Anal. Chem. 2003, 75, 6244-6250
Advantages of Perfectly Ordered 2-D Porous Pillar Arrays over Packed Bed Columns for LC Separations: A Theoretical Analysis P. Gzil,* N. Vervoort, G. V. Baron, and G. Desmet
Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan 2, 1050 Brussels, Belgium
A series of theoretical calculations is presented to quantify the gain in LC separation efficiency that can be expected if the traditionally used packed bed columns were replaced by columns with a perfectly ordered flow-through pore network. It is shown that a perfectly ordered 2-D array of porous cylindrical pillars could yield reduced plate heights as small as h ) 0.65 (for k′′ ) 0.75) to h ) 0.85 (for k′′ ) 2) and separation impedances as small as E ) 200 (for k′′ ) 0.75) to E ) 300 (for k′′ ) 2) without having to compromise on the porosity (E ) 0.4) and the retention capacity of the packed bed of spheres. Fitting the calculated van Deemter plots with Knox’s equation especially shows a strong decrease of the A-term contribution, hence confirming that the improved column performance indeed stems from the increased homogeneity of the packing. The presented results, hence, provide a clear quantitative support for Knox’s recent argumentation that the use of more uniform beds could greatly enhance the efficiency of pressure-driven LC. In pressure-driven LC (HPLC), still the chromatographic method of choice in most analytical labs, the best possible packed bed columns yield a minimal reduced plate height value around or slightly below h ) 2. Although there is no theoretical justification for the existence of such a lower limit, the large research efforts and improvements in the field of column packing and bead synthesis have not allowed breaking through this barrier. In a recent publication, Knox1 mentioned the use of the micromachining methods of the microelectronics industry as one of the feasible routes to devise LC columns that allow to breaking through this h ) 2 limit. His suggestion, in fact, follows upon the ground-breaking work of Regnier et al.,2,3 who first pointed out the possibility of micromachined packed columns to the chromatographic community. Knox1,4 deduced his arguments on the large potential advantage of 2-D ordered columns after having fitted the van Deemter curves of a large number of different packed and monolithic column systems with his well-established empirical correlation. * Corresponding author. Phone: (+)32-2-629-32-51. Fax: (+)32-2-629-32-48. E-mail:
[email protected]. (1) Knox, J. H. J. Chromatogr., A 2002, A 960, 7-18. (2) He, B.; Tait, N.; Regnier, F. Anal. Chem. 1998, 70, 3790-3797. (3) Regnier, F. J. High Resolut. Chromatogr. 2000, 23, 19-26. (4) Knox, J. H. J. Chromatogr., A 1999, A 831, 3-15.
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Figure 1. Schematic representation of a 2-D ordered porous pillar column.
h ) Aν1/3 +
B + Cν ν
(1)
He found that, even in the best possible packed bed columns, the major source of band-broadening is formed by the A term, representing the band broadening arising from the irregular structure of the mobile-phase zone. This finding then, of course, automatically implies that the use of columns with a perfectly ordered internal structure would strongly improve the efficiency of LC separations. In the present paper, we present a theoretical study conducted to quantify Knox’s argumentation. As a representative example, a column filled with a 2-D array of porous pillars is considered (Figure 1). Our interest in a porous pillar system, in contrast with the nonporous COMOSS structures of Regnier’s group, follows from our desire to compare both systems (2-D array and packed bed) on the basis of the same mass loadability (i.e., detection sensitivity) and retention capacity. In a nonporous array, the retention capacity and mass loadability are orders of magnitude smaller. To stay as close as possible to the traditional packed bed, the present study also only focuses on cylindrical pillars, although a wide variety of other cross-sectional pillar shapes (squares, hexagons, elongated ellipsoids, etc.) is conceivable.2 To our knowledge, the problem of manufacturing micromachined 2-D porous pillar arrays has not been addressed yet. The present work is, therefore, also intended as an incentive for groups working in fields such as the lab-on-a-chip (lithographic etching, replication in plastic materials, etc.) and the synthesis of porous 10.1021/ac034345m CCC: $25.00
© 2003 American Chemical Society Published on Web 10/22/2003
materials (sol-gel techniques, generation of porous silicon, etc.) to start developing microstructured porous pillar arrays. Given the recent advances in the field of lithographic etching, induced by the invention and commercialization of the Bosch or ASE process,5 allowing etching of micrometer-sized structures with a width to depth aspect ratio of 10 or more6 while maintaining a wall slope of 90° ( 0.1°, we believe the time is now ripe for it. The possibility to etch a pattern of 1-2-µm-sized structures, maintaining a nearly perfectly vertical wall slope over a height of 10-20 µm, implies that it should be possible to manufacture 2-D porous pillar columns having a depth of 10-20 µm. Because this depth is more than sufficient to guide a light beam through the column in the transverse direction, and since such a 2-D micromachined column could easily be made 1 cm or more wide, yielding an optical path of the same length, it should be obvious that a 2-D porous pillar array column could yield the same absorbance detection sensitivity as a conventional packed HPLC column. The column could also be directly interfaced to a micromachined U cell for on-column absorbance detection, as has recently been demonstrated.7 The injection of the sample also should not pose too many difficulties, because O’Neill8 and Chmela9 recently demonstrated the possibility of high resolution pressure-driven injection methods for on-chip LC. The use of electrokinetically driven injection methods via a perpendicularly etched injection channel also seems to be a viable option. The most important technological challenge related to the manufacturing of 2-D porous pillar columns seems to be the realization of a high-pressure resistant sealing of the channel parts and the connection tubings. In expectation of the availability of such microfabricated porous pillar arrays, we have investigated the problem from a purely theoretical perspective and have assessed their chromatographic performance by solving the full convection-diffusion material balances in a 2-D periodic domain, using a commercial computational fluid dynamics (CFD) software package (FLUENT v.6.1). CFD packages are ideally suited to calculate the velocity field in complex geometries and provide a unique opportunity to obtain detailed velocity data for flows through spaces that are inaccessible for experimental measurement methods. CFD simulations have, therefore, also become an increasingly used design tool in the lab-on-a-chip field10,11 and for the analysis of the flow pattern in chromatographic columns.12-15 Since the flow in an LC column is without any doubt purely laminar and is, hence, purely determin(5) Laermer, F.; Schilp, A. U.S. Patent 6,531,068, 2000. (6) Hynes, A. M.; Ashraf, H.; Bhardwaj, J. K.; Hopkins, J.; Johnston, I.; Shepherd, J. N. Sens. Actuators, A 1999, 74, 13-17. (7) Mogensen, K. B.; Petersen, J. P.; Hubner, J.; Kutter, J. Electrophoresis 2001, 22, 3930-3938. (8) O’Neill, A. P.; O’Brien, P.; Alderman, J.; Hoffman, D.; McEnery, M.; Murrihy, J.; Glennon, J. D. J. Chromatogr., A 2001, 924, 259-263. (9) Chmela, E.; Blom, M. T.; Gardeniers, J. G. E.; van den Berg, A.; Tijssen, R. Anal. Chem. 2002, 74, 3470-3478. (10) Ermakov, S. V.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 2000, 72, 35123517. (11) Molho, J. I.; Herr, A. E.; Mosier, B. P.; Santiago, J. G.; Kenny, T. W.; Brennen, R. A.; Gordon, G. B.; Mohammadi, B. Anal. Chem. 2001, 73, 1350-1360. (12) Schure, M. R.; Maier, R. S.; Kroll, D. M.; Davis, H. T. Anal. Chem. 2002, 74, 6006-6016. (13) Vervoort, N.; Gzil, P.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 843850. (14) Wu, Y. X.; Chin, C. B. Chromatographia 2002, 56, 679-686. (15) Gzil, P.; Baron, G. V.; Desmet, G. J. Chromatogr., A 2003, 991, 169-188.
Figure 2. Considered 2-D column geometries. (a) Porous cylinder array with equilateral triangular arrangement yielding ) 0.4. The relations between the different dimensions are given in the text. (b) Straight-running, infinitely wide channel. Channel thickness, d ) 1µm; stationary film thickness, df ) 0.75 µm. All porous zones are colored in red.
istic, the CFD results can be made fully accurate, provided care is taken that the obtained results are independent of the grid shape and size. CONSIDERED GEOMETRY AND CFD SOLUTION METHOD Figure 2a shows the considered 2-D cylinder array lay-out. We have opted for the equilateral triangular arrangement, characterized by the fact that the distances a and b are identical. A number of other cylinder arrangements has been considered as well (with a * b). Because these all yielded slightly larger theoretical plate height values than the a ) b case, we preferred to stick to the latter. A more comprehensive simulation study is, however, needed to decide on the optimal cylinder arrangement. To compare the 2-D array and the packed bed on the basis of the same column porosity, that is, on the basis of the same mass loadability and retention capacity, the relative dimensions of the pillar diameter and the 2-D array unit cell (delimited by the dashed rectangular border depicted in Figure 2a) have been selected such that the mobile phase zone makes up exactly 40% of the unit cell surface. For the equilateral triangular arrangement, this condition fixes the dimensions of the unit cell as given by: Z1 ) 1.2294dp and Z2 ) x3 Z1 ) 2.1294dp. In all the presented simulations, the total flow domain was 10 unit cells long (cf. the discussion of Figure 5, given later). Two different cylinder diameter values were considered: dp ) 3 µm and dp ) 1 µm. Prior to the calculations in the cylindrical array system, the adopted calculation procedure was first validated by performing a series of peak-broadening calculations in a system with a well-established analytical solution: a perfectly straight running channel formed by two parallel plates with spacing d and carrying a stationary phase layer with thickness df (Figure 2b). When setting up the problem for the CFD software, the stationary phase zones are treated as porous zones (with freely selectable flow resistance factor, always set at infinity in the present study) embedded in a continuous fluid zone with a velocity-inlet and a pressure outlet. The effect of the internal Analytical Chemistry, Vol. 75, No. 22, November 15, 2003
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Figure 3. Influence of the magnitude of the k1 and k2 adsorption reaction rate constants on the agreement between the elution profile in an infinitely wide rectangular channel calculated via the CFD method (black lines) and predicted by the analytical solution (red line) given in eq 10. Chromatographic parameters: k′′ ) 2, d ) 1 µm, df ) 0.75 µm, Dm ) 10-9 m2/s, and Ds ) 5 × 10-10 m2/s. In all cases, k2 was taken as k2 ) 0.5 k1. The deviation for the HETP value from the analytical solution was k1 ) 104, 139.33%; k1 ) 105, 13.91%; k1 ) 106, 1.36%; and k1 ) 107, 0.09%. The case of k1 ) 107 cannot be distinguished from the analytical solution (red line).
Figure 5. Evolution of the σ2 values (diamond symbols) with the distance for the ) 0.4 case. Showing entrance region and exit effect and comparison between simulated (square symbols) and expected mean break-through time (dashed line) calculated on the basis of eqs (3, 4), and (7). The full line for the σ2 values has only been given to visualize the existence of a linear region.
order to exactly obtain an effective stationary zone diffusion coefficient of Ds ) 5 × 10-10 m2/s. To represent the phenomenon of selective adsorption (retention), the species in the stationary zones were subjected to a second user-defined function, expressing the occurrence of a reversible chemical reaction transforming the freely diffusing species Y into a species Y* with identical physical properties but with a zero diffusivity, as to express its adsorbed state. k1
Y (freely diffusing) y\ z Y* (adsorbed) k 2
Figure 4. ∆P/L versus u0 plots for the considered dp ) 3 µm and dp ) 1 µm cases (dynamic fluid viscosity η ) 1.003 × 10-3 kg/(m s); relative array and pillar dimensions as shown in Figure 2a). The u0 data refer to the migration velocity of an unretained species with k′′0 ) 0.75.
porosity of the stationary zones can easily be investigated, because the software package has a built-in function that allows attributing a given, freely selectable, internal porosity to the porous zones. To approximate the conditions in a traditional packed bed column, a value of int ) 0.5 has been assumed in all presented calculations. To mimic the effect of the slow intraparticle diffusivity, a userdefined function (additional, user-written source code that is implemented in the solver) has been written to give the species entering the stationary zones a diffusion coefficient different (i.e., smaller) from that in the fluid zone. The resulting effective stationary zone diffusivity was measured by monitoring the transient concentration increase in the center of a single cylinder in response to a sudden concentration step on the outer surface of this cylinder. The transient response curve was then compared to the analytical solution16 to this problem. It was found that for the case of int ) 0.5, the diffusion coefficient of the species present in the porous zones had to be put at Dint ) 2.5 × 10-10 m2/s in 6246
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(2)
By changing the ratio K of the forward and backward rate constant, different adsorption equilibriums can be imposed, which in turn corresponds to a variation of the phase and zone retention factors (k′ and k′′). From the known external () and internal (int) porosities, the zone and phase retention factors can be directly calculated as (k′′0 is the zone retention factor of a nonadsorbing species):19
k′′ ) k′′0 (1 + K) with k′′0 )
1- int
(3)
and
k′ )
k′′ - k′′0 1 + k′′0
(4)
In our calculations, K was varied as indicated in Table 1 to impose different k′′-values. For the presently considered porous pillar system ( ) 0.4 and int ) 0.5), the zone retention factor for the nonadsorbing species can, using eqs (3 and 4), easily be shown to be given by k′′′0 ) 0.75. (16) Carslaw, H. S.; Jaeger, J. C. The Conduction of Heat in Solids; Oxford University Press: Oxford, U.K., 1959. (17) Giddings, J. C. J. Chromatogr. 1961, 5, 46-60. (18) Aris, R. Proc. R. Soc. 1959, A252, 538-550. (19) Knox, J. H.; Scott, H. P. J. Chromatogr. 1983, 282, 297-313.
Table 1. Values for the A, B, and C Constants Obtained by Fitting Knox’s Equation (Eq 1) to the 2-D Array Peak-Broadening Data int
K
k′′
k′
A
B
C
νopt
hmin
0 0.5 0.5 0.5
na 0 1 1.67
0 0.75 1.5 2
0 0 0.43 0.71
0.014 0.043 0.043 0.043
0.84 1.35 1.91 2.26
0.001 0.059 0.064 0.063
40 5 5 5
0.07 0.64 0.78 0.84
Retention times and theoretical plate height values were obtained by recording the passage of a plug of virtually injected tracer species as a function of time at a number of regularly spaced detection planes, and by numerically calculating the 0th-, first- and second-order moment of the obtained break-through curves using
∫ ) ∫
+∞
tR,i
0
+∞
0
∫ ) ∫
+∞
σt,i
2
Cit dt
0
+∞
0
Ci dt
- tR2
(5)
Li Li (1 + k′) ) (1 + k′′) u0 um
um 1 + k′′0
(6)
(7)
(8)
Comparing the k′ and k′′ values obtained from eqs 5 and 7 with the theoretically expected values calculated from eqs 3 and 4 yielded an additional control over the accuracy of the CFD calculations. These values never differed more than 0.3%. Theoretical plate height values were directly obtained from eqs 5 and 6 using
H)
σt,j2 - σt,i2
Lij (tR,j - tR,i)2
)
2
wherein um is the mean axial component of interstitial fluid velocity and wherein u0 is the migration velocity of the nonretained species. Both characteristic velocities are related by the zone retention factor k′′0 of the unretained species.
u0 )
(
DS 2Dm 1 + k′′ + u Dm 2 udf 2 (1 + 9k′′ + 25.5k′′ ) ud2 2 k′′ + (10) 2 2 210 Dm 3 (1 + k′′) DS (1 + k′′)
From eq 5, the phase and zone retention factors (k′ and k′′) can be calculated as
tR,i )
RESULTS AND DISCUSSION Peak Broadening in the 2-D Rectangular Channel. For a pressure-driven chromatographic system formed by two parallel, infinitely wide plates, Gidings17 and Aris18 have shown theoretically that
H)
Ci dt
Cit2 dt
and before running another 200 iterations. The change in the velocity field values and pressure drop was always within 0.3%, small enough to conclude that the solutions are grid-independent and sufficiently converged. To ensure that the calculated species transport rates are grid- and time-step-independent, a range of different grid sizes was explored. It was found that, for the entire simulation domain (i.e., 10 half-unit cells in series), the use of ∼25 000 cells was sufficient to achieve an accuracy of ∼0.2% on the resulting H value.
(9)
and were made dimensionless on the basis of the pillar diameter (h ) H/dp). All calculations were carried out on Dell PCs with Intel Xeon 2 Ghz processor, and equipped with 2 Gb RAM. The grids were generated with GAMBIT (v.2) software run on the same hardware configuration. To ensure that the velocity fields calculated by the CFD software were grid-independent, a velocity gradient adaptation of the grid was performed after the initial 100-300 iteration steps
Given that we wanted to test our simulation procedure for a system with ) 0.4, the stationary film thickness (df ) (3/4)d) is so large that the stationary zone diffusion contribution to the over-all band broadening can no longer be neglected, as is normally done for thin-film columns. The k′′Ds/Dm term in the diffusion part of eq 10 accounts19 for this additional thick-film effect. When developing the CFD calculation procedure, the selection of the absolute values of the forward and backward rate constants k1 and k2 (cf. eq 1) turned out to be very critical for the accuracy of the solution. When k1 and k2 are too small, the adsorption step occurs too slowly and influences the total rate of the chromatographic exchange process. In this case, an agreement with eq 10 cannot be obtained, because the latter has been established by assuming an infinitely rapid adsorption step. Figure 3 clearly shows that by increasing k1 and k2, the simulated break-through curves gradually converge exactly toward the break-through curve predicted by the analytical solution, hence, demonstrating the accuracy of the adopted CFD calculation procedure. The values of k1 and k2, however also should not be taken too large: above a given value of k1 and k2, the stability of the numerical calculation schemes requires the use of smaller integration time steps, which then, of course, leads to larger calculation times. As a general rule of thumb, it was found that k1 and k2 should be selected such that k1 ) 106 s-1. For the 2-D cylinder array, the influence of the selected k1 and k2 was verified by comparing cases with different k1 and k2 values, and taking the smallest k1 and k2 values for which the deviation from the exact h value (obtained by letting k1 and k2 go to infinity) was smaller than 1%. It was found that, as expected, the rule of thumb given above for the rectangular open-tubular channel remains valid for the 2-D cylinder array. Flow Resistance Data for the 2-D Pillar Arrays. For pressure-driven LC, the flow resistance of the column plays a crucial role in the overall separation performance. Generally, the flow resistance is calculated on the basis of the migration velocity (u0) of the nonretained species,20 using (20) Bristow, P. A.; Knox, J. H. Chromatographia 1977, 10, 279.
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φ ) dp2/Kv
(11)
wherein Kv is the column permeability.
Kv ) u0ηL/∆P
(12)
The perfectly linear relationship between the calculated pressure drop gradient (∆P/L) and the migration velocity of the nonretained species u0 (Figure 4) shows that the 2-D system satisfies D’Arcy’s law, as is theoretically expected for a laminar flow through a porous medium. From the slope of the curves, a column permeability of Kv ) 1.87 × 10-14 m2 (dp ) 3 µm) and Kv ) 2.08 × 10-15 m2 (dp ) 1 µm) is obtained. As theoretically expected from the fact that laminar flows always display an inversely quadratic relation between the pressure drop variable and the characteristic dimension,21 using eq 11, these column permeability values lead to the same flow resistance, which for the present case is given by φ ) 480 (i.e., based on a nonretained solute with k′′0 ) 0.75). The φ ) 480 value is ∼30% smaller than the theoretical value for the best possible uniform packed bed, for which φ ) 710 (calculated on the basis of the well-known Kozeny-Carman21 equation using ) 0.4 and for the same k′′0 ) 0.75 case). We believe that the smaller flow resistance of the 2-D array column can be explained from the perfectly ordered arrangement of the cylinders and from the well-streamlined pattern of the flow-through pore network. The fact that the two different dp cases yield the same dimensionless flow resistance value to within 0.2% should be considered as an additional validation of the adopted calculation method. Peak Broadening in the 2-D Pillar Array (E ) 0.4 Case). Figure 5 shows a typical plot of the evolution of the calculated σt2 values with the distance along the column. Between the second and the next to last detection plane, the σt2-values vary in a perfectly linear way with the distance, hence, reflecting that the H values obtained from this region are constant and fully converged. The deviation between the entrance and the first detection plane reflects the existence of an entrance effect, in agreement with the physical expectations. The deviation between the next to last detection plane and the detection plane at the outlet corresponds to an exit effect. No physical ground other than that the boundary conditions imposed at the outlet generate an error which propagates upstream can be suggested for this exit effect. The effect was especially important for the velocities below uopt (B term governed region). For all presented cases, the total domain (10 unit cells long) was, however, always sufficiently long to have at least four cells that were not significantly affected by the exit effect. Figure 6a shows the theoretical plate heights obtained as a function of the unretained solute velocity (u0) for the dp ) 3 µm case and for different values of the zone retention factor k′′. As can be noted, the plate height values for the nonporous pillar case, that is, for k′′ ) 0, are extremely small. In terms of a reduced plate height, the values are not larger than h = 0.1. This value is ∼6 times smaller than that cited by Knox1,4 for the best possible (21) Coulson, J. M.; Richardson, J. F. Chemical Engineering; Volume 2 - Particle Technology and Separation Processes; Pergamon Press: Oxford, U.K., 1991; Volume 2.
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Figure 6. (a) van Deemter plots (H versus u0) for different values of the zone retention factor k′′. Array and pillar dimensions as shown in Figure 2a for the dp ) 3 µm case. Other parameters: Dm ) 10-9 m2/s and Ds ) 5 × 10-10 m2/s. (b) Comparison between the dp ) 3 µm (square symbols) and the dp ) 1 µm case (triangular symbols) for k′′ ) 0.75 and k′′ ) 2. (c) Corresponding reduced van Deemter plots (h versus ν) for the dp ) 3 µm (square symbols) and the dp ) 1 µm data (triangular symbols) shown in (b) and a comparison between the simulated 2-D array results (symbols) and the fitted h curves (dashed lines) based on Knox’s equation (eq 1). The full circular symbols were obtained by putting A ) 0.5, B ) 2, and C ) 0.05 into eq 1 and, hence,23 represent the best possible packed bed column. The full line is only given to connect the different data points. The dashed lines in (c) represent the fitted Knox equation (eq 1) using the parameters given in Table 1.
packed column of nonporous, nonretaining glass beads. We believe that this large difference can only be explained from the perfectly ordered flow-through pore structure of the 2-D array, as opposed to the inevitably more irregular structure of the packed bed. This assumption is supported by the fact that Giddings22
estimates the contribution to band broadening of the intrachannel mass transfer (the single possible source of band broadening in a perfectly ordered system) to be of the order of h ) 0.1, that is, exactly of the same order as found in the current study for the 2-D array. For the porous pillar cases (with k′′ ) 0.75, k′′ ) 1.5, and k′′ ) 2), the plate heights are significantly larger than for the nonporous pillar case. The plate heights also show a slight dependence on the zone retention factor, as could be theoretically expected. Figure 6b compares the influence of the pillar diameter for the k′′ ) 0.75 and the k′′ ) 2 case. As expected, the 1-µm pillar system yields substantially smaller plate heights than the 3-µm pillar system. There is even a broad range of fluid velocities wherein submicrometer plate heights can be achieved. Normalizing the data by adopting the dimensionless plate height approach proposed by Giddings and Knox yields a (h, ν) plot wherein the dp ) 1 µm and dp ) 3 µm data perfectly coincide (cf. triangular and square symbols in Figure 6c), showing that the pillar diameter is a suitable dimension to normalize the theoretical plate heights, as is the particle diameter for the packed bed column. Furthermore, whereas 1-µm and 3-µm particles do not always yield the same packing quality in a real column, the regularity of the pillar arrangement in the presently considered idealized structure is, of course, independent of the pillar diameter, hence, explaining the perfect agreement between the (h, ν) curves for the dp ) 1 µm and dp ) 3 µm data. The most important feature of Figure 6c, however, is that the hmin values in the ordered 2-D array are ∼2.5 times smaller than in the best possible packed columns (for which A ) 0.5, B ) 2, and C ) 0.05 in eq 123). This obviously is a dramatic gain. The hmin values around h ) 0.65-0.85 are even smaller than those obtained in the best performing packed column CEC systems, for which typically, hmin ) 1.4,24 This finding indicates that part of the band broadening in packed column CEC also results from the heterogeneity of the flow-through pore structure and that an improved homogeneity of the column packing could also significantly enhance the performance of electrokinetically driven packed column separations. To further analyze the different contributions to the peak broadening in the 2-D pillar system, the (h, ν) curves shown in Figure 6c have been fitted with Knox’s equation (eq 1). The fitting was carried out with the solver function of Microsoft Excel 2000, considering A, B, and C as three independently variable parameters. The resulting parameter values are given in Table 1. The excellent agreement between the simulated 2-D array results (square and triangle symbols) and the fitted h-curves based on Knox’s equation (dashed lines) in Figure 6c shows that Knox’s equation also remains valid for 2-D pillar arrays. Considering the A, B, and C values given in Table 1, it can immediately be concluded that the increased order especially has an effect on the A term (A ) 0.043, i.e., ∼11 times smaller than the A ) 0.5 value cited23 for the best possible packed bed). This is in agreement with the theoretical expectations, because in an ordered 2-D array, only the intrachannel coupling can contribute to the band broadening, while the short-range interchannel, longrange interchannel, and transcolumn contributions are completely eliminated, since the ordered array can be considered a perfect (22) Giddings, J. C. Dynamics of Chromatography; Part I; Marcel Dekker Inc.: New York, 1965. (23) Knox, J. H. J. Chromatogr. Sci. 1980, 18, 453-461. (24) Knox, J. H.; Grant, I. H. Chromatographia 1987, 24, 135-140.
Figure 7. Separation impedance values (E) as a function of the unretained species reduced velocity ν (with k′′0 ) 0.75) for different values of k′′. Packed bed column data obtained by putting A ) 0.5, B ) 2, and C ) 0.05 into eq 1 and using φ ) 710 as the flow resistance factor. The 2-D array data are obtained using the h values from Figure 6c and using the φ ) 480 value derived from the pressuredrop data given in Figure 4.
transversal and axial repetition of the unit cell (cf. Figure 2a). The present results, hence, indicate that in packed column HPLC, about one-half of the plate height value is made up of the shortrange and long-range interchannel coupling contributions. The slight dependency of the C term on k′′ is in agreement with the theoretical expectations. The absolute values of the C term (∼0.06) are exactly of the same order as the best possible values23 in the 3-D packed bed system. The B term values are also close to the typical packed bed value of B ) 2 and display the perfectly linear k′′ dependency, as predicted by Knox and Scott.19 Separation Impedances. When a system is pushed to its pressure-drop limit, for example, to reach the maximum theoretical plate number or to achieve the minimal analysis time, its performance is determined by both H and Kv. The effect of both parameters has been very elegantly combined by Bristow and Knox20 into a single performance index, the so-called separation impedance, E.
E ) H2/Kv ) h2φ
(13)
Figure 7 shows a plot of E versus ν for the different considered k′′-values. Comparing the 2-D column data with the packed column correlations shows that an improved structuring of the packing could yield E values as small as E ) 200-300, which is typically 10 times smaller than the best possible packed column, for which generally a value of Emin ) 2000-3000 is cited.23,25 Since the E number is directly proportional to the minimal analysis time needed to perform a separation with given resolution,20 the 10fold decrease in E also enables a 10-fold decrease in the minimal analysis time, constituting a really dramatic gain. The obtained Emin values are similar and, in fact, are slightly better than the Emin values cited in the literature25,26 for the best possible stateof-the-art monolithic columns. It should however be noted that (25) Tanaka, N.; Kobayashi, H.; Nakanishi, K.; Minakuchi, H.; Ishizuka, N. Anal. Chem. 2001, 73, 420-429. (26) Tanaka, N.; Kobayashi, H.; Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Hosoya, K.; Ikegami, T. J. Chromatogr., A 2002, 965, 35-49.
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the small Emin values of these monolithic columns are essentially due to their larger porosity ( ) 0.6-0.85) and the correspondingly smaller flow resistance (see, for example, Kozeny-Carman’s law or the recently established modified ∆P correlation for monolithic columns13 to assess the strong decrease of φ with increasing ). Considering that the presently obtained Emin ) 200-300 values relate to the ) 0.4 case, it should be obvious that by simply considering ordered 2-D arrays with a larger porosity (e.g., ) 0.6), even smaller Emin values will be obtained. In this case, the retention capacity and the mass loadability of the array column, however, become significantly smaller than in the traditional packed bed case.
and support from the IUAP program on Supra-Molecular Chemistry and Catalysis of the Belgian Federal Government. P.G. is supported through a specialization grant from the Instituut voor Wetenschap en Technologie (IWT) of the Flanders Region (Grant SB/11419).
CONCLUSIONS The present study quantifies the (large) gain in LC separation performance that can be expected if it were possible to devise perfectly ordered chromatographic beds. Separation impedances 10 times smaller than the best possible packed bed HPLC column are potentially achievable without having to increase the column porosity. Because the reduction of the flow resistance stemming from the better streamlined flow-paths in the 2-D array is only ∼30%, the strong reduction of the separation impedance E is clearly essentially due to the strong reduction of the plate heights, a factor of ∼2.5 with respect to the best possible HPLC columns. The calculated hmin values around h ) 0.65-0.85 are even significantly smaller than those cited for the best performing packed bed CEC systems. The fact that these extremely small theoretical plate height values result from the increased order of the flow-through pore network can readily be derived from the 11-fold decrease of the A term: A ) 0.043 for the ordered 2-D array, as opposed to the A ) 0.5 value found for the best possible packed bed columns. It was, in fact, interesting to note that Knox’s equation, originally established for 3-D packed bed systems, also closely represents the peak broadening in ordered 2-D arrays. The present results also indicate that in a traditional packed column, about one-half of the theoretical plate height value is due to the short-range and long-range interchannel coupling contributions. Since the presently considered 2-D column system has the same porosity and retention capacity as a traditional column and could easily allow for optical path lengths of 1 cm and more, it is also important to note that the prospective dramatic increase in kinetic performance can be achieved while maintaining the same UV-vis absorption detection sensitivity as a traditional packed bed column. It might also very well be that other ordered column structures, different from the currently considered cylindrical pillars, might yield even better performance characteristics. More simulations are, however, needed to explore this possibility. Remaining problems possibly compromising the feasibility of 2-D porous pillar columns are to be found in the realization of properly sealed high-pressure connections, the mechanical stability of the produced large-aspect ratio pillars, the achievable machining accuracy, and the overall manufacturing cost, as compared to the current bead synthesis and column packing procedures. ACKNOWLEDGMENT The authors greatly acknowledge a research grant (FWO KNO 81/00) of the Fund for Scientific Research-Flanders (Belgium)
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Analytical Chemistry, Vol. 75, No. 22, November 15, 2003
NOTATION Symbols a, b
distances between the cylinder centers in the considered 2-D array (see Figure 2) (m)
Ci
spatially averaged species concentration measured at the ith detection plane (kg/m3)
d
channel diameter (m)
df
stationary phase thickness in rectangular channel (m)
dp
pillar diameter (m)
Dm
mobile zone diffusion coefficient (m2/s)
Ds
effective stationary zone diffusion coefficient, (m2/s); Ds ) 5 10-10 × m2/s in the presented calculations
E
separation impedance (/); see eq 13
H
height equivalent of a theoretical plate (m)
h
reduced theoretical plate height (h ) H/dp) (/)
k1, k2
forward and backward, respectively, rate constant for adsorption reaction in eq 1 (s-1)
k′
phase retention factor (/)
k′′
zone retention factor (/)
k′′0
zone retention factor of unretained species, (/)
K
adsorption equilibrium constant (K ) k1/k2), (/)
Kv
column permeability (m2); see eq 12
Lij
distance between two different detection planes i and j (m)
tR,ij
difference in mean passage time between two different detection planes i and j (s)
um
mean velocity of moving fluid ) velocity of nonpermeating solute (m/s)
u0
mean velocity of permeating, but nonretained, solute (m/s)
x
position in column (µm)
Z1, Z2
height and width of the unit cell of the ordered 2-D array (see Figure 2) (m)
Greek Symbols ν
reduced fluid velocity, based on u0 () u0dp/Dm) [/]
∆P
pressure drop (Pa)
external column porosity, [/]
int
internal porosity of the porous pillars, [/]; int ) 0.5 in the presented calculations
η
dynamic viscosity (kg/(m.s))
φ
flow resistance factor, [/]
Received for review April 3, 2003. Accepted July 8, 2003. AC034345M
