Accepted Manuscript

Title: Numerical study and theoretical performance limit ofinterconnected multi-capillary gas chromatography columnswith perfectly ordered pillar patterns

Authors: Sander Jespers, Frederic Lynen, Gert Desmet

PII: S0021-9673(17)31449-8DOI: https://doi.org/10.1016/j.chroma.2017.09.068Reference: CHROMA 358897

To appear in: Journal of Chromatography A

Received date: 11-8-2017Revised date: 26-9-2017Accepted date: 26-9-2017

Please cite this article as: Sander Jespers, Frederic Lynen,GertDesmet,Numerical studyand theoretical performance limit of interconnectedmulti-capillary gas chromatographycolumns with perfectly ordered pillar patterns, Journal of ChromatographyA https://doi.org/10.1016/j.chroma.2017.09.068

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1

Numerical study and theoretical performance limit of interconnected multi-capillary gas

chromatography columns with perfectly ordered pillar patterns

Sander Jespers(1), Frederic Lynen(2), Gert Desmet*,(1)

(1)Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan 2, 1050 Brussels, Belgium

(2)Universiteit Gent, Separation Science Group, Krijgslaan 281, B-9000 Gent, Belgium

*Corresponding author e-mail: gedesmet@vub.ac.be

Highlights

The band broadening in a novel type of microfabricated GC column is calculated numerically

The column structure performs as a bundle of parallel capillaries with regular intermixing

points

For a system pressure of 8 bar, the optimal inter-pillar distance could be determined to be at

75 µm

Under non-retained conditions, external-length based plate heights as low as 6 m can be

expected

Abstract

We present the results of a theoretical and numerical study of the chromatographic performance of

a novel type of microfabricated GC column. The column consists of an array of rectangular flow

diverters (pillars), creating a network of perfectly ordered, interconnected and tortuous flow-through

paths. Using van Deemter and kinetic plots of simulated band broadening data, we could

demonstrate that the proposed column structure performs as a bundle of parallel open-tubular

capillaries with rectangular cross-section, connected by a regular pattern of channel-intermixing

points that allow compensating for inevitable channel-to-channel differences in migration velocity

without adding any significant dispersion themselves. The established kinetic plots also allowed to

2

propose design rules for the optimal distance between the pillars as a function of the desired

separation efficiency and the available column pressure. The simulations also allowed establishing an

expression for the plate height as a function of the velocity of the carrier gas. Results are also

compared to the results of a recent experimental study.

Keywords

Microfabrication, computational fluid dynamics, performance limit, multi-capillary column, radially-

elongated pillars

1. Introduction

Since the first report on a microfabricated gas chromatography (GC) system by Terry et al. [1], many

research groups have shown great interest in the topic [2-6]. Various design and fabrication aspects

have already been investigated, going from the effect of the channel lay-out geometry [7] to the

optimization of turns [8] and proper coating techniques [2, 9, 10]. Most of this work considered

microcolumns with a square or rectangular open-tubular cross-section, because this is the shape

which is most straightforwardly obtained with the MEMS technology typically used to fabricate the

columns. However, numerical work by Poppe [11] showed this column format is producing more

dispersion than the classical open-tubular capillaries with circular cross-section. Alastair et al. [12]

therefore also investigated the possibility to produce microfabricated GC columns with a circular

cross-section.

A common drawback of all open-tubular systems (including the fused-silica drawn capillaries used in

conventional GC), is that any decrease in the diameter or width of the channel (to pursue a

proportional increase in the column efficiency) leads to a quadratic reduction of the column

permeability as well as the mass loadability of the columns. While the reduced column permeability

can in principle be countered by moving to high-pressure GC systems, the low loadability represents

a crucial limitation of the applicability of the capillary column format for the analysis of very volatile

components (air quality monitoring), ultra-trace compounds (residual solvent analysis in pharmacy,

detection of genotoxic compounds, impurities in water, persistent organic pollutants), or

metabolomics [13, 14].

To increase the loadability, the use of parallel capillary columns seems the most straightforward

solution. This has been investigated both for the conventional capillary format (so-called MCC

columns) as well as for microfabricated columns [15, 16]. However, inevitable small deviations in the

dimensions of the parallel channels can lead to extreme performance losses due to the so-called

polydispersity effect [17]. This is the additional dispersion arising from differences between the

3

migration velocities in the different parallel channels when there are no cross-channel inter-

connection points that would allow for a (partial) restoration of the experienced difference in

migration speed history.

A more advantageous trade-off between efficiency and loadability can be made by filling the

microfabricated channels with an array of micropillars. In this way, the column efficiency can be

controlled independently of the loadability, as the former is determined by the spacing between the

pillars, while the latter is determined by the overall channel width, fully similar to the situation in

packed bed chromatography [18, 19]. Additionally, the available contact surface is much larger in

these columns.

Originally tested for electro-chromatography by Regnier et al. [20], our group has, together with

some other groups, investigated the potential of pillar array columns (PACs) for liquid

chromatography [21-23]. Recently, the benefits of using radially elongated pillars (REPs) instead of

the more conventional square, circular, or diamond-shaped pillars was demonstrated. The most

important advantage of these REPs is that they reduce the effective axial diffusion (so called B-term

band broadening) in proportion to the square of the (very high) flow-through path tortuosity they

induce [24]. Furthermore, REPS are also less prone to the so-called side-wall effect [25].

In the present study, we investigate the potential use of REP array columns for GC using

computational fluid dynamics (CFD) simulations to establish a van Deemter expression describing

how their efficiency can be expected to evolve with the imposed flow rate under the specific

conditions of a gas flow. The van Deemter expression is subsequently used to calculate the kinetic

performance limit curve of the REP column and compare it with open-tubular capillary columns with

circular and square cross-sections. The geometry used for these simulations was the same as used in

a separate experimental study [see Supplementary Material SM for details]. A detailed view of a

segment of the investigated pillar array bed is shown in Figure 1. Briefly, the 6.195 mm wide channel

is filled with a grid of pillars (1.455 mm in the radial directions and 10 µm in axial directions) spaced

75 µm apart, resulting in rows of 4 pillars, creating 8 parallel flow paths. The relatively large

elongation of the pillars in the radial direction aims at suppressing the axial diffusion (B-term band

broadening) of the analytes (a large contribution to the plate height in GC). The inter-pillar distance

and the column length were chosen based on kinetic performance predictions derived from [26]. The

star-shaped structures in the middle of each pillar were added to eliminate the stagnant fluid zone

that is present when a fluid stream perpendicularly hits a wall. While the present work focused on

the specific case of a REP column with an interpillar distance of 75 µm, it is also shown how the

4

derived theory (van Deemter and kinetic performance limit) can be used to find the optimal

interpillar distance as a function of the required efficiency and of the available instrument.

2. Mathematical and numerical procedures

All simulations were performed with Ansys® Workbench version 16.2 from Ansys, Inc., purchased

from Ansys Benelux, Wavre, Belgium. Within this software platform all flow domains were drawn

with Ansys® Design Modeler and meshed with Ansys® Meshing. All simulations were performed with

Ansys® Fluent on Dell Power Edge R210 Rack Servers each equipped with an Intel Xeon x3460

processor (clock speed 2,8 GHz, 4 cores) and 16 Gb, 1333 MHz ram memory, running on Windows

server edition 2008 R2 (64-bit). The mesh size was chosen such that the shortest flow domain

contained 10 mesh cells. The mesh itself consisted of quadrilateral cells. To check mesh

independency, a mesh containing cells half the original size, resulting in a quadruple cell count, was

used. All simulations were done in 2D to reduce the required calculation time (including a top and

bottom to the channel would require roughly 10 -100 fold longer calculations).

Plate height measurements

The geometry consisted of 30 repetitions of one unit cell of the pillar bed (see Fig. 4, further on) for

all simulations except for the two highest velocities where 60 repetitions were required to reach a

steady-state plate height. First, the velocity fields were computed solving the Navier-Stokes

equations using the segregated pressure-based steady-state solver. For the spatial discretization, the

least squares cell based method was used to calculate concentration gradients, the coupled scheme

was used for pressure-velocity coupling, and the second order interpolation scheme for pressure and

second order upwind scheme for momentum. The left and right side of the geometry were defined

as symmetric boundaries, the boundary lines of the flat-rectangular pillars were defined as walls. The

inlet plane at the bottom of the geometry was put at a fixed flow rate (ranging from 0.1 – 0.8

mL/min), and the outlet plane at the top of the geometry was set to “outflow”.

Subsequently, the mesh cells of the first 5 µm of the channel were patched with 1% species. The

transient solver, with first order implicit temporal discretization and second order upwind scheme for

spatial discretization, was then used to solve the convection-diffusion equation yielding the transient

concentration field of species band migrating through the flow domain. A fixed time stepping

method with 15000 steps was used, the size of each step was chosen so that in each time step

roughly 1-2 mesh cells were traversed. Simulations of the steady-state velocity field in the

5

aforementioned geometries took about 1 hour, while the transient species concentration field

simulations took about 24 hours.

Determination of the effective diffusion coefficient

For this type of simulation, the geometry consisted of one unit cell (see Fig. 4 further on). Since the

geometry is constant with respect to height, only a 2D simulation was used. Boundary conditions

were set to symmetry for the side walls and to wall for the sides of the flat-rectangular pillars, the

inlet plane, and the outlet plane. The mass fraction of species is set to 1 % at the inlet wall and to 0 %

at the outlet wall. After setting the velocity in all directions to zero, the convection-diffusion equation

was solved under steady-state conditions. The calculation was stopped when the residuals remained

constant to within 10-6. At this moment, the mass flow rate at the outlet wall was recorded.

3. Results and discussion

3.1. Velocity field and peak dispersion in the pillar bed

An example of the velocity field obtained by solving the Navier-Stokes equation in the REP bed is

shown in Fig. 2. The overview (Fig. 2a), and especially the detailed view (Fig. 2b) of the flow field

clearly show the hydrodynamic conditions in the bed are nearly identical to those one would expect

in a bundle of (folded) open-tubular channels, because the velocity field corresponds exactly to the

analytical solution for the velocity field of a pressure-driven flow between two parallel plates (see

Fig. 2b), representing the equivalent of an open-tubular channel for the currently considered 2D-

simulations. It is only in the small areas between the pillars, where the fluid moves in the axial

direction (Fig. 2c), that the velocity field differs significantly from this analytical expression. As can be

noted from Fig. 2a, these zones only make up a very small part of the total flow region. The detailed

view of the velocities near the star-shaped flow-dividing structures added to the pillars (Fig. 2d)

illustrates how the addition of this structure prevents the formation of a significantly large stagnant

fluid zone that would originate from the fluid stream hitting a flat wall.

Figure 3 shows a series of snapshots of the solution of the diffusion-dispersion equation

corresponding to the velocity field in Fig. 2. Going from left to right, as the elapsed time increases,

the species band increases in width. Important to notice is that, due to the perfect ordering of the

pillar bed, the band stays perfectly symmetrical. A front-back cross section of the species band in the

4th panel of Fig. 3 is shown in Section SM3 of the Supplementary Material to illustrate the symmetry

in the direction of flow. The plate height corresponding to a certain velocity was determined by

monitoring the three first moments of the species band expressed as a function of the space-domain:

6

LH

x

x

2

(1)

with Δσx² and ΔL the difference between two consecutive time steps of:

2

0

1

0

22

MOM

MOM

MOM

MOM

x

(2)

0

1

MOM

MOML (3)

where the moments in the space-domain of the species band are given for each time step by:

dxxcMOM

(4)

Where c is the mass-fraction of species and α = 0, 1, or 2. Parameters x and L are respectively the

position and the distance in the net direction of flow (cf. the black arrow “F” in Fig. 3).

3.2. Establishing a van Deemter expression for REP columns in GC

In a previous publication [24], a Van Deemter expression was established for the plate height (Hx)

observed in the mean flow direction x (cf. Lx- arrow in Fig. 2a, with Hx=N/Lx) of a REP in the un-

compressed flow regime prevailing under LC conditions. It was also established how the Hx-plate

height relates to the plate height Hi found when expressing the plate height as a function of the

actual tortuous flow path followed by the fluid, i.e., the i-coordinate (cf. Li-arrow in Fig 2, with

Hi=N/Li).

m

x

x

m

x

x

i

i

i

x

Dk

udkk

u

DCu

u

BuC

u

BHH

2

22

22)1(105

)5.2591(2

(5)

With τ the flow path tortuosity (see Eq. (8) later on), ui the carrier gas velocity in the i-coordinate, ux

the carrier gas velocity in the x-coordinate, Dm the molecular diffusion coefficient of the species, k

the retention factor of the species, and d the distance between between two pillars.

Note that it was found that, when the REPs are sufficiently wide (aspect ratio>5 to 10), the B- and C-

constants in Eq. (5) can be directly replaced by their respective Golay expressions for the B- and C-

term dispersion in an open-tubular channel with the same cross-section as the one of the flow-

through paths in the chip (the expression on the right hand side of Eq. 5 represents the Golay-

7

expression for the flow between two parallel plates, i.e. the case relevant for the presently

considered simulation geometry).

Whereas Eq. (5) was constructed for LC conditions, it is hypothesized here that the plate height in a

REP column used under GC conditions would follow the same law as expressed by Eq. (5), but now

properly accounting for the carrier gas decompression by multiplying with Giddings’ decompression

factor [27]:

23

24

2

,

22

2

,

23

24

2)1(8

)1)(1(9

)1(105

)5.2591(2

)1(8

)1)(1(9

P

PP

Dk

udkk

u

D

P

PPCu

u

BH

pst

pstx

pstx

pst

x

x

x

(6)

where Dpst is the molecular diffusion coefficient at standard pressure (1 atm), ux,pst the gas velocity in

the x-coordinate at standard pressure, d the characteristic distance in the chip bed (in the case of

REPs this is the interpillar distance), k the retention factor (k=0 in the simulations), and P the ratio of

inlet over outlet pressure.

To validate Eq. (6), its predictions have been compared with the (Hx,ux)-data obtained from the CFD

simulations on the pillar array bed. However, first the unknown tortuosity factor τ needed to be

determined in an independent way. This was done by considering the diffusion-only mass transport

in the bed, which can be described using the integrated form of Fick’s first law of diffusion (Eq. 7):

x

cDJ

eff

(7)

with J the mass flux (kg/ m²s), ρ the fluid density (kg/m³), Deff the effective diffusion coefficient

(m²/s), c the mass fraction of species (kg/kg), and x the diffusion distance (m). Using Eq. (7) to

calculate Deff in the pillar array bed, the tortuosity τ is found from:

eff

mol

D

D (8)

Figure 4 shows the steady-state concentration gradient profile obtained using the CFD software to

calculate the diffusion-only mass transport obtained when switching off the flow and putting the

tracer mass fractions at the inlet and outlet plane at respectively c = 1% and c = 0% (thus Δc = 0.01).

From the mass flow (M) at the outlet reported by the CFD software, the mass flux J needed in Eq. (7)

is found using:

AJM (9)

8

where A is the total flow through area (1530 µm total width for the unit cell × 1 m which is the

default depth of the CFD software for 2D simulations) and ε is the bed porosity (ε = 0.88, calculated

as the fraction of fluid volume to the total volume of the unit cell). With this relation, and knowledge

of A and Δx from the geometry, and of ρ and Dmol (which are user input and in all simulations chosen

as 0.16 kg/m³ and 3.2·10-5 m²/s, respectively) the tortuosity τ of the design was found to be 9.2. This

value is slightly lower than the “geometrical” value τ=10 (obtained as = Li/Lx) one would normally

take as a first approximation for τ. The small difference is obviously due to the fact that the species

do not diffuse in straight lines through the corners around the REPs but follow some optimized path.

The plate heights recorded with the simulations (open squares), as well as the plate heights

calculated according to Eq. (6) with τ= 9.2 (full lines) are shown in Fig. 5. Both data sets are clearly in

very good agreement, and a minimal plate height of 6 µm is recorded. Although the simulations were

unable to reach the region of the van Deemter curve where the C-term becomes dominant (mostly

because we consider non-retained components and these require extremely high velocities to enter

sufficiently deep into the C-term region), the results can be seen as a validation of Eq. (6) for GC REP

columns in the range of velocities that is most relevant for practice. For what concerns the C-term

region, the velocity field calculations shown in Fig. 2 demonstrate the flow behaves exactly like in a

straight open-tubular channel over the vast majority of the geometry. This allows to assume that the

C-term in REP columns used in GC will also be given by the analytical expression derived from the

Golay-Aris theory for open-tubular channels (as is also the case in LC, demonstrated in [24]).

Furthermore, the expression for the carrier gas decompression used in Eq. (6) has been derived

without any assumption on the geometry of the flow paths [27], and should hence be the same for a

meandering as for a straight channel.

Another implication of the agreement in Fig. 5 is that the presence of the inter-mixing points (i.e., all

the regions where adjacent flow paths are in direct contact) does not significantly affect the

dispersion, as the band broadening can be fully represented by the expressions for a meandering

open-tubular channel, without the need to add an additional term for the intermixing points.

3.3. Comparison with experimental data

In figure 6, the theoretical van Deemter expression given by Eq. (6) is compared with experimental

data (given as a function of reduced plate height h versus the carrier gas-specific flow rate f

introduced by Blumberg [28]) obtained in a separate experimental study [SM] using an unretained

compound in a Lx = 70 cm long microfabricated column (Lx is external lengh, while the internal length

Li is about times larger, see Fig. 2a for the definition of the measurement direction of Lx and Li) with

the same pillar array packing as the one used in the present simulation study. As can be noted, the

9

agreement is very good, despite the many potential error sources for the experimental data (extra-

column band broadening, uncertainty on Dpst, …). However, due to the great difficulty in depositing a

uniform stationary phase layer on the inner surface of a channel with a square or rectangular cross-

section (pooling of the stationary phase in the channel corners), significant experimental deviations

from Eq. (6) can be expected under retained conditions.

3.4. Kinetic performance comparison

Using the van Deemter expression given by Eq. (6), the theoretical kinetic performance limit (KPL)

curve of GC REP columns can be calculated using the kinetic plot procedure described in [26]. A KPL

plot connects all possible combinations of length and velocity for which the considered

chromatographic material performs at its kinetic optimum, i.e., where the highest N is produced in

the shortest time. As such, it suffices to compare only the KPL curves of two chromatographic

supports (or operating conditions) to compare their complete kinetic performance, as any point not

in the curve corresponds to a sub-optimal condition. To calculate the KPL curves, we used the

information that the permeability of a REP column is equal to that of a square open-tubular capillary,

with the same dimensions as one of the flow-through paths in the chip, divided by τ² [24]. The

pressure drop required to force the carrier gas through a column with a flow path tortuosity τ and

length Lx at a velocity ux (velocity measured in the Lx-direction shown in Fig. 2a, while the ui-velocity

in the Li-direction is times higher) will thus be τ²-fold larger compared to the pressure drop required

to achieve the same velocity u = ux in a straight channel with length L = Lx. Figure 7 compares the KPL

curve of a GC REP column (red) with an interpillar spacing of 75 µm, τ= 9.2, and channels with square

cross-section (75 µm deep) to those of open-tubular capillary columns with the same square cross-

section (black squares) and with a circular cross-section (black lines). As this channel now contains a

top and a bottom, contrary to the infinite parallel plates considered until now, the constants 1, 9, and

25.5 in Eq. (6) were adjusted to a value of respectively 1.804, 10.196, and 25.892, in accordance with

Poppe [11]. First, it should be noticed that the KPL curve of the chip is identical to that of an open-

tubular capillary with square-cross section of 75 µm × 75 µm (open square in Fig. 7). This implies that

the tortuous flow paths in the REP column can be expected to provide exactly the same kinetic

performance as a bundle of perfect interconnected capillaries, with the same cross-section and with

the same length as the internal length Li (see Fig. 2a) as in the REP column. In other words, whereas

the REP column achieves a given efficiency N in an effective axial length that is times shorter than a

reference straight channel, the time needed to produce these plates is exactly the same because the

net effective velocity in the tortuous path in the REP is in turn times slower.

10

Next, comparing the KPL curve of the chip to that of an open-tubular capillary with circular cross-

section and a diameter of 75 µm it is clear that the chip performs slightly worse (at least in the

regime before reaching the vertical asymptote). This is of course due to the additional dispersion that

follows from the less homogeneous velocity field (stagnant zones in the corners) in square channels

compared to circular channels. However, as discussed earlier, the loadability of a 75 µm diameter

capillary is inevitably limited, whereas the loadability of the chip can be increased virtually without

limit by increasing the overall column width. The high loadability of the chips makes them particularly

well suited as second dimension column in a GC × GC set-up. Although recent publications have

demonstrated the benefits of using the same internal diameter for both columns [29, 30], many

practitioners still use short (0.5 – 2m long) and narrow (50 – 100 µm diameter) columns in the

second dimension to ensure a fast analysis of the fractions coming from the first dimension (typically

30 m long, 250 µm diameter) [31, 32]. This mismatch in diameters means that it is impossible to

simultaneously operate both columns at their optimal velocity in most GC × GC set-ups [33]. The total

channel width of the chips, however, can be adjusted so that both columns can be used at their

optimal velocity, while the dimensions of the individual tracks can be relatively small, granting the

required separation speed.

Finally, comparing the kinetic performance of the REP column to that an open-tubular capillary with

circular cross-section and a diameter of 250 µm (the current standard for GC separations, and

offering a similar loadability in terms of injection volume and applicable flow rate as the presently

considered REP column with 8 parallel 75m through-channels), a considerable difference is

observed. For instance, to generate 100,000 plates a kinetic optimized chip would require 3.5 s, while

a kinetically optimized capillary would require double that time. Also, using the open-tubular

capillary would only lead to shorter analysis times when more than 400,000 plates are required

(practical applications seldom require such high plate numbers).

The kinetic plot method can also be used to guide the design of (microfabricated) columns. An

example of this is shown in Fig. 8, where the kinetic plot curves for GC REP columns with τ=9.2 and

50, 75, and 100 µm interpillar distance and etching depth are compared for atmospheric outlet

conditions once with the maximum pressure of the system set at 8 bar, and once at 64 bar. Under

the constraints of a maximum system pressure of 8 bar, Fig. 8 shows that an interpillar distance of 75

µm yields the best compromise in terms of overall performance when the desired plate number is in

the range of 50,000-500,000. And when contemplating using a high pressure GC set up, one where

for example a maximum pressure of 64 bar would be available, the optimal interpillar distance would

be only in the order of 10 µm (see dashed lines in Fig. 8). This system would allow to achieve the

same N about 8 times faster than a single channel 75 m capillary or a 75 m REP column, and even

11

about 15 times faster than possible with a 250 µm circular capillary, that would have about the same

loadabilty as the REP column.

4. Conclusions

The band broadening in a radially elongated pillar (REP) array column (i.e., the ordered equivalent of

a packed bed column) under GC conditions can be represented using the same Golay-expressions as

those representing the B- and C-term dispersion in an open-tubular channel with the same cross-

section as the one of the flow-through paths in the REP column. The tortuosity of the path does not

add anything to the dispersion (it only increases the length of the internal flow path but this at the

same time also allows to increase the achievable plate number concomitantly). This also implies the

mixing points that are added between the adjacent streams to correct for any deviating migration

velocities in the individual parallel flow-through channels create no additional dispersion.

Because of the tortuous path, the minimal plate height Hx based on the axial length Lx can

theoretically be as low as 2.25 m in a column with an equivalent through-pore size of 75 m

(reduced plate height hx= 0.03). Calculating the plate height Hi based on the hydrodynamic length Li,

this corresponds to a value of 20.7 m, i.e. the same value as obtained in a straight open-tubular

channel (flow between two parallel plates in the presently considered 2D simulation study).

A kinetic plot analysis showed the tortuous flow paths in the REP column provide exactly the same

kinetic performance (tm vs N) as a single capillary with the same square cross-section and with the

same length as the internal flow-path length Li as in the REP column. As such, the REP column

intrinsically behaves as a bundle of perfect capillaries (with “perfect” referring to the case where

there is no channel-to-channel variation on the cross-sectional area such that the bundle produces

the same efficiency as a single capillary). This implies the REP column can achieve a given efficiency N

in an effective axial length that is times shorter than a reference straight channel with the same

total flow-path length, while the time needed to produce these plates is exactly the same because

the net effective velocity (in the net axial direction) in the tortuous path in the REP is in turn times

slower than in the straight channel.

5. Acknowledgement

S.J. gratefully acknowledges Research grant from the Research Foundation Flanders (FWO

Vlaanderen).

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12

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Figure captions

Figure 1. (a) overview of the micropillar array bed containing rows of 4 RE pillars; (b) detailed view of

the REP positioning in the bed. The blue arrows (F) schematically represent the flow path through

one unit cell of the bed.

Figure 2.(a) Example of the simulated velocity field inside the pillar array bed (velocity increases

linearly from dark blue to red). The Li and Lx arrows schematically indicate respectively the actual

distance a molecule travels in the direction of the internal and tortuous flow path and the distance it

travels in the net direction of flow. (b) Velocity vectors of the flow between two pillars. The arrows

perfectly fit onto the analytical solution for the velocity field between two parallel plates. (c) Detailed

view of the velocity field at a pillar junction. (d) Detailed view of the velocity field around one of the

flow-dividing structures.

Figure 3. Solution of the diffusion-dispersion equation of a species band in the REP bed as a function

of the time (s) at ux = 2 cm/s. The black arrow indicates the net direction of flow through the bed. The

color scale linearly represents the mass fraction of species (red representing the maximum value,

blue representing a mass fraction of zero), the numerical values corresponding to red and blue are

adjusted at each time step to increase the visibility of the entire plug.

Figure 4. Simulated concentration profile, expressed as mass fraction of species (red = 1%, blue = 0%,

increasing linearly) in one unit cell of the pillar bed under steady state conditions and zero flow. The

inlet and outlet sections (horizontal thick black lines) are kept at a constant species mass fraction

(resp. 1% and 0%). This is also where the steady-state mass flow M is being recorded. The side walls

(thick horizontal lines) are kept at symmetry.

Figure 5. Plate height (Hx) in the x-coordinate as a function of the carrier gas velocity in the x-

coordinate and at stand pressure (ux,pst) for an unretained compound in the GC REP column. Data

points: simulated data. Full line: theoretical predictions according to Eq. (6) with τ= 9.2, k= 0, Dpst=

3.2·10-5 m²/s, and d= 75 µm.

Figure 6. Reduced plate height (h) of an unretained compound as a function of the carrier gas-

specific flow rate (f) [poole]. × theoretical predictions according to Eq. (6); ◇ experimental data

obtained on a 70 cm long microfabricated column with the same pillar array packing presented in

[ref=submitted elsewhere].

Figure 7. Comparison of the theoretical kinetic performance limit curves of open-tubular capillary

columns with circular cross-section (black lines) and with square cross-section (black squares), with

that of a GC REP column (red) for the case of an unretained compound. The characteristic distance

(internal diameter for capillary columns, interpillar distance for REP columns) is given above the

curves.

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Figure 8. Comparison of the kinetic performance limit curves for GC REP columns with different

interpillar distance and etching depth. For all cases τ=9.2, po= 101325 Pa, T= 100 °C, and taking

helium as the carrier gas. Full lines: Δpmax= 8 bar; dashed lines: Δpmax= 64 bar.

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