Intraparticle and interstitial flow in wide-pore superficially porous and fully porous particles

Accepted Manuscript Intraparticle and interstitial flow in wide-pore superficially porous and fully porous particles Mark R. Schure, Robert S. Maier, Taylor J. Shields, Clare M. Wunder, Brian M. Wagner PII: DOI: Reference:

S0009-2509(17)30537-7 http://dx.doi.org/10.1016/j.ces.2017.08.024 CES 13767

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

8 June 2017 9 August 2017 27 August 2017

Please cite this article as: M.R. Schure, R.S. Maier, T.J. Shields, C.M. Wunder, B.M. Wagner, Intraparticle and interstitial flow in wide-pore superficially porous and fully porous particles, Chemical Engineering Science (2017), doi: http://dx.doi.org/10.1016/j.ces.2017.08.024

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Intraparticle and interstitial flow in wide-pore superficially porous and fully porous particles Mark R. Schurea,∗, Robert S. Maierb,∗, Taylor J. Shieldsc , Clare M. Wunderd , Brian M. Wagnerc a

Theoretical Separation Science Laboratory Kroungold Analytical Inc., 1299 Butler Pike, Blue Bell, PA 19422 USA b

Department of Chemical Engineering and Materials Science University of Minnesota, Minneapolis, MN 55455-0132 USA c

Advanced Materials Technology, Inc. Suite 1-K, Quillen Building, 3521 Silverside Rd, Wilmington, DE 19810 USA d

Department of Chemical and Biomolecular Engineering University of Delaware, Newark, DE 19716 USA

Abstract Using a synthetic particle model, viscous fluid flow through packed beds of porous particles is directly simulated at the pore scale (1000˚ A) using the lattice Boltzmann method to characterize intraparticle and interstitial flow. Synthetic particle models are derived from synthesis conditions, scanning electron and focused ion-beam microscopy. A fully porous particle (FPP) and a superficially porous particle (SPP), derived from the FPP, both with the same external surface, are studied. Packed beds of random packings and body-centered cubic packings of the SPP and FPP models were generated by a Monte Carlo procedure that employs random translation and rigid-body rotation of the particles. Detailed velocity distributions are presented for the interstitial and intraparticle regions of the packed beds and within the particles. These results confirm that porous particle packed beds are heterogeneous systems which require extensions to classical theory for correctly predicting the resistance to flow. It is shown that SPPs require less pressure than FPPs to maintain the same flow velocity. For the random SPP and FPP packed beds, the particle hull mass flux is ≈ 10% of the interstitial flux and ≈ 3% of the total volumetric flux in the flow direction through the SPP hull. The calculated intraparticle pore velocities confirm that an internal flow, characteristic of “perfusion” chromatography, exists within the porous shell that can enhance biomolecular separations.

∗

Corresponding authors, e-mail: [email protected] and [email protected]

Preprint submitted to Elsevier

August 29, 2017

1

1. Introduction

2

Many of the newer particles utilized in liquid chromatograhy (LC) use core-shell particles, also

3

known as superficially porous particles (SPPs) (DeStefano et al., 2008; Schuster et al., 2012; Wagner

4

et al., 2012; Schuster et al., 2013; Schure and Moran, 2017; Wagner et al., 2017). These particles

5

contain a nonporous core with a porous shell on the outside. This is in contrast to fully porous

6

particles (FPPs) which have been traditionally used in LC, and “pellicular” particles (Horv´ath and

7

Lipsky, 1969), which have thicker shells than modern SPPs and were introduced in the late 1960s.

8

Many questions still exist about SPPs, including understanding the flow-field inside and outside of

9

the particle in wide-pore SPP materials. The work reported herein attempts to explain some of these

10

questions.

11

A number of large-scale simulation studies have examined the microscopic details of solute trans-

12

port through a packed bed of nonporous particles (Maier et al., 1998, 2000; Kandhai et al., 2002;

13

Maier et al., 2003; Bijeljic et al., 2004; Sullivan et al., 2005). This technology has also been used

14

to study chromatographic column processes (Schure et al., 2002; Schure and Maier, 2006; Hlushkou

15

et al., 2007; Khirevich, 2011). In all of these studies, the chromatographic particle is a nonporous

16

sphere enabling the study of fluid-phase mass transport; the positions of these particles are explicitly

17

stated in the packed bed by using particle packing software to construct the bed.

18

No transport models have been developed previously that explicitly define a detailed pore structure

19

at the particle level. In cases where transport is to be discussed for porous particles, stochastic, coarse-

20

grained effects are utilized to predict transport under random diffusion (Spaid and Phelan Jr, 1997;

21

Kandhai et al., 2002; Daneyko et al., 2015). In this and all other scenarios used in simulating packed

22

beds for studying LC, it has been assumed that fluid flow ceases in the particle phase as the pore size

23

is much too small, typically ≤ 400˚ A, to permit flow through any of the particle pore structure. This

24

would justify the assumption of a purely diffusive transport of solutes into and out of the intraparticle

25

pore space.

26

It is well-recognized that particles with large connecting porous networks, often with pore sizes

27

on the order of ≥ 5000˚ A, have a significant internal convective flow (Lloyd and Warner, 1990; Afeyan

28

et al., 1991; Rodrigues et al., 1993; Frey et al., 1993; Rodrigues, 1997; de Neuville et al., 2014; Wu

29

et al., 2015); this is often referred to as “perfusion” and is characteristic of wide-pore materials. In one

30

of these works (Frey et al., 1993) particles were referred to as “gigaporous” when the pore diameter

31

to particle diameter ratio is > 0.01. The particles studied here fit that definition. 2

32

Previous studies have not examined the microfluidic details of the surface and internal flow of wide-

33

pore particles with pore diameters ≥ 1000˚ A. If there is flow into (and out of) wide-pore particles, this

34

would enhance diffusion. The extent of convective flow in and around the pore surface of a porous

35

particle is extremely hard to ascertain by experiment, although single particle measurements have

36

been made (Pfeiffer et al., 1996).

37

A number of investigations have examined the internal flow in porous particles. For example,

38

in a well-known study (Neale et al., 1973), Brinkman’s equation (Brinkman, 1947, 1949; Durlofsky

39

and Brady, 1987) was used as the basis to predict the permeability of porous particle beds. These

40

authors calculated bed permeability as a function of the interstitial porosity for various values of β, the

41

nondimensional particle permeability. Their results show that for β > 100, the particle permeability

42

has little effect on the bed permeability, and this is the range in which the simulations in this paper

43

are relevant.

44

Guiochon and coworkers (Gritti et al., 2007) measured the porosity and permeability of SPP

45

and FPP packed columns. Although they found the SPP column less permeable than the FPP, the

46

differences in diameter and interstitial porosity of their SPP and FPP columns complicate a direct

47

comparison. The Kozeny-Carman (KC) equation (Carman, 1937; Giddings, 1965; Neue, 1997; Quinn,

48

2014) was also used to predict permeability based on particle size and interstitial porosity. (Note

49

that this approach would predict the same permeability for SPP and FPP particles of the same

50

diameter in columns of the same interstitial porosity.) They found the SPP column permeability less

51

than predicted by theory and less than the prediction for the FPP column. They suggested that

52

the greater rugosity of the SPP, which is a measure of the deviation from a purely spherical surface

53

and also referred to as surface roughness, might play a role in understanding the permeability, but

54

they did not propose a specific physical explanation. Yet in another study (Ismail et al., 2016), the

55

SPP and FPP particle diameters and interstitial porosities were more similar than those studied by

56

Guiochon and coworkers, mentioned above, and it was found that the SPP column was 22% more

57

permeable than the FPP column.

58

A porous particle can be regarded as a rough sphere by ignoring intraparticle flow. The effect

59

of surface roughness on packed bed pressure drop has been studied in several contexts. It has been

60

found (Crawford and Plumb, 1986) that the pressure drop increases with increasing surface area

61

and roughness of nonporous particles. However, it was demonstrated (Eisfeld and Schnitzlein, 2001)

62

that increased porosity associated with rough particles counteracted this effect. Others (Nemec and 3

63

Levec, 2005; Allen et al., 2013) have suggested that this effect can be attributed to departures from

64

sphericity rather than surface roughness. Nonetheless, the surface roughness literature highlights the

65

importance of accounting for porosity and surface area within the porous particle structure.

66

The influence of SPP morphology on pore-level flow can be studied with high resolution fluid

67

mechanics if a model of the SPP or FPP can be formulated. For studying wide-pore materials,

68

computer simulation of the model can yield useful and realistic information on fluid flow. One of

69

the benefits of simulation is that differences can be eliminated in particle surface geometry when

70

comparing SPP and FPP packed beds. This makes it possible to evaluate theoretical approaches,

71

such as the KC equation, for predicting porous particle flow properties. Differences in permeability

72

between SPP, FPP and solid-sphere beds are discussed in the work presented here, including the

73

contrast of some theoretical predictions with calculations reported herein.

74

In this paper a synthetic particle model derived from synthesis conditions, electron microscopy

75

and particle simulation techniques, described below, are used to study the detailed flow near, into and

76

out of the particle. Both SPP and FPP morphologies can be produced as sphere models, which are

77

included as supplemental files. The surface area and pore volume of the sphere models are in close

78

agreement with experimental particles made in the laboratory. Calculation of a high-resolution flow

79

field in the interstitial and intraparticle pore regions of the models is therefore considered a simulation

80

of flow through the experimental particles. Specific attention is given here to understanding the

81

permeability, mass and volumetric flux and detailed flow field that exist both internal and external

82

to the particle.

83

2. Experimental and Computational Methods for Particles

84

2.1. Experimental methods and techniques

85

A number of silica SPPs were prepared using the layer-by-layer method (Hayes et al., 2014) to put

86

porous shells on nonporous cores. The porous shell is composed of silica sol particles with average

87

radius rsol = 0.0575 µm. The average core radius rc = 1.65 µm and the thickness of the shell, Ls , is

88

≈ 0.58 µm. These parameters are utilized for the construction of the synthetic particle, as described

89

below in section 2.2. The particle diameter is the mean value of 70 particles, each with two orthogonal

90

measurements taken utilizing scanning electron microscopy (SEM) (Reimer, 2013). SEM images were

91

obtained using a Zeiss (Jena, Germany) Auriga 60 high resolution SEM with integrated focused ion

92

beam (FIB) (Yao, 2007) at the University of Delaware (Newark, DE).

4

93

A number of these parameters are summarized in Table 1 for both experimental and synthetic

94

particles. For the experimental particles, dividing the standard deviation by the mean gives a relative

95

variation in mean diameter of 4.6%, a relatively narrow size distribution. One of the SPP systems

96

was chosen for further study, with label 1030-27-D-SPP, a 1000˚ A mean pore diameter SPP.

97

Experimental surface areas and pore volumes were measured with a Micromeritics (Norcross, GA)

98

Tristar II instrument using the N2 BET adsorption method for surface area determination (Brunauer

99

et al., 1938) and the BJH method for pore volume determination (Barrett et al., 1951). The pore

100

volume was determined using the desorption branch of the N2 isotherm. The method of calculating

101

the surface area, porosity and hull radius of synthetic particles is described below in section 2.3. The

102

synthetic particle hull radius is rh ≈ 2.15 µm for all particles in Table 1. The agreement between

103

synthetic and experimental particle diameter is within 3.7% which was deemed acceptable.

104

Two experimental views of the SPP are shown in Figure 1 using SEM and FIB for the cross-

105

sectional view. Porosity calculations from FIB slice image analysis data predict an average porosity

106

of ≈ 50%. This was the starting point for synthetic particle construction.

107

2.2. Synthetic particle construction

108

Synthetic SPP and FPP particles are created by a two-step process. First, a homogeneous random

109

packing of sol particles is generated, sufficiently large to contain the SPP or FPP hull. Second, the

110

SPP core particle is inserted into this bulk phase, and sol particles lying outside the SPP or FPP hull

111

or inside the core particle are removed.

112

The bulk phase can be created by any method for generating random, close-packed spheres. In

113

the present work, 27,000 sol particles were gradually compressed within periodic boundary conditions

114

(PBCs) to create cube-shaped packs of specified porosity (50%, 45% and 36%). Since these packs

115

were slightly smaller than the SPP hull, periodic images of the packs were tiled in a periodic array

116

to create a bulk phase, or superstructure, that would completely contain the SPP or FPP hull. Any

117

sol particles that intersected or lay outside the hull were then removed, if the radial distance from

118

the center satisfied r > rc + LS − rsol . For the SPP, a core particle was inserted at the center of the

119

superstructure, and sol particles were removed where r < rc . Sol particles satisfying rc < r < rc + rsol

120

were retained in the model, giving the core particle a rough texture. The synthetic sol particle radius

121

for all SPP and FPP is rsol = 0.0575 µm and the synthetic core radius of the SPP is rc = 1.65 µm

122

corresponding to experimental values.

123

Defects in the particle surface are then introduced by selectively removing particles from the outer 5

Experimental particle designation

Surface area m2 /g

Pore volume cc/g

Layers

Diameter µm

Stnd. Dev. µm

1030-27-D-SPP

7.94

0.230

≈4

4.46

0.206

Synthetic particle designation

Surface area m2 /g Sm

Pore volume cc/g Vp,m

Surface area % deviation from experiment

Pore volume % deviation from experiment

Intraparticle porosity sp

1030-27-D-50-0-SPP 1030-27-D-50-0-FPP

8.26 22.4

0.200 0.470

-3.97

11.9

0.560 0.510

1030-27-D-45-0-SPP 1030-27-D-45-0-FPP

8.75 22.5

0.180 0.390

-10.2

20.6

0.520 0.460

1030-27-D-36-0-SPP 1030-27-D-36-0-FPP

9.47 22.6

0.150 0.280

-19.3

35.7

0.450 0.382

1030-27-D-50-10-SPP 1030-27-D-50-10-FPP

7.74 21.8

0.220 0.492

2.56

3.76

0.601 0.521

1030-27-D-45-10-SPP 1030-27-D-45-10-FPP

8.22 21.9

0.200 0.410

-3.56

12.2

0.560 0.470

1030-27-D-36-10-SPP 1030-27-D-36-10-FPP

8.92 21.9

0.170 0.290

-12.4

26.7

0.490 0.390

Table 1: Properties of the experimental and synthetic models for this study. The first three numbers and letters (103027-D) in the particle designation are the lab notebook entry of the experimental particle. The next two entries for the synthetic particle are the 27,000 particle cubic pack interstitial porosity (as a percentage) used in particle building and the percentage of particles removed from the first two outer layers. The abbreviation SPP and FPP are as per the text. The porosities, sp , are determined using the random sampling algorithm described in the text.

6

124

surface inwards. A number of removal algorithms have been developed that vary in sophistication.

125

Both the SPP and FPP structures are produced simultaneously so that both contain exactly the same

126

surface defects. All software is written in FORTRAN-90, C++ and MATLAB for calculations and the

127

POVRay scripting language is utilized for particle visualization.

128 129

2.3. Synthetic particle metrics 2.3.1. Hull radius

130

The hull radius is an artificial boundary used to separate the interstitial region from the particle

131

region. The hull radius rh is estimated by sorting the distance of the sol sphere centers from the

132

origin and storing the 300 largest distances. To these distances, rsol is added, and the mean and

133

standard deviation of these are determined. The mean of these distances is the hull radius, rh , which

134

is ≈ 2.15 µm. The standard deviation is less than one-thousandth of the mean.

135

2.3.2. Porosity

136

The intraparticle porosity is the ratio of porous volume to a specified particle volume. When

137

discussing FPPs, the intraparticle porous volume is unambiguous. With SPPs there are two ratios:

138

one which uses the hull volume, Vh = 43 πrh3 , and the other includes just the shell volume, Vs = Vh − Vc ,

139

where the core volume is Vc = 43 πrc3 . The latter is necessary to relate the experimental surface area

140

and pore volume to that of the model particles. We use the symbol p when the specified volume is

141

the hull volume and sp when the shell volume is used as the specified volume. Note that for FPPs,

142

the shell volume is the hull volume and sp = p . Another intraparticle porosity, 0p , used in section

143

2.5, is the ratio of the total pore volume in all particles of the packed bed to the total box (column)

144

volume.

145

146

The intraparticle porosity, p , may be computed as the complement of the ratio of solid volume of the SPP or FPP, identified as Vt,s , to hull volume Vh as:

p = 1 −

Vt,s Vh

(1)

147

3 3 For FPPs Vt,s = nsol 43 πrsol and for SPPs Vt,s = nsol 43 πrsol + Vc . This works fine for sol particles with

148

no overlap, as is the case with FPPs.

149

Alternatively, p can be determined for FPPs and SPPs by choosing random points in the particle

150

volume and taking the fraction of points not within the solid material. Typically 500,000 random

151

points are chosen with the hull volume. For SPPs, sp is determined by sampling just the shell region

7

152

to determine the pore volume. Typically, 100,000 random points are sampled between the hull and

153

core region for sp .

154

2.3.3. Pore volume

155

The mass specific pore volume, Vp,m , is:

Vp,m

sp · Vs = (Vc · ρc ) + (1 − sp ) · Vs · ρs

(2)

156

where ρc and ρs are the density of the core and shell, respectively. For FPPs Vc = 0, Vs = Vh and

157

sp = p . If the density of the core and shell are equal, ρc = ρs = ρcs , and Equation 2 reduces to

Vp,m =

sp · Vs (Vc + (1 − sp ) · Vs ) · ρcs

158

where the density of the solid material ρcs = 2.20 g/cc (Iler, 1979).

159

2.3.4. Surface area

(3)

160

The total surface area of the particle structure is the sum of all of the individual sphere surface

161

areas corrected for any overlap. If a chosen sphere has overlapping spheres in its volume, only the

162

non-overlapped area is included in the calculation of surface area. The procedure for determining the

163

overlapped area includes tiling 10242 nearly equispaced points on each sphere surface (von Laven,

164

2015) and determining the fraction of points, ξi , which are not included in any overlapping sphere

165

volume. The mass specific surface area is n P

Sm =

Si · ξi

i=1

(Vc + (1 − sp ) · Vs ) · ρcs

(4)

166

Again, for FPPs Vc = 0, Vs = Vh and sp = p . The surface area per particle is given as Si = 4πri2

167

noting that the porous particle is comprised of n spheres including the core sphere when present.

168

2.4. Model particles

169

The comparison of the experimental and synthetic particles given in Table 1 shows that the

170

synthetic particle 1030-27-D-50-10-SPP closely matches the experimental particle surface area and

171

porosity. This supports investigation into the detailed microfluidic environment around and in these

172

particles, which is described below. The visual agreement in Figure 1 between experimental and

173

simulated particles was further justification to use these models for fluid mechanics calculations.

174

Although particle models cannot be produced that match exactly with the experimental particle, at 8

175

the sol particle level, this ultimate accuracy may not be required to produce a model sufficiently

176

realistic enough to study flow in wide-pore porous materials.

177

The SPP has 12,209 sol particles plus one core particle, and the corresponding FPP particle

178

contains 24,014 randomly packed sol particles. The sol particles in the outer layers of the FPP have

179

the same etching step as the SPP and are arranged identically to the SPP. The internal porosity

180

of the SPP particle is 31.2% when the core is included in the particle volume. The porosity of the

181

FPP particle is 53.9% using Equation 1. This corresponds closely to the 50% native pack porosity

182

that was used to construct the SPP. Removing sol particles from the surface results in increasing the

183

particle porosity. There is typically ≈ 3% or less difference between sp calculated using Equation 1

184

and the randomly sampled particle algorithm described above; the number of sampling points appears

185

to make only minor differences.

186

2.4.1. Local particle porosity profile

187

Figure 2 shows the local particle porosity in the SPP and FPP as a function of radial distance

188

from the particle center. The two particles have the same sol geometry in the region surrounding

189

the SPP core. Hence, the local particle porosity profiles are identical just beyond the core radius.

190

The SPP local porosity is elevated at the core because some sol particles that intersect the core were

191

removed, leaving slightly larger pore spaces. The FPP local porosity fluctuates near the center due

192

to the small volume of the radial sampling regions.

193

The inset of Figure 2 shows the local porosity profile in the region close to the hull, located at

194

r ≈ rh . The local porosity decreases from unity at the hull to a value of ≈ 0.5 at a radial position

195

one sol diameter from the hull. If the effective radius of the SPP were decreased by one sol particle

196

radius, i.e. reff = rh − rsol , the volume excluded by reducing the radius would be roughly 90% pore

197

and 10% solid. This volume would be the equivalent of 350 sol particles or about 3% of the total sol

198

mass. In this case, the volume would be considered part of the interstitial region, and the interstitial

199

porosity, , would increase from 34.1% (based on rh ) to 38.8% (based on reff ). Figure 2 can be used

200

to estimate how the intraparticle and interstitial local porosities would change if the hull radius were

201

defined differently.

202

2.5. Construction of synthetic particle packed beds

203

Packed beds of porous particles are constructed by a two-step process. First, a packed bed of

204

spheres is generated by a hard-sphere Monte Carlo method (Tildesley and Allen, 1987). Second, each

9

205

sphere in the packed bed is replaced by a porous particle of equivalent hull radius, and the bed is

206

then further compressed by a similar Monte Carlo method.

207

The first step in the process has been described in earlier work (Maier et al., 2003). Briefly, each

208

major iteration performs one or more Monte Carlo sweeps in which each sphere is assigned a random

209

translation. The trial move is accepted if it does not result in overlap. The sweeps are followed by

210

compression (scaling) of the bounding box, which concludes the major iteration. The bounding box

211

may have a combination of impenetrable and periodic faces, or it can be described by some other

212

simple geometry, such as a cylinder.

213

The second step of the process begins by replacing the packed spheres with SPPs or FPPs. Each

214

porous particle is given a random rigid-body rotation that preserves the internal geometry of the

215

sol particles. Because the porous particle hull is a bounding sphere around the sol particles, simply

216

substituting them for solid spheres leaves loose contacts between neighboring particles. Therefore,

217

further iterations of the Monte Carlo method are performed on the porous particle packed bed.

218

The random move for each particle consists of both a rigid-body rotation and translation. In-

219

dividual pairs of sol particles belonging to neighboring porous particles must be tested for overlap.

220

Overlap testing is made more efficient by restricting attention to sol particles that populate the exte-

221

rior of the particles. During these iterations, the porous particles undergo relatively small rotations

222

and translations, and the compression typically eliminates gaps on the order of a sol radius between

223

neighboring particles. Although the neighboring particle hulls overlap slightly, the sol particles do

224

not. Because the relative particle motion is so limited, the interstitial region of the resulting porous

225

particle bed remains similar to the original packed bed of spheres.

226

A random packing of porous particles was simulated by replacing the spheres in a close-packed

227

bed of 125 solid spheres contained in a cubic bounding box with periodic boundaries. This is shown

228

in Figure 3A. The dimension of the original bounding box is 9.35 · rh with 36% porosity. After further

229

Monte Carlo compression, the bounding box dimension was reduced by 1% or ≈ 3 sol radii. The

230

resulting interstitial region is 34% of the bounding box volume, compared with 36% shown in Figure

231

3B.

233

A body-centered cubic (BCC) packing of porous particles was constructed by placing a porous √ particle at the center of a BCC unit cell. The initial unit cell size was 4 · rh / 3. One porous particle

234

is located at the center and one at each unit cell corner. After compression, the unit cell size was

235

reduced by ≈ 1%, slightly more than one sol radius. The resulting interstitial region is 29% of the

232

10

236

unit cell volume, compared with 32% for the original unit cell.

237

A question remains whether the SPP or FPP could be packed more densely. This is difficult to

238

assess because the porous particle surface is not uniform, but it would depend to a certain extent on

239

the ability of the porous particle to rotate into more compact positions. It is possible that starting with

240

a looser bed of solid spheres would give the SPP or FPP room to explore more compact orientations.

241

2.5.1. Packed bed porosity

242

Table 2 summarizes the box size, the number of particle structures present in the bed and the

243

three porosity values characteristic of the solid-sphere and porous particle packed beds utilized in the

244

following calculations. The total porosity, T , is the sum of the intraparticle porosity, 0p , which is

245

the ratio of void space in the particle shells to the total bed volume, and the interstitial porosity, ,

246

which is the ratio of the volume between particle hulls and the total bed volume of the column. For

247

a solid-sphere packing, the bed porosity and interstitial porosity are the same.

248

Porous particles substantially increase the total porosity. The total porosity of the random bed

249

solid-sphere packing is 36%, which is the random close packing limit (Torquato et al., 2000; Song

250

et al., 2008) for spheres. By replacing the solid spheres with SPPs, the total porosity increases to

251

55%. By replacing the solid spheres with FPPs, the total porosity increases to 70%. These increases

252

in total porosity are due primarily to the void space in the porous particles, since their interstitial

253

volume differs by only 2% to 3% from the solid-sphere bed.

254

Although porous particles increase the total bed porosity, their void volume does not dominate the

255

interstitial volume. For the random SPP packed bed, the intraparticle and interstitial pore volumes

256

are 38% and 62% of the total bed volume, respectively. For the random FPP packed bed, the

257

corresponding ratios are 49% and 51%. Increasing the sol packing density within the porous particle

258

is not likely to have much impact on these ratios. Assuming the sol could be packed to 64% density,

259

the intraparticle share of the total bed volume would increase by 3% and 4% for the SPP and FPP,

260

respectively.

261

2.5.2. Nearest-surface distributions

262

The nearest-surface distribution (NSD) is a probability density function for the distance from

263

a point in the fluid phase to the nearest solid surface and is an indicator of the pore size. The

264

NSD for the random SPP packed bed is shown in Figure 4. The NSD is separated into two parts;

265

an intraparticle and an interstitial distribution. The combined area beneath the intraparticle and

266

interstitial distributions is equal to unity, based on relative pore volumes of 38% and 62%, respectively. 11

Packing

Box size µm

random SPP random FPP random sphere BCC SPP BCC FPP BCC sphere

19.89 19.89 20.08 4.896 4.896 4.959

Number Total of porosity particles T 125 125 125 2 2 2

0.550 0.696 0.360 0.517 0.674 0.320

Bed porosity interstitial intraparticle  0p 0.341 0.341 0.360 0.294 0.294 0.320

0.208 0.355 0.000 0.223 0.380 0.000

Table 2: The bed packing designations used in this study derived from 1030-27-D-50-10-SPP and 1030-27-D-50-10-FPP.

267

However, the two distributions are nearly disjoint, reflecting the different characteristic length scales

268

of the particle pore structure, on the order of dsol , and the particle diameter, on the order of 2rh .

269

The intraparticle distribution ranges from zero to approximately one sol diameter. This range is

270

much wider than a homogeneous and infinite random sphere packing where the upper limit is closer

271

to a sol particle radius. The intraparticle distribution has a higher limit here because the hull forms

272

an open boundary and the distance from the hull to the nearest sol particle, in many cases, exceeds

273

a sol particle radius.

274

The intraparticle distributions for SPP and FPP are compared in the inset to Figure 4. The area

275

under the FPP curve is normalized to unity, while the area under the SPP curve reflects its lesser

276

pore volume. Both the FPP and SPP have the same large-pore distributions because the outermost

277

region of the particles, near the hull, are identical. The FPP distribution has a higher density of small

278

pores than the SPP because it has a porous region of close-packed sol corresponding to the solid SPP

279

core region.

280

The SPP interstitial distribution in Figure 4 is also representative of the FPP interstitial distri-

281

bution, since the two packed beds have identical construction. In the interstitial region, the distance

282

to the nearest sol particle ranges from zero to approximately eight sol diameters. Note the interstitial

283

distribution is actually an approximation. It is based on the NSD for the solid-sphere random pack-

284

ing, which gives the distance from interstitial points in the fluid to the particle hull surfaces. The

285

additional distance from the hull surfaces to the nearest sol particles was approximated by shifting

286

this NSD one half sol diameter to the right (roughly the average distance from the hull to the nearest

287

sol particle). This shift is responsible for the discontinuity in the distribution shown in Figure 4. The

288

true interstitial distribution would be continuous and increasing from zero to a peak near half of a 12

289

sol diameter, while the remainder of the true distribution would be similar to the approximation.

290

3. Fluid mechanics

291

3.1. Lattice Boltzmann method

292

Fluid motion in the pore space is described by the incompressible Navier-Stokes equations with

293

appropriate initial and boundary conditions (White and Corfield, 2006). Fluid flow was simulated

294

under the assumptions of steady, saturated flow at small Mach number using the lattice Boltzmann

295

(LB) method. The LB method solves the discrete-velocity Boltzmann equation for the fluid mass and

296

momentum density distributions (Sterling and Chen, 1996; Maier et al., 1998; Maier and Bernard,

297

2010; Kr¨ uger et al., 2016). The method is a pseudo-transient, explicit scheme for recovering Navier-

298

Stokes behavior in the low Mach-number limit. The LB equations are solved on a regular 3-D grid

299

with the D3Q19 method (Maier et al., 1998; Wagner, 2008; Kr¨ uger et al., 2016), in which physical

300

space is discretized on a computational grid and velocity space is represented at each point by a set

301

of 19 direction vectors. The computational grid is then superimposed on the packed bed geometry.

302

The LB relaxation parameter which controls the rate at which local equilibrium is imposed at

303

each time step was set to unity. This parameter establishes the ratio of the lattice grid spacing

304

and time step in the method, typically referred to as the “lattice viscosity.” The desired physical

305

fluid parameters, including the dynamic viscosity, 1.0 × 10−3 m2 s−1 , are obtained by setting the grid

306

spacing to achieve the necessary spatial resolution (see below) and solving for the time step length

307

that makes the lattice and physical Reynolds numbers equivalent.

308

The method is made parallel by domain decomposition of the computational grid; sub-domains

309

are mapped to computer processors using the MPI Cartesian communicator topology (MPI Forum,

310

2016; Gropp et al., 1999). Border cell data is exchanged between adjacent sub-domains at each LB

311

time step. Only grid cells within the pore space are part of the flow calculation.

312

Flow was driven by a uniform pressure gradient, and no-slip conditions were enforced at all solid

313

boundaries. Gravitational forces were not included. PBCs were used to wrap the flow around to the

314

opposite face of the simulation box. The flow simulation was initiated at zero velocity and iterated to

315

a steady state. Application of the method to porous media flows and its accuracy has been evaluated

316

in a previous work (Maier and Bernard, 2010).

317

The samples of porous media simulated in this work are relatively small because finely-spaced

318

computational grids are required to accurately model fluid dynamics in the pore spaces. This is

319

especially true for porous particles because the flow must be resolved in the sol pore space and in the 13

320

interstitial region surrounding the particles. Periodic boundaries are therefore an attractive method

321

for simulating flows in small samples without the entrance or exit effects associated with external

322

boundaries. Although the sample is periodic on the length scale of the box, there is no repetition of

323

macropore geometry within the sample, and each porous particle has a random rotational orientation.

324

The flow simulations are intended to describe velocity distributions within and around the particles.

325

Previous results show that for monodisperse spheres of diameter d, the velocity distribution becomes

326

well-converged when the grid spacing, ∆x, is decreased from d/20 to d/40 as shown in Figure 13, p.248 of

327

(Maier and Bernard, 2010). In the present work, ∆x = dsol /24, where dsol is the sol particle diameter.

328

Thus, the velocity distributions for the sol pore space have adequate resolution, while distributions

329

for the larger macropore spaces have higher resolution. Flow was simulated in the random SPP and

330

FPP packs using grid dimensions of 4150 × 4150 × 4150. The BCC SPP and FPP packs use grid

331

dimensions of 1024 × 1024 × 1024, the random sphere pack uses grid dimensions of 180 × 180 × 180

332

and the BCC sphere pack has dimensions of 176 × 176 × 176. These packs are described in Table 2.

333

3.2. Pressure and permeability

334

The particle Reynolds number is defined as Resol = vdsol /ν, where v is the mean pore velocity and

335

ν the kinematic viscosity. Flows were simulated at Resol = 1 × 10−3 . The Darcy flux, or superficial

336

velocity, is defined as q = εv. The permeability, k, as defined in Darcy’s Law (Brinkman, 1947;

337

Giddings, 1965; Bear, 2014; Quinn, 2014) is q = −(k/µ)∇p, where ∇p is the pressure gradient and µ

338

is the dynamic viscosity. In this work, the pressure gradient is always aligned with the z-axis, so the

339

permeability is calculated simply as k = qz µ/(dp/dz).

340

Higher pressure is required to drive the mean pore velocity at constant Resol with porous particles.

341

For the random packings, the required pressure is about 44% (SPP) and 100% (FPP) higher than the

342

solid-sphere packing, as shown in Table 3. For the SPP and FPP BCC packings, the pressure is 36%

343

and 96% higher than the BCC solid-sphere packing, respectively. Considering both types of porous

344

particles and both packing arrangements, SPPs require less pressure than FPPs to maintain the same

345

flow velocity conditions.

346

347

Although the required pressure is higher for porous particles, it is not nearly so high as predicted by the KC equation, given by

dp/dz = [180µ(1 − T )2 qz ]/[d2eff 3T ] 348

(5)

for spherical particles of diameter deff . While the required pressure in the solid-sphere packing is well 14

Packing

qz (m/s)

random SPP 4.75 × 10−3 random FPP 6.05 × 10−3 random sphere 3.13 × 10−3 BCC SPP 4.47 × 10−3 BCC FPP 5.86 × 10−3 BCC sphere 2.78 × 10−3

dp/dz (kg/m2 s2 )

k (m2 )

3.78 × 108 5.31 × 108 2.62 × 108 4.16 × 108 6.01 × 108 3.06 × 108

1.26 × 10−14 1.14 × 10−14 1.20 × 10−14 1.07 × 10−14 9.75 × 10−15 9.10 × 10−15

Nondimensional average pore velocity: particle interstitial 0.067 0.059 0 0.069 0.059 0

1.57 1.98 1.00 1.71 2.22 1.00

Table 3: Bed velocity, qz , pressure gradient, dp/dz, permeability from Darcy’s equation, k, and nondimensional pore velocity in the direction of the pressure gradient for simulated solid and porous particle packed beds. All cases are simulated at Resol = 1.00 × 10−3 and the average pore velocity in the direction of the pressure gradient is v z = 8.694 × 10−3 m/s. The average pore velocity within the porous particles and in the interstitial region is made nondimensional by dividing by v z .

349

predicted by Equation 5, the pressures in the SPP and FPP packings are far lower than predicted

350

(see Table 4). These predictions used deff = 6/SV , where SV is the ratio of total surface area to

351

solid volume in the bed. The departure of the simulation results from the predictions indicates

352

that packings of porous particles are not equivalent to homogeneous packings of sol particles. Porous

353

particles admit flow preferentially through the interstitial region and are therefore better characterized

354

as heterogeneous media. The homogeneity assumption underlying Equation 5 has been discussed in

355

related works (Schlueter and Witherspoon, 1994). Packing p random FPP random SPP excl. core random SPP incl. core BCC FPP BCC SPP excl. core BCC SPP incl. core

0.540 0.571 0.310 0.540 0.571 0.310

Particle deff 1.15 × 10−7 3.16 × 10−7 3.16 × 10−7 1.15 × 10−7 3.16 × 10−7 3.16 × 10−7

Bed kp

kN EN

k

5.47 × 10−17 5.63 × 10−16 3.48 × 10−17 5.47 × 10−17 5.63 × 10−16 3.48 × 10−17

1.12 × 10−14 1.35 × 10−14 1.10 × 10−14 6.36 × 10−15 8.10 × 10−15 6.24 × 10−15

1.14 × 10−14 1.26 × 10−14 1.26 × 10−14 9.75 × 10−15 1.07 × 10−14 1.07 × 10−14

Table 4: Comparison of particle and bed permeabilities. The simulation permeability, k, theoretical permeability with Brinkman’s correction (Neale et al., 1973), kN EN , and particle permeability, kp , using the KC equation are given with both random and BCC packing. The particle porosity, p , and the effective diameter of the sol particles, deff , taking into account the surface area of the core, are used to calculate kp .

356

The permeabilities of the SPP and FPP packed beds do not differ greatly from the solid-sphere

357

packed bed given in Table 3. On one hand, this is surprising because the porous particles have a

358

much higher specific surface area which correlates with increased resistance to flow in a homogeneous 15

359

particle packing. It is unsurprising, however, because the interstitial porosity of the packs are similar,

360

as shown in Table 2, and therefore can accommodate similar volumes of flow at the same pressure.

361

Even so, the similarity is limited.

362

Previously, it was noted that the gap between the porous particle hull and the sol particles is on

363

the order of the sol radius. This additional void space between the particles allows the SPP and FPP

364

to accommodate more flow through the interstitial region than their interstitial void fraction would

365

otherwise suggest. This capacity advantage is counterbalanced by the need to pass a greater volume

366

of fluid through the interstitial region to achieve the same superficial velocity as the sphere packing.

367

The additional fluid in the particles cannot be pushed through the particle quickly, and it is instead

368

pushed around the particles at a faster rate.

369

It is also interesting to note that the SPP bed permeability is slightly greater than the solid-sphere

370

bed, while the FPP bed permeability is slightly lower. The SPP bed permeability is approximately

371

10% greater than the FPP. A similar result was reported (Ismail et al., 2016) where the SPP column

372

permeability was 22% greater than the FPP. The balance between increased interstitial capacity and

373

flow volume thus works slightly in favor of the SPP packing and slightly against the FPP packing.

374

But the advantage is small and depends on the specific packing density of the SPP particles relative

375

to the solid spheres.

376

Theoretical predictions of FPP permeability using the approach in (Neale et al., 1973) are accurate

377

in the present case for the random bed. Table 4 compares the simulated and predicted bed perme-

378

abilities and these differ by < 2% for the random FPP packing. The prediction for the BCC FPP

379

packing is less accurate, differing by 25% from the simulation. Theoretical predictions of permeability

380

are known to be less accurate for BCC packings than for random packings. In a well-known work on

381

flow in crystalline packings (Zick and Homsy, 1982), the drag coefficient for BCC solid spheres differs

382

by over 25% from the KC equation, but it agrees within 2% with our result in Table 3. Note that the

383

drag coefficient for BCC spheres is given as FD = (2/9)rh2 /(1 − )/162.9.

384

The prediction of SPP random bed permeability is more problematic than the FPP case because

385

estimation of kN EN depends on an estimate of the particle permeability. For an FPP composed of

386

monodisperse sol, it is straightforward to compute kp from the sol diameter and particle porosity.

387

For a SPP, the porosity can be defined either to include or exclude the core particle, leading to

388

different values of both the particle and bed permeability. Table 4 gives predictions of the SPP

389

bed permeability for both of these cases. Excluding the core particle volume results in a 7% over16

390

prediction of the random SPP bed permeability, while including it leads to a 12% under-prediction;

391

these are compared to the simulation value k. The former prediction is preferred, and the rationale

392

for excluding the core particle when computing the internal particle porosity is that the KC equation

393

is not intended to predict the permeability of a heterogeneous system. The calculation of permeability

394

for a SPP with Brinkman-level correction has been presented, (Masliyah et al., 1987) but was not

395

computed for this work.

396

3.3. Fluid velocity distributions

397

The fluid velocity distribution in a packed bed is a density function describing the relative mass

398

(or volume) of fluid moving at a given velocity, shown in Figure 5. In a packed bed, most of the fluid

399

is in close proximity to a solid surface. Assuming a pressure-driven flow, this fluid has relatively low

400

velocity. Therefore, the velocity distribution in a packed bed typically has a sharp peak (mode) near

401

v = 0 and decreases exponentially with increasing v. Packed beds of porous particles have the same

402

general distribution as solid-sphere beds, but the mode is more pronounced because the surface area

403

is large compared to a solid-particle bed.

404

In porous particle beds, all solid surfaces are internal to the particles because the hull is defined as

405

a sphere surrounding the sol with radius rh . Therefore, the interstitial region comprises the volume

406

that has no solid surfaces. The intraparticle fluid velocity distribution is mostly slow flow and has

407

little overlap with the interstitial velocity distribution.

408

The interstitial flow region of the porous particle bed is geometrically similar to the solid-sphere

409

bed (of equivalent radius rh ). For this reason it was thought that the flow distributions might also be

410

similar. This is incorrect because the interstitial region in a bed of solid spheres is bounded by the

411

sphere surfaces, but for a bed of SPPs or FPPs the interstitial region is bounded by the (imaginary)

412

hull. The fluid gap between the hull and the sol surfaces is on the order of the sol radius, and this

413

gap allows flow at the shell surface to approach the average pore velocity.

414

3.4. Mass flow and flux through the hull

415

Flux through the porous particle hull is the component of fluid momentum density normal to the

416

surface, f = ρv · n, with units of kg/m2 s, where v is the velocity at a point on the surface and ρ is the

417

fluid density. Conservation of mass implies that the integral of flux over the hull surface S is zero Z

Z f dA /

S

dA = 0 S

17

(6)

418

The average hull flux magnitude, f¯h , is therefore defined as f¯h =

Z

Z |f |dA /

S

dA

(7)

S

419

where S is the hull surface of all particles in the packed bed. The average total mass flow into a single

420

particle hull is then defined as Z f¯h Fh = · dA 2N S

421

(8)

with units of kg/s and with N equal to the number of particles.

422

Hull flux magnitudes are, in general, less than the superficial flux magnitude given by ρqz . This is

423

because fluid velocities near the hull surface are less than the average pore velocity, and their vectors

424

are nearly tangent to the surface. For the random SPP packed bed, the hull flux is ≈ 9% of the

425

interstitial flux. For the random FPP packed bed, the hull flux is ≈ 10% of the interstitial flux and

426

both are in the direction of the pressure gradient.

427

Flow into the SPP is significant despite the relatively small flux magnitude. Approximately one

428

third of the bed volume flows into the particles in the time required to travel one SPP diameter

429

at the superficial velocity, which is approximately one millisecond for Resol = 1.00 × 10−3 . The

430

corresponding nondimensional expression is Fh 1 2rh Fbh = · · ρ L3 q

(9)

431

where the first term on the right is the volume flow rate into the hull, the second term is the reciprocal

432

bed volume and the third term is the characteristic time to travel one particle diameter. Fbh is therefore

433

the number of bed volumes passing into the SPP hulls during the time required to travel one SPP

434

diameter at the superficial velocity. These values are 0.34 and 0.37 for the random and BCC SPP

435

packings, respectively. Fh , Fbh and f¯h are tabulated in Table 5.

436

Although higher pressure is needed to drive flow to the standard Resol through the FPP bed and

437

the resulting mass flow is greater (Tables 3 and 5), the difference in mass flow is proportional to the

438

difference in pressure. For the same pressure gradient, mass flow rates into the SPP and FPP will be

439

the same and the superficial velocity will be greater in the SPP bed.

440

Higher pressure is also needed to drive flow through the BCC bed compared to the random bed.

441

The BCC packing is denser, and higher pressure is required to achieve the standard Resol . However,

18

442

the mass flow rate into the particles is actually slightly lower in the BCC bed than the random

443

packing, despite the elevated pressure. The explanation is that the mass flow rate into the particle is

444

proportional to the interstitial flux, which is the product of the interstitial porosity and pore velocity,

445

and this product is lower in the BCC packing. Packing

ch F

f¯h (kg/m2 s)

3.29 × 10−1 2.88 × 10−1 3.74 × 10−1 3.13 × 10−1

4.27 × 10−1 6.07 × 10−1 4.02 × 10−1 5.72 × 10−1

Fh (kg/s)

random SPP 1.26 × 10−11 random FPP 1.78 × 10−11 BCC SPP 1.17 × 10−11 BCC FPP 1.69 × 10−11

ch , and average flux magnitude, f¯h , through the particle hull area. Table 5: Mass flow, Fh , nondimensional mass flow, F −3 All cases are simulated at Resol = 1 × 10 .

446

3.5. Flux distribution on the particle hull surface

447

The flux distribution over the hull surface is a density function describing the probability of a

448

given flux value being observed on the surface. Flux distributions were calculated for each porous

449

particle and averaged into the overall flux distribution.

450

The overall flux distribution has a Laplacian shape as shown in Figure 6. It has a sharp mode

451

at fh = 0 and decreases exponentially with distance from the origin. Figure 6 also shows the flux

452

distributions for the two individual particles with the least and greatest flux magnitude. These differ

453

in amplitude near the origin but are otherwise similar. The flux magnitude of these two particles is

454

23% less and 22% greater than the average flux magnitude for all particles, respectively.

455

3.6. Volumetric flux in the direction of the pressure gradient

456

Although a significant volume of fluid crosses the porous particle hull, the interstitial region

457

carries the vast majority of fluid through the bed. Considering flow only in the direction of the

458

pressure gradient, the interstitial volumetric flux is ≈ 97% of the total. The volume flowing into the

459

particles and the volume flowing through the interstitial region are both large. This apparent paradox

460

emphasizes the fact that the two are fundamentally different measurements.

461

Total hull area scales with the bed volume, L3 , while the bed cross section scales with L2 . Fluid

462

mass flow is proportional to surface area in both cases, so particle hull flow scales with L3 and bed

463

flow scales with L2 . The ratio of hull flow to bed flow will therefore change with the bed dimensions,

464

even if the Reynolds number remains constant. 19

465

Intraparticle flux is defined as the volumetric flux through the particles in the direction of the

466

pressure gradient. It is the difference between flux through the entire bed and flux through the

467

interstitial region. Table 6 gives the volumetric flux in the direction of the pressure gradient for the

468

intraparticle region, the interstitial region and the entire bed. Each of the flux values is defined as

469

the volumetric flow through the cross-sectional area of the bed. Packing

Volumetric flux (m/s) intraparticle interstitial bed (total)

random SPP random FPP random sphere BCC SPP BCC FPP BCC sphere

1.21 × 10−4 1.82 × 10−4 0 1.34 × 10−4 1.96 × 10−4 0

4.66 × 10−3 5.87 × 10−3 3.13 × 10−3 4.36 × 10−3 5.67 × 10−3 2.78 × 10−3

4.70 × 10−3 6.06 × 10−3 3.13 × 10−3 4.49 × 10−3 5.86 × 10−3 2.78 × 10−3

Area (m2 ) 3.96 × 10−10 3.96 × 10−10 4.03 × 10−10 2.40 × 10−11 2.40 × 10−11 2.46 × 10−11

Table 6: Volumetric flux in the direction of the pressure gradient for the intraparticle region, interstitial region and bed. Flow in each bed has the same mean pore velocity, corresponding to Resol = 1 × 10−3 . Area is the cross-sectional area of the bed.

470

3.7. Internal fluid velocity

471

The pore velocity magnitude is the length of the velocity vector, ||v||, at a given point or averaged

472

over a given volume, and it is used to characterize flow within the porous particle as a function of

473

radial distance from its center. Figure 7 shows the average velocity magnitude as a function of radial

474

position inside the particle hull. The average magnitude is normalized by the overall mean pore

475

velocity, v z .

476

The average velocity magnitude at the SPP hull surfaces is ≈ 30% of the average velocity mag-

477

nitude in the entire bed volume. Moving from the hull toward the center, the average magnitude

478

decreases exponentially over a distance of one sol diameter and remains approximately constant over

479

the remaining radial distance to the core. The magnitude of this relatively constant velocity value is

480

approximately two orders of magnitude smaller than the average surface magnitude. Individual SPP

481

particles deviate from this profile, but the difference is small. The particle with the greatest internal

482

velocity magnitude has a profile that varies from 20% to 30% higher than the average profile. The

483

particle with least internal velocity magnitude varies from 20% to 35% below the average.

484

The FPP and SPP results are similar, observing that the region of diminished pore velocity in

485

the FPP spans nearly the entire particle radius. The mean pore velocity is slightly lower in the FPP, 20

486

as shown in Table 3, and the velocity range is slightly wider, as illustrated by the minimum and

487

maximum curves in Figures 7A and 7B. The extent to which a wider velocity range within the shells

488

of particles affects zone broadening in a chromatography column is not clear.

489

3.7.1. Reverse flow

490

In a pressure-driven flow through a packed bed of spheres, ring vortices may form at the down-

491

stream side of a sphere at very small Reynolds numbers. This phenomena is well known experimentally

492

in creeping flow past tangent spheres (Van Dyke and Van Dyke, 1982) and from a flow simulation in

493

a crystalline array of spheres (Maier et al., 2000). These vortices contribute a small amount of reverse

494

flow, which is perhaps best described as a local recirculation caused by contracting-then-expanding

495

flow. The phenomena is essentially absent in a random array of spheres, at least on a length scale

496

comparable to the pore size, because the random arrangement destroys the symmetry necessary for

497

such flow structures. Nevertheless, random sphere packs do have very small amounts of reverse flow.

498

However, these are to be found in pores oriented transversely to the pressure gradient, where the local

499

flow direction forms a very slight angle against the general flow.

500

In a BCC sphere packing, if solid spheres are replaced by porous particles, the sol structure

501

inhibits the development of pore-scale vortices. However, the interstitial region still has symmetrical

502

contracting and then expanding flow, and the resulting back pressure induces reverse flow in the

503

downstream side of the SPP sol, shown in Figure 8. This is in roughly the same location where the

504

ring vortex would appear between solid spheres. However, in the case of a random pack of SPPs, flow

505

against the gradient is resisted in the sol layers.

506

Also shown in Figure 8 are the FPP velocity field and probability density. While the FPP and

507

SPP interstitial flows are similar, the FPP velocity field is more collimated in the vortex region; the

508

reverse flow is more tightly focused, and its magnitude is less than in the vortex region of the SPP.

509

3.8. Perfusion in the model particle

510

In early work describing perfusion chromatography (Afeyan et al., 1991), an equation was devel-

511

oped (van Kreveld and van den Hoed, 1978) that equates the pore velocity to the increase in effective

512

pore diffusion coefficient Deff :

Deff = D + v · dp /2

(10)

513

where D is the solute pore diffusion coefficient in the absence of flow, dp is the particle diameter and

514

v is the velocity of fluid in the pore. To estimate Deff in a typical chromatography column, we assume 21

515

a mean pore velocity of v = 1.5 × 10−3 m s−1 . From Table 3 we note the ratio of mean perfusion

516

velocity (intraparticle) to overall mean pore velocity is 0.067 in a random SPP packing. We therefore

517

extrapolate the mean perfusion velocity in a typical chromatography column as v = 1.0 × 10−4 =

518

0.067 × 1.5 × 10−3 m s−1 . Inserting this into Equation 10, along with dp = 4.30 × 10−6 m, gives

519

Deff = D + 4.3 × 10−10 m2 s−1 or Deff = D + 4.3 × 10−6 cm2 s−1 .

520

This suggests that for small proteins where D ≈ 1 × 10−6 cm2 s−1 , Deff is enhanced by more than

521

a factor of four under these conditions. Because the solid phase mass transport efficiency is inversely

522

proportional to the effective pore diffusion coefficient (Giddings, 1965; Afeyan et al., 1991; Frey et al.,

523

1993; Carta and Jungbauer, 2010), the efficiency term increases thus lowering the solid phase plate

524

height by a factor of four. This is significant for biomolecule separations where diffusion coefficients

525

are small. Larger flow rates and interstitial velocities will increase v and cause an increase in Deff .

526

In the previous analysis, the value of D will typically be less than 1 × 10−6 cm2 s−1 , typical

527

of biomolecules. For example, the free solution diffusion coefficient of lysozyme with a molecular

528

weight of ≈ 14 kDa is ≈ 1 × 10−6 cm2 s−1 (Tanford, 1961; Mattisson et al., 2000). This suggests

529

that the pore diffusion coefficient will be smaller than this value (the pore diffusion coefficient is

530

always less than the free solution diffusion coefficient), and the effective pore diffusion coefficient will

531

be increased significantly by convection. The increase in convection-aided effective pore diffusion

532

becomes significant for molecules of higher molecular weight (and lower diffusion coefficients) than

533

lysozyme when the molecular size and weight is larger and all other conditions are held constant. When

534

the molecular size is a significant fraction of the pore size, restricted diffusion becomes important and

535

lowers chromatographic efficiency (Carta and Jungbauer, 2010; Wagner et al., 2017). Under the

536

conditions discussed here, the pore velocity may be large enough to help reduce this effect.

537

4. Discussion and Conclusions

538

Porous particles have much larger specific surface area than solid particles and higher pressures

539

are required to drive the flow to a specified pore Reynolds number. However, the pressure required to

540

achieve a specified volumetric flux is comparable for solid, SPP and FPP particles of equivalent hull

541

radius because the flux also depends on the bed volume. In fact, the SPP actually requires a slightly

542

lower pressure than solid spheres to achieve a specified superficial velocity. This result is reflected

543

in the fact that bed permeabilities hardly differ among the solid, SPP and FPP packed beds. FPP

544

permeability predicted by Brinkman’s equation is accurate, though the predicted SPP permeability

22

545

is less accurate due to the heterogeneous structure.

546

The SPP and FPP particles contain a significant fraction of the bed void volume, but the interstitial

547

void space carries the majority of fluid through the bed. The average fluid speed at the particle surfaces

548

is significant, but only a fraction of the flow is directed into the particle. The velocity magnitude

549

drops exponentially with internal distance from the surface. The average intraparticle pore velocity

550

within the shell is approximately two orders of magnitude less than the interstitial pore velocity.

551

Although the average intraparticle velocity is small in the direction of the pressure gradient,

552

volumetric flow into the particles is significant. Approximately one-third of the bed void volume flows

553

into the particle surfaces in the time required for fluid to travel one particle diameter at the superficial

554

velocity. Thus, the rate of fluid exchange is substantial near the particle surfaces.

555

One of the least understood aspects of LC is the convection and diffusion of large solute molecules

556

passing near, into and out of pores. For example, the rate of diffusion of proteins into and out of

557

pores limits the resolution of chromatographic separation due to the small diffusion coefficient which

558

is characteristic of larger molecules (Carta and Jungbauer, 2010). When the molecular size of the

559

solute is of the order of the pore diameter or characteristic length, even slower diffusion occurs into

560

and out of the pores. This effect is hydrodynamic in origin and is referred to as restricted diffusion

561

(Brenner and Gaydos, 1977; Dechadilok and Deen, 2006; Carta and Jungbauer, 2010).

562

Although the Brinkman level of approximation can be extended to the prediction of average

563

intraparticle velocities, direct simulation methods, such as the LB method used here, allow inspection

564

of the flow field at the pore level and, ultimately, the determination of effective solute diffusion as a

565

function of position in the porous particle. The authors are currently using this approach to model

566

the motion of finite-size particles driven by convection. This will hopefully facilitate the design of

567

new particles specifically for analytical bioseparations and may also explain the penetration depth

568

effect that appears in some adsorbents where the porous shell thickness seems to have a minor effect

569

on retention of medium to large biomolecules. A subsequent article will investigate aspects of mass

570

transport of finite-size solutes along with residence time and depth penetration in SPPs and FPPs.

571

Pore-level fluid simulation also can assist in the understanding of shear forces in polymer flows.

572

In size-exclusion chromatography (SEC) (Striegel et al., 2009b), degradation of polymers has long

573

been known to occur (Striegel, 2008; Striegel et al., 2009a,b), but the mechanisms of degradation are

574

inferred from related phenomena (Harrington and Zimm, 1965) and not directly observed. Under a

575

zero-concentration model of polymer flow, such that the presence of the polymer is assumed not to 23

576

affect the shear field, polymer extensional forces in the pore vicinity can be estimated using the present

577

methods. This would facilitate comparison with known physical data on bond scission (Caruso et al.,

578

2009).

579

In recent years, confocal laser scanning microscopy (CLSM) has provided particle-level details in

580

packed capillaries for particles with diameters greater than 1 µm (Reisinga et al., 2016). For sol

581

particles ≤ 100 nm in diameter, CLSM may not have the necessary resolution, so FIB and other

582

techniques for particle-level detail are required. Short of having a sol particle-resolved model for the

583

porous particle, packing methods, as used here, can be effective model builders for porous particles.

584

When model building software controls the shell thickness and particle defect generation, it is possible

585

to examine particle morphologies that not only model new experimental particles, but can be used as

586

particle models to explore new pore geometry schemes.

587

5. Symbol Glossary D

Solute pore diffusion coefficient

deff

Effective particle diameter

Deff

Solute effective pore diffusion coefficient

dp

Particle diameter

dsol

Sol particle diameter

f

Flux momentum density normal to particle surface

f¯h

Average hull flux magnitude

Fh

Mass flow into the hull

Fbh

Number of bed volumes passing into SPP hulls for finite time

k

Darcy (bed) permeability

kp

Intraparticle permeability

L

Length of cubic box used in velocity calculations

Ls

Porous shell thickness of SPP

n

Number of spheres per particle

N

Number of particles per bed

nsol

Number of sol particles per spherical particle

n

Surface normal

q, qz

Superficial velocity (Darcy flux)

24

r

radial distance

rc

Core radius

reff

Effective particle radius

rh

Hull radius

ri

Radius of the ith particle

rsol

Sol particle radius

Resol

Particle Reynolds number

Si

Surface area per particle

Sm

Mass specific surface area

SV

Surface area to volume ratio

v

Fluid velocity at a point

||v||

Euclidean norm of the vector v

vx , vy , vz

Velocity components in x, y and z

v

Mean pore velocity

vz

Mean velocity in the flow (z) direction

Vc

Core volume

Vh

Hull volume

Vp,m

Pore volume per mass of particles

Vs

Shell volume

Vt,s

Particle solid volume

∆x

Grid spacing

β

Nondimensional particle permeability



Interstitial porosity

p

Intraparticle porosity using the particle hull as the total volume

sp

Intraparticle porosity using the shell as the total volume

0p

The ratio of total pore volume to the total box volume

T

Total porosity

µ

Dynamic viscosity

ν

Kinematic viscosity

ρc

Density of the core 25

ρcs

Density of the core and shell

ρs

Density of the shell material

ρ

Density of fluid

ξi

Fraction of points used for surface area calculation

∇p, dp/dz Pressure gradient ∇v

588

Velocity spatial derivatives

Acknowledgments

589

The support of the National Institutes of Health under grant R44-GM108122-02 is gratefully

590

acknowledged. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the Uni-

591

versity of Minnesota for providing resources that contributed to the research results reported within

592

this paper. Additional thanks go to the W. M. Keck Center for Advanced Microscopy at the University

593

of Delaware for access to their electron microscope facility.

594

Supplementary material

595

Two files are given which have the coordinates to the synthetic sphere packs. These are 1030-27-

596

D-50-10-SPP and 1030-27-D-50-10-FPP shown in Table 1. The header information is contained in

597

the file header-info.

598

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599

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600

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32

Figure 1: Visualization of the experimental (A, B) and synthetic (C, D) SPP. A. SEM B. FIB cross-section C and D. Different external views showing the synthetic particle 1030-27-D-50-10-SPP. The blue color in D shows the second, deeper layer of the sol particles.

33

Figure 2: Local particle porosity vs. nondimensional radial distance from the particle center for SPP and FPP particles. The radial distance is made nondimensional by dividing r by the sol diameter dsol . The broken vertical line denotes the SPP core particle radius. The inset provides greater detail of the porosity in the outermost spherical shell.

34

Figure 3: A. Random packing of 125 spheres in a cubic bounding box with periodic boundaries and B. Random packing of 125 SPP in a cubic bounding box with periodic boundaries.

35

Figure 4: Main plot: Intraparticle and interstitial nearest-surface distributions for the random SPP packed bed. The probability density is plotted as a function of the nearest-surface distance normalized by the sol diameter. The sum of the two distributions is normalized to have unit area. Inset: FPP and SPP intraparticle distributions. The FPP distribution is normalized to unit area and the SPP distribution has area proportional to its pore volume relative to the FPP. The SPP intraparticle distributions (red curves) in the main plot and the inset are the same distributions with different normalizations.

36

Figure 5: Main plot: Velocity distributions in the FPP (solid line) and SPP (dashed line) random beds. P (v) denotes the relative volume of fluid with velocity v as a fraction of the bed pore volume. v/hvi denotes the z-component of fluid velocity normalized by the mean value, which is the same for FPP and SPP. Red, green and blue curves denote the intraparticle, interstitial and combined distributions. Inset plots show the full range of the interstitial and intraparticle distributions.

37

Figure 6: A. Hull flux distributions for random SPP packed bed. B. Distribution of surface flux momentum density 2 normal to the particle R surface (f = ρv · n) in the FPP and SPP beds (kg/m s). P r(f ) is the normalized distribution density, such that P r(f )df = 1.

38

Figure 7: Relative velocity as a function of depth in the particle for A. SPP and B. FPP. These results derive from a random packing of 125 spheres in a cubic bounding box with PBCs. In these figures, r is the radial distance from the center of the core, rh is the hull radius and dsol is the shell thickness.

39

Figure 8: Velocity in the direction of the pressure gradient (left to right) in BCC packed beds for A. SPP and B. FPP. The probability density is plotted against vz /v z . The inset shows contours of the velocity field and flow vectors in the x-z plane. Colors map to velocity as: red, vmin < v < −0.01 v; orange, −0.01 v < v < 0; black, 0; yellow, 0 < v < 0.01 v; blue, 0.01 v < v < 0.1 v; gray, 0.1 v < v < v; and green, v < v < vmax .

40

*Highlights

Highlights:  Models of fully porous and superficially porous particles are constructed  Extensive fluid flow is calculated within and around these particles in packed beds  Velocities within these wide pore particles show significant internal flow  Perfusion chromatography can be performed with these particles  Course-grained theory agrees with the fluid velocities