (2012) 1152–1155.

[31]

645

T. Titze, A. Lauerer, L. Heinke, C. Chmelik, N.E.R. Zimmermann, F. Keil, D.M. Ruthven, J. Kärger, Angew. Chem. Int. Ed. 54 (2015) 14580–14583.

TE D

640

Higgins, S. Weiss, C.F. Landes, ACS Nano 9 (2015) 9158–9166.

M AN U

639

SC

637

[32]

E.M. Renkin, J. Gen. Physiol. 38 (1954) 225–243.

647

[33]

W.M. Deen, AIChE J. 33 (1987) 1409–1425.

648

[34]

G. Guiochon, A. Felinger, D.G. Shirazi, A.M. Katti, Fundamentals of Preparative

650 651 652

and Nonlinear Chromatography, 2nd ed., Academic Press, Amsterdam, The

AC C

649

EP

646

Netherlands, 2006.

[35]

H.S. Fogler, Elements of Chemical Reaction Engineering, 4th ed., Prentice Hall,

Upper Saddle River, NJ, 2006.

653

[36]

D. Hlushkou, A. Svidrytski, U. Tallarek, J. Phys. Chem. C 121 (2017) 8416–8426.

654

[37]

P. Dechadilok, W.M. Deen, Ind. Eng. Chem. Res. 45 (2006) 6953–6959.

30

ACCEPTED MANUSCRIPT

655

[38]

656

S.-J. Reich, A. Svidrytski, D. Hlushkou, D. Stoeckel, C. Kübel, A. Höltzel, U. Tallarek, Ind. Eng. Chem. Res. 57 (2018) 3031–3042.

[39]

Y. Guo, K.H. Langley, F.E. Karasz, Phys. Rev. B 50 (1994) 3400–3403.

658

[40]

S.G.J.M. Kluijtmans, J.K.G. Dhont, A.P. Philipse, Langmuir 13 (1997) 4982–4987.

659

[41]

S.-J. Reich, A. Svidrytski, A. Höltzel, J. Florek, F. Kleitz, W. Wang, C. Kübel, D. Hlushkou, U. Tallarek, J. Phys. Chem. C 122 (2018) 12350–12361.

SC

660

RI PT

657

[42]

G. Guiochon, F. Gritti, J. Chromatogr. A 1218 (2011) 1915–1938.

662

[43]

V. González-Ruiz, A.I. Olives, M.A. Martín, Trends Anal. Chem. 64 (2015) 17–28.

663

[44]

R. Hayes, A. Ahmed, T. Edge, H. Zhang, J. Chromatogr. A 1357 (2014) 36–52.

664

[45]

W. Tong, X. Song, C. Gao, Chem. Soc. Rev. 41 (2012) 6103–6124.

665

[46]

S. Correa, E.C. Dreaden, L. Gu, P.T. Hammond, J. Control. Release 240 (2016)

667

364–386. [47]

668

G. Prieto, H. Tüysüz, N. Duyckaerts, J. Knossalla, G.-H. Wang, F. Schüth, Chem.

TE D

666

M AN U

661

Rev. 116 (2016) 14056–14119. [48]

D. Enke, R. Gläser, U. Tallarek, Chem. Ing. Tech. 88 (2016) 1561–1585.

670

[49]

M.T. Kreutzer, F. Kapteijn, J.A. Moulijn, Catal. Today 111 (2006) 111–118.

671

[50]

J.M. Badano, C. Betti, I. Rintoul, J. Vich-Berlanga, E. Cagnola, G. Torres, C. Vera,

673 674

AC C

672

EP

669

J. Yoria, M. Quiroga, Appl. Catal. A: General 390 (2010) 166–174.

[51]

K.K. Unger, Porous Silica – Its Properties and Use as a Support in Column Liquid

Chromatography, Elsevier, Amsterdam, 1979.

675

[52]

C.J. Brinker, G.W. Scherer, Sol–Gel Science, Academic Press, NewYork, 1990.

676

[53]

K. Nakanishi, N. Tanaka, Acc. Chem. Res. 40 (2007) 863–873.

677

[54]

A. Feinle, M.S. Elsaesser, N. Hüsing, Chem. Soc. Rev. 45 (2016) 3377–3399.

31

ACCEPTED MANUSCRIPT

[55]

J. Landers, G.Y. Gor, A.V. Neimark, Colloids Surf. A 437 (2013) 3–32.

679

[56]

J.R. Kremer, D.N. Mastronarde, J.R. McIntosh, J. Struct. Biol. 116 (1996) 71–76.

680

[57]

P. Gilbert, J. Theor. Biol. 36 (1972) 105–117.

681

[58]

T. Müllner, K.K. Unger, U. Tallarek, New J. Chem. 40 (2016) 3993–4015.

682

[59]

I. Arganda-Carreras, Image Analysis, 2018. https://sites.google.com/site/

[60]

685

K. Hormann, V. Baranau, D. Hlushkou, A. Höltzel, U. Tallarek, New J. Chem. 40 (2016) 4187–4199.

M AN U

684

iargandacarreras/software/ (accessed Nov, 2018).

SC

683

RI PT

678

686

[61]

I.C. Kim, S. Torquato, J. Chem. Phys. 96 (1992) 1498–1503.

687

[62]

K.A. Cychosz, R. Guillet-Nicolas, J. García-Martínez, M. Thommes, Chem. Soc.

688 [63]

691

Langmuir 34 (2018) 9936–9945. [64]

692

TE D

690

A. Svidrytski, A. Rathi, D. Hlushkou, D.M. Ford, P.A. Monson, U. Tallarek,

S.A. Schuster, B.M. Wagner, B.E. Boyes, J.J. Kirkland, J. Chromatogr. A 1315 (2013) 118–126.

EP

689

Rev. 46 (2017) 389–414.

[65]

F. Gritti, G. Guiochon, J. Chromatogr. A 1217 (2010) 1604–1515.

694

[66]

M. Barrande, R. Bouchet, R. Denoyel, Anal. Chem. 79 (2007) 9115–9121.

695

[67]

F. Gritti, G. Guiochon, J. Chromatogr. A 1218 (2011) 907–921.

696

[68]

R.S. Maier, M.R. Schure, Chem. Eng. Sci. 185 (2018) 243–255.

697

[69]

C.P. Haas, T. Müllner, R. Kohns, D. Enke, U. Tallarek, React. Chem. Eng. 2 (2017)

AC C

693

698 699

498–511. [70]

J. Kärger, Microporous Mesoporous Mater. 189 (2014) 126–135.

32

ACCEPTED MANUSCRIPT

[71]

702

D. Enke, J. Kärger, R. Gläser, Catal. Sci. Technol. 5 (2015) 3137–3146. [72]

703 704

E.L. Margelefsky, R.K. Zeidan, M.E. Davis, Chem. Soc. Rev. 37 (2008) 1118–

RI PT

701

M. Goepel, H. Kabir, C. Küster, E. Saraçi, P. Zeigermann, R. Valiullin, C. Chmelik,

1126. [73]

705

J.M. Camposa, J.P. Lourenço, H. Cramail, M.R. Ribeiro, Prog. Polymer Sci. 37 (2012) 1764–1804.

SC

700

[74]

S.L. Suib, Chem. Rec. 17 (2017) 1169–1183.

707

[75]

X.-Y. Yang, L.H. Chen, Y. Li, J.C. Rooke, C. Sanchez, B.L. Su, Chem. Soc. Rev.

708

46 (2017) 481–558.

M AN U

706

[76]

L.E. Blue, J.W. Jorgenson, J. Chromatogr. A 1218 (2011) 7989–7995.

710

[77]

J.T. Fredrich, A.A. DiGiovanni, D.R. Noble, J. Geophys. Res. 111 (2006) B03201.

711

[78]

Y. Bernabé, M. Li, Y.-B. Tang, B. Evans, Oil Gas Sci. Technol. 71 (2016) 50.

AC C

EP

TE D

709

33

ACCEPTED MANUSCRIPT

FP10

FP30

dmeso [nm] a

9.4

16.8

16.0

23.9

SBET [m2 g–1] a

109.5

86.1

419.3

101.6

Vpore [cm3 g–1] a

0.25

0.32

1.46

0.59

ε0 [‒] b

0.29

0.37

0.75

0.37

µ [nm] c

9.8

16.4

30.8

k [‒] c

2.28

2.46

2.14

τ0 [‒] d

4.28

2.84

a

2.40

1.32

2.21

Table 2. Connectivities of the topological skeleton of the reconstructions.a

nt/nj [%]

FP10

FP30

79.5

81.2

85.4

88.5

15.2

14.3

11.9

9.6

AC C

nq/nj [%]

CS16

EP

CS9

721 722 723

55.8

Mean mesopore size dmeso, specific surface area SBET, and pore volume Vpore based on nitrogen physisorption analysis. Void volume fraction ε0 (accessible porosity for point-like tracers) extracted from the reconstructions. c Mean chord length µ and homogeneity factor k from CLD analysis of the void space in the reconstructions. d Tortuosity τ0 for point-like tracers from pore-scale diffusion simulations. b

719 720

RI PT

CS16

SC

CS9

M AN U

713 714 715 716 717 718

Table 1. Geometrical and transport properties of the mesoporous silica samples.

TE D

712

nx/nj [%]

5.3

4.5

2.7

1.9

Z

3.25

3.23

3.17

3.13

a

Percentage of nodes connecting 3, 4, or >4 branches (nt/nj, nq/nj, and nx/nj, respectively) as well as the resulting average pore connectivity (Z).

34

ACCEPTED MANUSCRIPT

724

Figure legends

725 Fig. 1. Basic architecture of the employed core–shell silica particles, which are prepared by

727

fusing a mesoporous silica layer onto a solid (impermeable) silica particle.

RI PT

726

728

Fig. 2. Nitrogen sorption isotherms (left) and derived pore size distributions (right) for the

730

core–shell particles (A) and the fully porous particles (B).

SC

729

M AN U

731

Fig. 3. (Top) 3D sections from the electron tomographic reconstructions (solid in grey) of

733

the larger-pore core–shell and fully porous particles (samples CS16 and FP30,

734

respectively). (Bottom) Selected 2D slices (white‒solid, black‒void) from the

735

reconstructions of samples CS16 and FP30.

736

TE D

732

Fig. 4. Closed, inaccessible porosity in the reconstructions of samples CS9 and FP30

738

(highlighted in blue and red, respectively).

739

EP

737

Fig. 5. CLDs for the void space of the four reconstructions normalized by the respective

741

mean mesopore size of the samples (dmeso) as obtained from nitrogen physisorption

742

analysis. Added for comparison is the void space CLD for the mesoporous skeleton of a

743

hierarchical, macroporous–mesoporous silica monolith (sample Si26 with dmeso = 25.7 nm)

744

derived in previous work [38].

AC C

740

745

35

ACCEPTED MANUSCRIPT

Fig. 6. (A) Diffusion coefficient Deff,K(λ), normalized by the diffusivity of point-like

747

tracers, Deff,K(λ = 0) = Dm/τ0, as a function of λ = dtracer/dmeso, the ratio of tracer to mean

748

mesopore size. (B) Tracer-size-dependent accessible porosity ε(λ) normalized by ε(λ = 0) ≡

749

ε0. The corresponding values of τ0 and ε0 are summarized in Table 1.

RI PT

746

750

Fig. 7. Topological skeleton of the reconstructions as a function of λ = dtracer/dmeso, the ratio

752

of tracer to mean mesopore size. Data for silica monolith samples from our previous work

753

[38] (Si12, Si21, and Si26) are added for comparison. (A) Number of nodes in the branch–

754

node network connecting three branches (triple-point junctions, nt) normalized by the

755

corresponding number for point-like tracers. (B) Statistics for nodes connecting four

756

branches (quadruple-point junctions, nq).

M AN U

SC

751

757

Fig. 8. Evolving pore networks representing pore space accessibility for different values of

759

λ = dtracer/dmeso, the ratio of tracer to mean mesopore size, in the reconstructions of samples

760

CS16 (dmeso = 16.8 nm) and FP30 (dmeso = 23.9 nm).

EP

761

TE D

758

Fig. 9. 3D renderings of the void space in the reconstructions of samples CS9 (dmeso = 9.4

763

nm) and FP30 (dmeso = 23.9 nm). Orange circles highlight narrow and constricted pores in

764

the core–shell material.

AC C

762

36

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Highlights Morphology of mesoporous particles is reconstructed using electron tomography.

•

Reconstructions serve as geometrical models in simulations of hindered diffusion.

•

Expressions for diffusive hindrance factors and accessible porosity are derived.

•

Transport dynamics is related to structural features of the reconstructions.

•

Bottle-necking issues are discussed for different particle preparation protocols.

AC C

EP

TE D

M AN U

SC

RI PT

•