Influence of the Pillar Shape on the Band Broadening and the

Anal. Chem. 2004, 76, 3716-3726

Influence of the Pillar Shape on the Band Broadening and the Separation Impedance of Perfectly Ordered 2-D Porous Chromatographic Media J. De Smet,* P. Gzil, N. Vervoort, H. Verelst, G. V. Baron, and G. Desmet

Department of Chemical Engineering, Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels, Belgium

We report on a computational study assessing the effect of the pillar shape in perfectly ordered porous chromatographic media. Using computational fluid dynamics to compare the band broadening and flow resistance characteristics of a large number of different pillar shapes, it is found that the most axially elongated shapes yield the best chromatographic performance and that diamonds are to be preferred over ellipsoids. The former pack into a more uniform pore space and display a smaller Cs value, whereas the latter pack into a locally constricted pore space and therefore generate a considerably larger flow resistance. For the presently considered case of a densely packed array (E ) 0.4), changing the pillar shape from a cylinder to a more elongated diamond, for example, reduces the minimal plate heights from hmin ) 0.84 to hmin ) 0.72, the C factor from C ) 0.062 to C ) 0.050, and the separation impedance from Emin ) 330 to Emin ) 220, without affecting the number of interchannel coupling points. Despite of the undisputed success packed-bed HPLC has had ever since its introduction in the late 1960s,1,2 the chromatographic community has not stopped searching for alternative column formats and separation media2,3 capable of yielding more theoretical plates in a shorter time. In recent years, this search is spurred by the ever-increasing demand4 for peak capacity and throughput rate coming from the diverse “omics” fields and from pharmaceutical R&D laboratories engaged in combinatorial drug screening. Inspired by the large success of capillary electrophoresis (CE), one of the first heavily pursued alternatives for packed-bed HPLC was open-tubular LC. This option was, however, fairly rapidly abandoned as it turned out that the increased separation speed and plate number limit did not weigh up against the lack of retention capacity and the very poor mass loadability of the micrometer-sized capillaries needed to realize this gain. A more promising, and indeed more successful, approach (also inspired by the success of CE) did not involve a change in column * Corresponding author. Tel.: (+)0.32.2.629.33.27. Fax: (+)0.32.2.629.32.48. E-mail: [email protected]. (1) Huber, J. F. K. J. Chromatogr. 1969, 7, 85-90. (2) Kirkland, J. J. J. Chromatogr. Sci. 1969, 7, 7-14. (3) Rozing, G. LC‚GC Eur. 2003, 6a, 14-19. (4) Stevenson, R. Am. Lab. 2001, 8, 50-56.

3716 Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

architecture but relied on a change of the mobile-phase driving force. Replacing the parabolic velocity profile of the pressuredriven flows by the more flat velocity profile of electrokinetically driven flows gave birth to the CEC concept, combining the best of both worlds, i.e., the selectivity and mass loadability of LC with the large plate numbers of CE. Although quite successful, CEC to date still struggles with a number of tough-to-beat problems such as frit reproducibility and the variation of the flow rate caused by slight changes of the electrolyte concentration or the presence of macromolecular sample impurities.5-7 These problems kept the interest in alternative column formats for pressure-driven LC alive and left room for the breakthrough of the monolithic column. This radically new column concept was first developed using polymer chemistry.8,9 Silica-based monoliths were developed quite rapidly hereafter, and the first generation of commercialized products is available.10-12 Due to their increased porosity, silica monoliths yield a dramatic reduction of the separation impedance, which in turn can be translated in shorter analysis times, in larger plate number limits, or both. Looking into the future of LC, as was done by Knox in one of his last papers,13 he suggested that LC column manufacturers should try to significantly increase the homogeneity of their columns. One of the potential routes to increase the packing uniformity far beyond that of the currently used packed-bed and monolithic silica columns would be the use of the micromachining methods currently applied in the microelectronics industry and the lab-on-a-chip field. This idea was in fact already launched a few years earlier by Regnier et al.,14-17 when presenting their (5) Unger, K. K.; Huber, M.; Hennessy, T. P.; Hearn, M. T. W.; Walhagen, K. Anal. Chem. 2002, 74, 200A-207A. (6) Bartle, K. D.; Carney, R. A.; Cavazza, A.; Cikalo, M. G.; Myers, P.; Robson, M. M.; Roulin, S. C. P.; Sealey, K. J. Chromatogr., A 2000, 892, 279-290. (7) Jiskra, J.; Claessens, H. A.; Cramers, C. A. J. Sep. Sci 2003, 26, 13051330. (8) Hjerte´n, S.; Liao, J. L.; Zhang, R. J. Chromatogr. 1989, 473, 273-275. (9) Svec, F.; Frechet, J. M. Anal. Chem. 1992, 64, 820-822. (10) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. Anal. Chem. 1996, 68, 3498-3501. (11) Sinz, K.; Cabrera, K. Int. Labmate 2001, 25 (7), 17. (12) Tanaka, N.; Kobayashi, H.; Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Hosoya, K.; Ikegami, T. J. Chromatogr., A 2002, 965, 35-49. (13) Knox, J. H. J. Chromatogr., A 2002, 960, 7-18. (14) He, B.; Regnier, F. J. Pharm. Biomed. Anal. 1998, 17, 925-932. (15) Regnier, F. E. J. High Resolut. Chromatogr. 2000, 23, 19-26. (16) Slentz, B. E.; Penner, N. A.; Lugowska, E.; Regnier, F. Electrophoresis 2001, 22, 3736-3743. 10.1021/ac049873s CCC: $27.50

© 2004 American Chemical Society Published on Web 05/19/2004

revolutionary COMOSS column concept. In their approach, chromatographic beds are no longer obtained by packing individual pillars but by in situ etching a 2-D array of micromachined pillars, using a lithographic mask to precisely define the position and the dimension of the pillars. Unfortunately, the pillars in the COMOSS columns are nonporous. With only the outer pillar surface available for adsorption, COMOSS columns offer phase ratios and mass loadabilities that are orders of magnitude smaller than in packed-bed HPLC columns. It is therefore easy to understand that, before a micromachined column will ever be able to solve timely separation problems, a solution needs to be found to produce micromachined pillars with the same internal pore structure as the porous pillars currently employed in HPLC. As it is difficult to contemplate in advance how large the potential gain of such a micromachined, perfectly ordered porous medium would be, we recently made a computational fluid dynamics study of the chromatographic performance of a hypothetical, perfectly ordered 2-D array of porous cylinders.18 It was found that, owing to the perfectly ordered and homogeneous flowthrough pore structure, reduced plate heights as small as h ) 0.75 (for k′′ ) 2), and separation impedances as small E ) 300 (for k′′ ) 2) can be expected, without having to compromise on the porosity ( ) 0.4) and the retention capacity of the packed bed of spheres. In the present study, we explore the possibilities of one of the other potential advantages of micromachined 2-D packings: apart from yielding a maximal control over the bed uniformity, lithographic etching techniques also introduce the opportunity to leave the traditional cylindrical or spherical shape of packed-bed and monolithic silica packings for a panoply of alternative pillar or pillar shapes. Currently, there is no theoretical basis for the selection of the optimal pillar shape, and it would be interesting to find out which one of the possible design rules (maximal specific outer surface, minimal pore size, maximal pore size homogeneity) is the most important. The interest in this problem was also aroused by the work of Knox13 and Regnier,14-17 who both proposed a number of interesting candidate pillar shapes (squares, hexagons, ellipsoids). As they had to resort to purely qualitative argumentations, without any exact quantitative information on the velocity field and the species diffusion rates, it was, however, difficult to put forward any of these possible pillar shapes as the best. To investigate this problem more closely, we have made a systematic comparison of the influence of the pillar shape upon the band broadening and the separation impedance of a hypothetical, perfectly ordered porous pillar array column, devoid of any other band-broadening sources such as injection and detection. As already implicitly contained in its name, a porous pillar array column should be regarded upon as an ordered 2-D array of large aspect ratio pillars of a microporous material, with the same micropore size and specific internal surface area as the traditionally employed particulate HPLC materials. For a 3-D schematic of this concept, the reader is referred to ref 18. The reader should note that the use of micromachining techniques does not automatically imply that the width or the length of the columns has to be miniaturized: when etching, for example, a circular porous pillar (17) Slentz, B. E.; Penner, N. A.; Regnier, F. J. Sep. Sci. 2002, 25, 1011-1018. (18) Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 62446250.

array channel on the outer edge of a 4-in. wafer, channels with a length of ∼30 cm can be obtained (currently even 8-in. wafers are already being used in the microelectronics industry). Porous pillar array columns obviously are certainly not available yet, and it might still take quite a number of years before anyone can produce them with a sufficient accuracy, but it is hoped that the present study will inspire people active in the field of micro- and nanostructured materials to start solving the present problems. To obtain the required band-broadening data, a well-validated18,19 computational fluid dynamics software package is used to compute the velocity field and species diffusion rates in both the mobile- and stationary-phase zone. To emphasize the effect of the pillar shape, other design variables such as the bed porosity, the domain size, and the number of trans-channel coupling points are kept identical. As has been pointed out by Regnier et al.,14,17 trans-channel coupling points are needed to overcome the excessive band broadening originating from small pore-to-pore width variations. We recently confirmed this on a hypothetical segmented column structure,19 showing that even a flow-through pore variability of only a few percent can lead to a dramatic plate height increase if the number of trans-channel coupling points is too small. Given the difficulties of making any a priori assumptions on the flow-through pore variability of a real, micromachined porous pillar array column, we think it is critical for any exercise on the determination of the ideal chromatographic packing shape that the number of trans-channel coupling points remains constant. CONSIDERED PILLAR AND PACKING GEOMETRIES Figure 1 shows the top view of the representative 2-D unit cells of each of the different considered pillar shapes. Each of these unit cells should be considered as the unit building block from which a whole porous pillar array column can be constructed. Although many other staggering patterns are conceivable, the presently considered equilateral triangular staggered conformation, wherein the center of each pillar is placed on the corner points of an equilateral triangle with side ddom (see Figure 1), has been preferred for its isotropicity and the concomitant close resemblance to the packed bed of spheres. Figure 1 also shows the definition of the most relevant geometrical parameters. Sticking to the definition of the domain size usually adopted in the silica monolith field,20 it is straightforward to take the axial width of the unit cell as the domain size (ddom). The length of each unit cell (ldom) is then, due to the adopted equilateral triangular staggering, simply equal to (3)1/2ddom. The two simplest measures characterizing the different pillar shapes are their maximal axial and lateral width, respectively denoted by dax and dlat. As will become clear in the Results and Discussion section, grouping both values into the axial elongation parameter R (R ) dax/dlat - 1) will allow arranging the obtained plate heights into a logical order. The different values for R, varying from zero for the cylinders to R ) ∞ for the parallel plate case, are listed in Table 1. In a first rough approach, the pore space can be characterized by the shortest distance between two adjacent pillars. In the regions of the unit cell where only one pore is present, a value for d1 is determined, and in the regions where the flow splits into two pores, a value for d2 is determined. (19) Gzil, P.; Baron, G. V.; Desmet, G. J. Chromatogr., A 2003, 991, 169-188. (20) Leinweber, F. C.; Tallarek, U. J. Chromatogr., A 2003, 1006, 207-228.

Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

3717

Table 1. Overview of Some of the Most Important Geometrical and Chromatographic Performance Parameters of the Different Considered Pillar Shapes shape

R

Aa

Ba

Ca

φ

Cs,theob

τ

hmin

Emin

cylinders (a) hexagons (b) ellipse 1.5 (c) ellipse 1.7 (d) diamonds 1.5 (e) diamonds 1.7 (f) diamonds 2 (g) limited diamonds (h) parallel plate (i)

0.000 0.155 0.500 0.732 0.500 0.732 1.000 1.887 ∞

0.042 0.048 0.019 0.018 0.031 0.029 0.017 0.014 -0.001

2.26 2.24 2.28 2.28 2.29 2.25 2.29 2.24 2.29

0.063 0.061 0.061 0.057 0.055 0.053 0.050 0.042 0.031

481 444 530 664 501 433 452 470 346

0.048 0.046 0.044 0.041 0.039 0.037 0.035 0.027 0.018

1.221 1.239 1.139 1.124 1.149 1.120 1.101 1.052 1.000

0.83 0.83 0.78 0.76 0.77 0.74 0.72 0.63 0.53

331 304 324 380 294 238 228 192 98

a

Obtained by fitting the Knox equation (eq 2) to the h values shown in Figure 3a. b Calculated via eq 6.

Figure 1. Different considered pillar geometries and calculated velocity magnitude fields: (a) cylinders, (b) hexagons, (c) ellipsoids with R ) 0.50, (d) ellipsoids with R ) 0.73, (e) diamonds with R ) 0.50, (f) diamonds with R ) 0.73, (g) diamonds with R ) 1.0, (h) touching diamonds with R ) 1.89, and (i) parallel plates.

One of the most straightforward cross-sectional pillar shapes is the perfect cylinder (case a), yielding a packing pattern corresponding most closely to that of a packed bed of spheres and constituting the minimal specific outer surface case. Another straightforward pillar shape is the equilateral hexagon (case b), having only a slightly larger specific outer surface as the cylinder but yielding a much more uniform pore space after packing. The ellipsoidal pillars (cases c and d) have been considered for their more streamlined hydrodynamic shape if compared to the 3718 Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

cylinders and the hexagons of cases a and b. The two presently considered ellipsoids have a more or less arbitrarily selected axial elongation of, respectively, R ) dax/dlat - 1 ) 0.5 (case c) and R ) dax/dlat - 1 ) 0.73 (case d). Smaller R values lead to ellipsoids differing too little from the cylindrical and hexagonal pillars, whereas ellipsoids with larger R values were discarded because they pack into too locally constricted pore spaces; i.e., the d2 values very rapidly tend to zero when R is increased above the value used in case d. Pillars with a diamond-like cross-sectional shape (cases e-h) obviously pack into a much more uniform pore space and are therefore ideally suited to compare different degrees of axial elongation, going from the more compact shape obtained for R ) 0.5 (case e), over two intermediate cases with R ) 0.73 (case f) and R ) 1 (case g), up to the maximal diamond elongation case of R ) 1.87 (case h). As can be noted, a further increase of R would finally result in a parallel plate system (case i). Square pillars were not considered as they cannot be fitted onto the equilateral triangular staggered grid under the constraint of ) 0.4 without overlapping. It should also be noted that cases h and i no longer allow for a trans-channel coupling. They therefore should merely be considered as reference cases, although it could be argued that there will still be a considerable degree of transchannel species exchange in the limiting diamond case (case h), as the very thin porous zones near the touching points of the pillars only constitute a negligible mass-transfer resistance. To focus on the pillar shape only, all considered unit cells were designed to have the same external bed porosity, domain size, and staggering conformation. It should be obvious that these constraints impose a fixed linear relation between the characteristic pillar dimensions and the domain size, such that all characteristic dimensions scale together with the domain size. The imposed constraints furthermore also imply that the different pillar shapes have the same cross-sectional area. The number of transchannel coupling points also automatically remains the same in each case. By taking the bed porosity equal to the typical packedbed value of ) 0.4, it is furthermore also ascertained that the pillar arrays have the same mass loadability and retention capacity as the traditionally employed packed-bed HPLC columns. Two series of different pillar sizes have been considered. In series 1, all cases had a domain size equaling that of the unit cell for a cylinder pillar packing with dc ) 1 µm; in series 2, the domain size was three times larger, hence corresponding to the domain size of a cylinder pillar packing with dc ) 3 µm. All pillars were assumed to have an internal porosity of int ) 0.5. Combining this

with the fact that the external porosity in all cases was equal to ) 0.4, it can easily be verified that the zone retention factor k′′0 of the unretained species (i.e., for species with a solute distribution coefficient K ) 0) is given by

k′′0 ) ((1 - )/)int ) 0.75

(1)

CALCULATION PROCEDURES (1) Velocity Field and Species Dispersion Calculation. In each considered case, the flow domain consisted of a series connection of 10 of the unit cells depicted in Figure 1. A commercial computational fluid dynamics (CFD) software package (FLUENT v.6.1) was used to solve the full convection-diffusion material balances across the entire flow domain. The result of these simulations was a series of species break-through curves recorded at 10 regularly spaced detection planes. The methods to determine plate height values from these break-through curves were already elaborated in ref 18, together with the procedures adopted to ensure the grid-size and time-step independency of the obtained data. So-called velocity inlet and pressure outlet conditions were respectively imposed at the front and end planes of the flow domain. Along the sidewalls, a zero normal concentration gradient condition was imposed. With this symmetry condition, the considered flow domain behaves as if it were embedded in an infinitely wide structure. For the same reason, the parts of the sidewalls occupied by the fluid zone were subjected to a slip flow boundary condition (zero normal velocity gradient) to calculate the velocity field. At the surfaces of the porous pillars, a no-slip boundary condition (u ) 0 at the wall surface) was imposed to account for the flow arresting effect of the solid pillar surfaces. Further setting up the problem in the CFD program, the pillar zones were defined as porous zones with an infinitely large flow resistance. Each individual pillar zone was subsequently also defined as being embedded in a continuous fluid zone, defined as the remaining surface area of the unit cells. As also already explained in ref 18, the software package was extended with a number of self-written numerical routines to simulate the diffusion and adsorption processes inside the porous pillars. A first user-defined function was written to mimic the effect of the slow intrapillar diffusivity. The function was used to attribute the species entering the stationary-phase zones a diffusion coefficient different (i.e., smaller) from that in the fluid zone. In all presented cases, the stationary zone diffusion coefficient Ds was always put at 5 × 10-10 m2/s, whereas the mobile zone diffusion coefficient was always set at Dm ) 1 × 10-9 m2/s. The liquid-phase viscosity was always put at η ) 10-3 kg/(m‚s). A second user-defined function was used to subject the species zone to an adsorptive reaction with equilibrium constant K when present in the porous zone. In all simulations, a value of K ) 5/3 was used. In combination with the unretained zone retention factor of k′′0 ) 0.75 (cf. eq 1), this leads to a zone retention factor of k′′ ) k′′0(1 + K) ) 2, a typical value for both packed-bed and monolithic column LC. All simulations were carried out on Dell PC’s with dual Intel Xeon 2-Ghz processors and equipped with 2 Gb RAM. The grids needed to discretize the conservation equations were generated with GAMBIT (v. 2) software, run on the same hardware

configuration. As in our previous work,18,19,21 the following validation checks were carried out for each simulation: does the observed peak migration speed agree with the expected value based on the zone retention factor and the mean x-velocity, is the observed tortuosity independent of the fluid velocity, and, most importantly, are the obtained plate height and flow resistance values independent of the calculation grid size and time step? To interpret the band-broadening results, the well-known Knox equation22

h ) Aν1/3 + B/ν + Cmν + Csν

with

C ) Cm + Cs (2)

has been fitted to the obtained plate height values, using the standard fitting routine of Microsoft Excel, based on a Newtonian search with forward derivatives. (2) Theoretical B and Cs Term Contribution. A key role in the interpretation of the theoretical plate height values will of course be played by the B and Cs term contributions, as these are the two terms in the van Deemter equation for which a theoretical basis exists.22 As, for example, established by Knox,23 the B term contribution to the band broadening is given by

1 + γsz.k′′ 1 + k′′

Btheo ) 2

(3)

Since all different considered cases have been calculated with γsz ) Ds/Dm ) 0.5, k′′ ) 2, and k′′0 ) 0.75, it can easily be verified that Btheo ) 2.28,4 independently of the pillar shape or size. As can be assessed from its physical meaning,24 the Cs term on the other hand will depend strongly on the pillar shape. The Cs term contributions of the different considered pillar shapes can be calculated using the general procedure described by Giddings.24 Briefly this method consists of solving

∇2θ ) -1

(5)

over the entire pillar cross section, with θ ) 0 at the outer surface and with ∂θ/∂n ) 0 on the internal symmetry-planes, and by subsequently calculating xθ as the surface area-averaged θ value. According to Giddings’ derivation, the Cs term is then given by

deq2 Dm

HCs ) Csu0

with

Cs ) 2θ

k′′(1 + ) k′′0) 1 (6) (1 + k′′)2 γsz

Equation 5 can be solved analytically for a number of simple geometries. For a cylinder (2-D) and a sphere (3-D), eq 5 respectively yields xθ ) 1/32 and xθ ) 1/60. To solve eq 5 for the other considered pillar shapes, a self-written numerical integration routine, based on a fourth-order Runge-Kutta integration scheme has been used. The thus obtained xθ values were (21) Vervoort, N.; Gzil, P.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 843850. (22) Knox, J. H. J. Chromatogr., A 1999, A 831, 3-15. (23) Knox, J. H.; Scott, H. P. J. Chromatogr. 1983, 282, 297-313. (24) Giddings, J. C. Dynamics of Chromatography-Part I; Marcel Dekker Inc.: New York, 1965.


3719

Figure 2. Plot of φ versus d2/(ddomτ) for the different considered pillar shapes. The two full lines have been added to emphasize the general trend.

subsequently used in eq 6 to calculate the theoretical Cs values for the presently considered case of k′′0 ) 0.75, k′′ ) 2, and γsz ) 0.5. As can be noted from Table 1, the Cs,theo values decrease with increasing axial elongation R. When considering the diamonds and ellipsoids with R ) 0.5 and R ) 0.73, the diamonds obviously yield smaller Cs values than the ellipsoids. This is a consequence of the fact that the Cs constant in eq 6 is in fact a shape factor, related to the mean distance the species have to travel between the pillar border and its center.24 For the diamonds, this distance is slightly smaller than for the ellipsoids, as the sharp tips of the diamonds constitute a region with very short diffusion distances. In the ellipsoids, such sharp tip regions are absent.

versus the linear velocity (u0) of the unretained species, perfectly linear relationships were obtained (data not represented here). From the slope of these lines, it is straightforward to calculate the resulting column permeabilities Kv, using

RESULTS AND DISCUSSION (1) Comparison of the Velocity Fields. Figure 1 shows the computed velocity magnitude maps for each of the considered cases. The average velocity being identical in each case, it can immediately be noted from the color intensity scale that the more elongated shapes generally yield a much more uniform distribution of the flow-through pore velocities than the more compact shapes. It is also interesting to note that the more blunt pillar shapes (circles and ellipsoids) create a stagnant fluid zone (blue region) near the front and rear ends of the pillars, whereas these zones are much less pronounced or even absent for the pillars with a sharp tip (diamonds and hexagons). As will become clear further on, this difference, however, has no significant impact on the flow resistance or the band broadening. As the simulation software yields a map not only of the velocity magnitude but also of the x component of the velocity, it is straightforward to determine the tortuosity τ of the different packings by calculating the ratio of the mean velocity magnitude to the mean x velocity.21 The obtained values are listed in Table 1. As can be noted, τ gradually decreases with increasing axial elongation of the pillar structures, in full agreement with one’s physical expectations. (2) Comparison of the Flow Resistances. For each simulation, the Fluent software also returns a value for the total pressuredrop ∆P across the entire flow domain. Plotting these values

As a validation check, it has been verified that the calculated φ value for the parallel plate case corresponds to the theoretically expected value. They agreed to within 0.3%. The obtained φ values for the other shapes are summarized in Table 1. The most striking result obviously is that, despite their presumed advantageous hydrodynamic shape, the ellipsoids display a much larger flow resistance than the more blunt cylinders and hexagons. Overall, the data display quite a large variation and are, for example, totally uncorrelated to the axial elongation parameter R or the pore size d1. It is in fact difficult to correlate the flow resistances to any of the geometrical parameters. Using the mean pore diameter (data not presented here), the only discernible trend is that most of the systems with a large mean pore diameter (the ellipsoids and the cylinders for example) display a larger flow resistance than the systems with the smallest mean pore diameter value (the parallel plate and the diamonds with large R). This obviously is in conflict with one’s physical expectations, and the mean pore diameter has therefore been discarded as a possible correlation basis. The most physically underpinned trend we could discern is based on the ratio of the minimal pore size d2 and the bed tortuosity τ (Figure 2). To generalize the expression, d2 has, furthermore, been made dimensionless by dividing it by the domain size ddom. The use of d2 in this correlation is justified by the fact that it is representative for the pore size of the diamonds over nearly the entire unit cell, whereas it is also characteristic

3720


Kv ) u0ηL/∆P

(7)

These values were then in turn transformed into a flow resistance value φ, using

φ ) deq2/Kv

(8)

for the ellipsoid unit cell regions with the largest flow velocity () regions with the largest pressure drop). The use of the tortuosity in the correlation reflects the fact that in a more tortuous bed both the effective flow path length and the velocity magnitude needed to achieve a given linear x velocity are increased. Both effects obviously lead to a larger flow resistance, cf. the contribution of τ in Kozeny-Carman’s law25 for packed beds. The decrease of φ as a function of d2/(ddomτ) in Figure 2 allows understanding the disproportionately large flow resistance of the ellipsoidal pillars as a consequence of their poor packing ability. When fitted onto the corner points of an equilateral triangular grid, the ellipsoids pack into a much less uniform flow-through pore space than the diamonds or the hexagons, for example. The dependency on d2 now suggests that it is especially the tendency to form locally constricted flow-through pores (cf. Figure 1c,d) that causes the large flow resistance of the ellipsoidal pillars. Another important conclusion which can be drawn is that, although both systems have the same external porosity, the flow resistances of the porous pillar arrays are ∼2 times smaller than that of the packed bed of spheres, for which usually values around φ ) 750-800 are reported.26 We think the significantly smaller flow resistances of the porous pillar arrays are again a consequence of the local pore constriction effect, which is much more pronounced in a 3-D packing, where the pillars are in physical contact, than in a 2-D packing. (3) Comparison of Plate Height and Separation Impedance. As has been convincingly argued by Giddings24 and Knox,27 the performance of different chromatographic systems should preferably be compared in reduced (i.e., dimensionless) coordinates. When considering packings with a widely differing morphology, a proper reduction basis is difficult to find.20 In the present paper, we used the equivalent cylinder diameter (deq), defined as the diameter of the cylinder having the same crosssectional area Ap as the pillar shape under consideration:

deq ) x(4/π)Ap

(9)

As it translates the different pillar shapes into an equivalent cylinder diameter, the use of deq allows for the most direct comparison with the packed bed of spheres. Figure 3 shows the reduced plate height curves for the different pillar shapes and for series 1 and 2 (corresponding respectively to the cases with deq ) 1 µm and deq ) 3 µm). As can be noted from the coincidence of the data from series 1 and 2, deq obviously provides a suitable reduction basis. This is simply due to the self-similarity of the unit cell structures for series 1 and 2. Given the linear relation between deq and the other characteristic cell dimensions, the same degree of h-curve conformity would be obtained when any of the other possible characteristic unit cell dimensions would have been used as the reduction basis. It should also be noted that, under the present geometrical constraints, the condition of an identical deq also (25) Coulson, J. M.; Richardson, J. F. Chemical Engineering Volume 2: Pillar Technology and Separation Processes; Pergamon Press: Oxford, 1991. (26) Grutshka, E.; Snyder, L. R.; Knox, J. H. J. Chromatogr. Sci. 1975, 13, 2537. (27) Knox, J. H. J. Chromatogr. Sci. 1980, 7, 453.

corresponds to the case of an identical domain size, which is the reduction base typically used to interpret the plate height values of silica monoliths.12,20 As in all considered unit cells, the ratio between deq and ddom is equal to 1.229; the deq-based reduced plate height values presented in Figure 3 can easily be transformed in a ddom-based plate height by dividing them by this factor. For all but the parallel plate case, the solid and dashed lines in Figure 3a represent the best possible fit to the Knox equation (eq 2). The dashed line running through the parallel plate data on the other hand has been calculated on the basis of the wellestablished analytical solution originally derived by Giddings28 and Aris:29

H)

(

)

2 Ds 2Dm 2 (1 + 9k′′ + 25.5k′′ ) ud2 1 + k′′ + + u Dm 210 Dm (1 + k′′)2 2

ud f 2 k′′ (10) 3 (1 + k′′)2 Ds

The excellent agreement between the analytical solution and the computed CFD data can be considered as a validation for the employed CFD methods. The use of dashed lines for cases h and i in contrast to the full lines used for cases a-g refers to the fact that in these cases the radial equilibration between adjacent flowthrough pores is excluded. In a real column, where small defects cannot be avoided, the band broadening in cases h and i can, hence, be expected to deviate strongly from the predicted theoretically values. As can be noted from Figure 3a, eq 2 can be fitted quite well onto the pillar array data for all considered pillar shapes. The A, B, and C values returned by the fitting routine are given in Table 1. The obtained B values obviously agree quite well with the value (B ) 2.28) predicted by the theoretical B term expression given in eq 3. The reader should note that the slight deviations do not stem from numerical errors in the CFD algorithm, but are due to the fact that eq 2 does not fit exactly to the obtained h curves and that better fitting equations exist. Switching the A term power in eq 2 from n ) 1/3 to n ) 2/3, for example, yields B values that all fall within the range of B ) 2.27-2.31. Further considering the obtained Knox parameters, it is also interesting to note that the fitted C values are very similar to the C values usually reported for the best possible packed-bed columns.22 The A values on the other hand are much smaller than in the packed-bed case, for which typically A ) 0.5-1.22 As already noted in ref 18, this drastic reduction of the A term constitutes the essence of the advantage of perfectly ordered LC columns; i.e., the increased order nearly completely eliminates the main A term band-broadening sources such as the eddy-diffusion and the short- and large-scale interchannel interactions. This nearcomplete absence of the A term band broadening also explains the very small minimal plate heights of the van Deemter curves in Figure 3a. As these are of the order of hmin ) 0.6-0.8, they obviously are significantly smaller than the hmin ) 2 value usually considered as the ultimate performance limit for the packed bed (28) Giddings, J. C. J. Chromatogr. 1961, 5, 46-60. (29) Aris, R. Proc. R. Soc. 1959, A252, 538-550.


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Figure 3. (a) Reduced pate heights as a function of the reduced velocity for the different considered pillar shapes and sizes (b series 1, deq ) 1 µm; O series 2, deq ) 3 µm). The full lines represent the best fit to the Knox equation (eq 2), except for the parallel plate case, where the full line represents the analytical solution given in eq 10. Same case reference as in Figure 1. (b) Corresponding separation impedances E. The dashed lines used for cases h and i represent the fact that, unlike in the other cases, these structures do not allow for a radial equilibration between adjacent flow-through pores.

of spheres.22 Comparing the plate heights in Figure 3a with those typically obtained in a silica monolith, the advantage of the ordered pillar array column is even more pronounced. As the plate heights in silica monoliths are usually reduced on the basis of the domain size, a fair comparison requires that the h data in Figure 3a should first be divided by a factor of 1.23 (cf. the above remark on the relation between deq and ddom). This obviously further reduces the hmin values for the perfectly ordered pillar arrays to about hmin ) 0.5-0.65. These values are roughly 5 times smaller than in the best possible silica monolith columns, for which typically hmin ) 2.5 according to the Minakuchi correlation.30 Interpreting now the band-broadening data in Figure 3a in terms of the pillar shape, it can immediately be noted that the different curves can be ordered according to the axial elongation R of the pillars. Starting from the cylindrical pillars (smallest R value), the plate height curves gradually shift downward with increasing R and converge toward the parallel plate case (largest (30) Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Soga, N.; Tanaka, N. J. Chromatogr., A 1998, 797, 133-137.

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R value). Considering the separation impedance (Figure 3b), defined as

E ) H2/Kv ) h2φ

(11)

The correlation with R is partly lost, as the ellipsoids with R ) 0.73, for example, yield significantly larger separation impedance than the cylinders and hexagons. As can be understood from the data in Figure 2, the large separation impedance of the ellipsoids is a direct consequence of their disproportionately large flow resistance, in turn resulting from the locally constricted pore space into which the ellipsoids tend to pack. The relation between the axial elongation parameter and the main characteristics of the van Deemter and separation impedance curves is more closely investigated in Figure 4. As can be noted, both hmin and C lie in a more or less monotonically decreasing order when plotted against R. The fact that slightly different hmin and C values are obtained for the diamonds and ellipsoids with the same R (R ) 0.5 and R ) 0.73), however, shows that R does not catch all geometrical subtleties. As will become

Figure 4. Plot of hmin (a), C (b), and Emin (c) versus the axial elongation parameter R. The full lines correspond to the best-fitting second-order trend line obtained with MS Excel. The dashed horizontal line represents the parallel plate limit (R ) ∞). For the data in (c), the two data points for the ellipsoids (]) were not included in the fitting to emphasize the deviation of the ellipsoids from the other shapes.

clear in section 4 of Results and Discussion, the diamonds owe their smaller C values to the fact that they have a significantly smaller Cs,theo value (cf. section 2 of Calculation Procedures), which in turn follows from the fact that the diamond shape yields a slightly shorter mean intrapillar diffusion distance than the ellipsoids. In Figure 4 c, the “abnormally” large separation impedance of the ellipsoids now immediately jumps to the eye.

(4) Detailed Analysis of the Band-Broadening Data. To investigate to which extent the observed plate heights are influenced by the mobile and the stationary zone mass transfer, the value of the theoretically calculated B and Cs contributions has been subtracted from the total plate height. This yields a value for hmob, defined as

hmob ) h - Btheo/ν - Cs,theoν

(12)


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Figure 5. Variation of mobile-phase band-broadening contribution (hmob) with ν for the different considered pillar shapes. The dashed lines used for cases h and i represent the fact that, unlike in the other cases, these structures do not allow for a radial equilibration between adjacent flow-through pores.

with Btheo and Cs,theo calculated via eqs 3 and 6, respectively. As can be noted from Figure 5, the hmob curves for cases a-f nearly perfectly coincide in the ν > 10 range and lie upon a straight line with slope n = 0.8 in a log-log plot. It is only for the most elongated diamonds (with R ) 1 and R ) 1.89) and for the parallel flat plat case that a gradually smaller hmob contribution is obtained. Interestingly, the slope of the curves also gradually increases with R, going from a slope of n = 0.8 for cases a-f to a slope of exactly n ) 1 for the parallel plate case. The latter value is in agreement with the theoretical expression in eq 10, showing that hmob is linearly proportional to ν in the large ν range. The nearly perfect coincidence of the hmob values for cases a-f, at least in the ν > 10 range, implies that the (relatively large) differences between the flow fields only have a negligible effect on the total band broadening. In the ν < 10 range, a more complex behavior is noted and the differences between the pillar shapes are more prominent. Comparing the touching diamond system (case h) with the parallel plate system (case i), it is interesting to note that, although both systems have a nearly identical velocity field and pore geometry, their hmob contribution is clearly different. This difference can only be attributed to the fact that in case h the stationary zone thickness is not uniformly and symmetrically distributed in the direction perpendicular to the pore axis. The magnitude of this asymmetric distribution effect is, however, not very large, roughly corresponding to a difference of only h ) 0.02-0.1. On a total value of h ) 0.5 to h ) 1.5 (cf. Figure 3a), this is quite insignificant. The asymmetric distribution effect seems to play the largest role in the large ν range and vanishes nearly completely near ν ) νopt. The fact that the overall C values depend quite strongly on the pillar shape (cf. Table 1), whereas the hmob values are nearly completely independent of it, can only be explained from the fact that the major band-broadening source lies in the stationary zone and not in the mobile zone. A similar conclusion can be made by calculating the difference between the C and Cs,theo values listed in Table 1. Doing so, it is found that the difference (isolating the effect of the mobile-phase mass-transfer resistance) depends only very weakly upon the pillar shape (variation below 5% for all cases a-h) and, furthermore, only makes up about one-fourth to one3724 Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

third of the total C term. It can hence be concluded that the differences between the C term-dominated range of the van Deemter curves in Figure 3a are essentially due to the differences in the pillar shape and not to the differences in the pore space geometry or the flow field, which inevitably change together with the pillar shape. The above observations on the dominant effect of the stationary-phase mass-transfer resistance can also be understood from the fact that in the presently considered densely packed system the mean diffusion distance in the mobile zone is much shorter than in the stationary zone, augmented by the fact that the species spent twice as much time in the stationary zone than in the mobile zone (as a consequence of the k′′ ) 2 conditions), and also augmented by the fact that the diffusion rate of the sample species in the stationary zone is assumed to be only half of that in the mobile zone. The fact that the plate heights are fully dominated by the Cs term also implies that there is not much room for further improvement beyond the presently considered 2-D pillar array case. The h and E curves already lie very close to the parallel plate case, constituting the ultimate performance limit of LC. In contrast with packed-bed or monolithic silica columns, where the packing heterogeneity or the pore sizes are so large that the major source for band broadening lies in the A and Cm term, all “avoidable” band-broadening sources are nearly completely eliminated in the perfectly ordered porous pillar arrays, leaving the “unavoidable” stationary-phase mass-transfer resistance to be the single main band-broadening source. (5) Extrapolation of the Results. Considering the use of more elongated staggering patterns than the presently considered equilateral triangular staggering, it would be possible to elongate the individual pillars beyond the current upper limit of R ) 1.88. From the decrease of E and h with R noted in Figure 4, it can be inferred that such extremely elongated pillars would yield lower h and E curves than those shown in Figure 3. However, as it is will be impossible to surpass the parallel plate case, it can readily be anticipated that the h and E curves for these more elongated pillars will lie somewhere between the current most elongated pillar cases and the parallel plate case. When considering such

more axially elongated pillars, it should, however, be noted that the number of trans-channel coupling points would be reduced below the present number. The resulting chromatographic performance would then be more strongly influenced by the degree of uniformity with which the flow-through pores can be etched, such that any conclusion on the improved performance of a more axially elongated staggering could be overshadowed by the increased pore-to-pore variability problem. The recent experiments of Slentz et al.17 in a series of PDMS-replicated COMOSS columns seem to point in this direction. Considering the possibility of using arrays with an increased external porosity, it is straightforward to expect that the influence of d2 upon the flow resistance will be less critical. In this case, the ellipsoidal shape will be less disadvantaged, and the order of the E curves will correspond more closely to the order of the h curves. As it can also be expected that for very open structures (say ) 0.9) the band broadening and separation impedance will be nearly completely independent of the pillar shape, it can be inferred that the difference between the different pillar shapes will gradually decrease, from the relatively strong influence of the presently considered ) 0.4-case to a nearly nonexisting influence for the most open pillar arrays. Although the present study relates exclusively to a 2-D model where the pore connectivity obviously is quite different from that in a 3-D particulate column, it is thought that the conclusions on the advantage of axially elongated pillars and on the possibility to closely approach the parallel plate limit with a particulate system provided it is perfectly homogeneous will also hold in 3-D (although it is difficult to imagine how axially elongated pillars could be produced and perfectly aligned in a 3-D system). CONCLUSIONS Provided they could be manufactured with a sufficient degree of accuracy, ordered porous pillar array columns would provide a powerful alternative to the silica monolith columns. Whereas the current monolithic silica columns can achieve their small separation impedances (around E ) 200-400) only at the expense of an increased porosity, i.e., by sacrificing part of the mass loadability and retention capacity of the column, perfectly ordered porous pillar arrays could yield such small separation impedances at the typical packed-bed porosity of ) 0.4. Furthermore, the plate heights in the ordered arrays would be dramatically smaller than in the silica monoliths. Comparing both systems on the basis of an identical domain size, the ordered pillar systems are expected to yield reduced plate heights as small as h ) 0.5 for a retained solute with k′′ ) 2 and Ds ) 5 × 10-10 m2/s, i.e., easily 5 times smaller than the current silica monoliths. This observation emphasizes the fact that increasing the uniformity of a chromatographic packing is a much better approach to increase the peak capacity and throughput than increasing the porosity. Obviously, the former is a much easier task than the latter, but as with most things in life, there can be no gain without pain. We hope the present results will motivate researchers active in the microstructuring of porous materials and ceramics to start conceiving potential manufacturing routes yielding more uniform chromatographic beds. The present study has now demonstrated that pillar beds with an axially elongated shape will yield the best chromatographic performance and that diamonds are to be preferred over ellipsoids.

In perfectly ordered, densely packed columns, the Cs contribution is left as the single remaining important band-broadening source. It is hence obvious to find that the shaping of the stationary phase has a significant impact on the resulting chromatographic performance. Switching from cylinders (axial elongation, 0) to diamond with axial elongation, R ) 1, for example, allows one to reduce the minimal plate height from hmin ) 0.83 to hmin ) 0.72, the C factor from C ) 0.063 to C ) 0.050, and the separation impedance from Emin ) 331 to Emin ) 228, without affecting the number of trans-channel coupling points. The latter is a critical prerequisite in overcoming the pore-to-pore width variability, which will inevitably occur in any real-world micromachined packing. The presently considered perfectly ordered 2-D porous pillar arrays obviously constitute the ultimate performance limit one can ever expect from a particulate material packing. The predicted E numbers for the pillar arrays (offering the same number of radial equilibration points as a packed-bed HPLC column) are ∼10 times smaller than the best possible packedbed HPLC column and are only a factor of 2 away from the parallel plate limit, which is, provided there would be absolutely no poreto-pore width variability, the theoretically ideal chromatographic system. ACKNOWLEDGMENT The authors greatly acknowledge a research grant (FWO KNO 81/00) of the Fund for Scientific ResearchsFlanders (Belgium). P.G. is supported through a specialization grant from the Instituut voor Wetenschap en Technologie (IWT) of the Flanders Region (Grant SB/11419). GLOSSARY Ap

cross sectional area of a pillar (m2)

d

channel diameter (m)

d1

interpillar distance in unit cell areas where only a single pore is present (m)

d2

interpillar distance in unit cell areas where two pores are present (m)

dax

maximal axial distance in the pillars (m)

dc

diameter of the cylindrical pillars (m)

ddom

unit cell domain size (m)

deq

equivalent cylinder diameter (m), see eq 9

df

stationary zone thickness in rectangular channel (m)

dlat

maximal lateral distance in the pillars (m)

Dm

mobile zone diffusion coefficient (m2/s)

Ds

effective stationary zone diffusion coefficient (m2/s); Ds ) 5 × 10-10 m2/s in the present calculations

E

separation impedance (/), see eq 11

H

height equivalent of a theoretical plate (m)

h

reduced theoretical plate height (h ) H/deq) (/)

HCs

theoretical plate height contribution of stationary zone mass-transfer resistance (m)

hmin

minimal reduced theoretical plate height (/)

hmob

reduced theoretical plate height contribution of mobile zone mass-transfer resistance (/)

k′′

zone retention factor (/) Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

3725

internal porosity of the porous pillars (/); int ) 0.5 in the presented calculations

k′′0

zone retention factor of unretained species (/)

K

adsorption equilibrium constant (K ) k1/k2) (/)

Kv

column permeability, (m2), see eq 7

φ

flow resistance factor (/)

L

total flow domain length (m)

γsz

ratio of Ds to Dm (/)

ldom

unit cell domain length (m)

η

dynamic viscosity (kg/(m‚s))

u

mean velocity of nonpermeating solute (m/s)

ν

reduced fluid velocity, based on u0 () u0deq/Dmol) (/)

u0

mean linear velocity of permeating, but nonretained solute (m/s)

θ

dimensionless parameter used in eq 5 (/)

τ

tortuosity (/)

int

Greek Symbols R

axial elongation parameter (/)

∆P

pressure drop (Pa)

Received for review January 21, 2004. Accepted April 15, 2004.

external column porosity (/)

AC049873S

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Influence of the Pillar Shape on the Band Broadening and the

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