Anal. Chem. 2009, 81, 943–952
High-Speed Shear-Driven Flows Through Microstructured 1D-Nanochannels Joris Vangelooven,† Wim De Malsche,†,‡ Frederik Detobel,† Han Gardeniers,‡ and Gert Desmet*,† Department of Chemical Engineering, Vrije Universiteit Brussel, B-1050 Brussels, Belgium, and Research Program Mesofluidics, MESA+ Institute for Nanotechnology, Enschede, The Netherlands A new flow type for the conduction of rapid chromatographic and macro-molecular separations in 1D nanochannels is reported. It combines the pressure-drop-less operation of shear-driven flows with the meandering flow paths that are present in ordered arrays of micro- and nanopillars. Similar to shear-driven flows in open channels, the achievable fluid velocity is quasi unlimited and is not affected by a pressure- or voltage-drop, while the axial dispersion in the microstructured pillar arrays remains surprisingly low. In the present paper, we report on a series of flow resistance and band broadening experiments that have been conducted to characterize the hydrodynamical properties of this new flow type. In addition, theoretical computational fluid dynamics (CFD) simulations have been performed to explain the observations. Good agreement between theory and experiment was obtained. The past decade has witnessed a gradual shift of the scale at which the separation possibilities of so-called laboratory-on-a-chip or microfluidic devices are being explored and exploited.1 In the early years of the laboratory-on-a-chip era, the main applications were open-tubular capillary electrophoresis (CE) and capillary electrochromatography (CEC), typically using channels with a diameter in the order of 5 to 50 µm.2,3 Since then, the microfluidic channel dimensions were gradually decreased (often in one dimension, sometimes in two). Pioneering work in this area has been done by, among others, Harrison et al.4 and Ewing et al.,5 all leading the way to the fabrication and the use of submicrometer deep channels. In recent years, channels with an even more fabulously small depth, in the order of only a few nanometers, have, for example, been studied by Cao et al.6 and Nichols et al.7 The introduction of such ultrathin channels enabled the discovery * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: (+).32.2.629.32.51. Fax: (+).32.2.629.32.48. † Vrije Universiteit Brussel. ‡ MESA+ Institute for Nanotechnology. (1) Eijkel, J.; van den Berg, A. Microfluid. Nanofluid. 2005, 1, 249–267. (2) Seiler, K.; Harrison, D. J.; Manz, A. Anal. Chem. 1993, 65, 1481. (3) Jacobson, S. C.; Hergenro ¨der, R.; Koutny, L. B.; Warmack, R. J.; Ramsey, J. M. Anal. Chem. 1994, 66, 1107. (4) Harrison, D. J.; Fluri, K.; Effenhauser, C. S.; Manz, A. Science 1993, 261, 895–897. (5) Gavin, P. F.; Ewing, A. G. Anal. Chem. 1997, 69, 3838. (6) Cao, H.; Yu, Z. N.; Wang, J.; Tegenfeldt, J. O.; Austin, R. H.; Chen, E.; Wu, W.; Chou, S. Y. Appl. Phys. Lett. 2002, 81 (1), 174–176. (7) Nichols, K. P.; Eijkel, J. C. T.; Gardeniers, H.J.G.E. Lab Chip 2008, 8, 173–175. 10.1021/ac801691e CCC: $40.75 2009 American Chemical Society Published on Web 12/23/2008
of a number of novel separation effects,8 but also a number of new side-effects were observed, such as the tedious filling and wetting of the channels.9 Another notable trend over the last years has been the use of microscale obstructions (or pillars), inserted in the channels as an alternative particle packing to extend the separation possibilities of microfluidic devices and/or to introduce additional mass loadability and capacity, such as is needed in chromatographic or adsorptive separation processes.10-12 With the advancement in microfabrication technology, pillars or obstacles with dimensions in the order of some 50 nm can now be produced.13 This very small scale has given rise to the exploration of very diverse and original separation mechanisms. One interesting approach of using nanostructured obstacles in a separation channel is to use them as a molecular sieve for DNA.14 Han and Craighead15 also exploited the advantages of nanoscale obstructions by designing an entropic trap device for the size-dependent separation of DNA-strands. Other reports proposed novel separation mechanisms based on lateral displacement sorting principles of particles and large macromolecules, either using the hydrodynamic size of molecules and their size dependent physical exclusion of certain flow paths in symmetrical channels16,17 or using the geometrical asymmetric Brownian ratchets to facilitate diffusion in the orthogonal direction.18,19 Until now, the flow through the micro- and nanochannels and obstruction arrays has nearly exclusively been generated by means of an electrical field. In channels with a size of several micrometers, electrically driven flows are independent of the channel dimensions. As a consequence, they allow to circumvent the pressure-drop limitation on the velocity that is encountered if a microfluidic device is operated in a pressure-driven mode. However, attempting to reduce the channel dimensions into the submicrometer range, one cannot escape from the overlap of the (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)
Kovarik, M. L.; Jacobson, S. C. Anal. Chem. 2008, 80, 657–664. Pennathur, S.; Santiago, J. Anal. Chem. 2005, 77, 6772–6781. He, B.; Regnier, F. J. Pharm. Biomed. Anal. 1998, 17, 925–932. Peterson, D. S. Lab Chip 2005, 5, 132–139. De Malsche, W.; Eghbali, H.; Clicq, D.; Vangelooven, J.; Gardeniers, H.; Desmet, G. Anal. Chem. 2007, 79 (15), 5915–5926. Kaji, N.; Tezuka, Y.; Takamura, Y.; Ueda, M.; Nishimoto, T.; Nakanishi, H.; Horiike, Y.; Baba, Y. Anal. Chem. 2004, 76 (1), 15–22. Fu, J. P.; Schoch, R. B.; Stevens, A. L.; Tannenbaum, S. R.; Han, J. Y. Nat. Nanotechnol. 2007, 2, 121–128. Han, J.; Craighead, H. G. Science 2000, 288, 1026–1029. Huang, L. R.; Cox, A.; Sturm, J. C. Science 2004, 304, 987–990. Blom, M. T.; Chmela, E.; Oosterbroek, R. E.; Tijssen, R.; van den Berg, A. Anal. Chem. 2003, 75 (24), 6761–6768. van Oudenaarden, A.; Boxer, S. G. Science 1999, 285, 1046–1048. Rousselet, J.; Salome´, L.; Ajdari, A.; Prost, J. Nature 1994, 370, 446–448.
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Figure 1. (a) WYKO-image of one of the studied arrays, consisting of cylindrical pillars (h ) 300 nm, d ) 7 µm, domain size ) 10.4 µm) in a pillar bed with an external porosity of 60%. Height is depicted in a different length scale to emphasize the extreme flatness of the pillars. (b) A SEM image of the same pillar bed.
double-layer regions occupying the channel walls.20 In this double layer, the velocity is significantly smaller than in the plug flow part that can be established far away from the walls if the channel is sufficiently wide. As a consequence, the mean velocity that can be achieved in a channel with double layer overlap is much smaller than in a wide channel. In 1999, our group21 proposed a radically novel flow driving principle to circumvent the limitations of pressure- and electrically driven flows through microfluidic channels. This shear-driven flow is driven by a (longer) moving glass substrate, which drags the liquid through a (shorter) etched, open, flat-rectangular channel in silicon.22 The advantages and disadvantages of shear-driven flow have already been discussed extensively elsewhere.23,24 The generated flow has a linear velocity profile and hence a mean velocity which in all cases is equal to one-half of the moving wall velocity, independent of the channel depth and the nature of the liquid. In previous studies, we always obtained the same mean velocity (one half of the moving wall velocity) for any combination of a hydrophobic/hydrophilic liquid and hydrophobic/hydrophilic walls.22 After all, the Navier-Stokes equation is also independent of the surface properties, provided the no-slip condition prevails, and our past experimental observations22-25 showed that this condition still prevails in channels with a hydrophobic C18 coating, where the contact angle differs significantly. In a recent study of the flow and chromatographic band broadening in shear driven flows, the solid channel spacers which were previously used to maintain a fixed distance between the moving and the stationary wall25 were replaced by microstructured spacers to minimize the contact area between the moving and the stationary wall plates. The employed injection principle cannot prevent the injected sample from also entering the spacer regions, and we surprisingly observed that the band broadening in the microstructured spacer region was of the same (very low) degree as in the unobstructed channel area. The present paper therefore reports on a hydrodynamic study of shear-driven flows in microstructured channels, describing the nature of this hitherto unreported flow type and investigating the origin of the observed low dispersion.
a silicium enriched-nitride layer on a silicon chip by low pressure chemical vapor disposition (LP-CVD). Patterning a resist layer followed by a Reactive Ion Etching (RIE) step leaves 250 nm high pillars of silicium enriched-nitride. After etching, the wafers are diced into 10 mm × 20 mm rectangular pieces (Disco DAD-321, blade width 50 µm) that fit exactly into the stationary holder system, which was already described extensively in previous publications.23-27 Figure 1 shows a SEM and a WYKO scan27 of one of the fabricated pillar arrays. In all produced arrays, the center of each pillar was positioned on the corner points of an equilateral triangular grid (Figure 1b). After their preparation, the etched channels and pillars were inspected using a WYKO depth profiling system (Veeco Instruments, U.K.). Experimental Setup. The experimental setup and the procedures used in this work have previously been described.23,25,28,29 One important change has been made though: instead of using full spacers at the sides of the open channels, the fixed distance between moving and stationary wall is now maintained using the etched pillars forming a “packed” pillar bed. These pillars hence form a 250 nm deep channel, filled with micropillars extending between the fused-silica glass wall, upon which they are resting, and the bottom of the etched silicon wafer upon which they are attached. To evaluate the bandbroadening in the nanochannels, the movement of 100 µm wide fluorescent tracer plugs of Rhodamine 6G (5 mM), dissolved in pure HPLC-grade methanol (Cas no. 6756-1, Sigma-Aldrich, Belgium), has been studied by recording realtime images of the flowing plugs with a CCD camera (Charged Coupled Device, Orca-ER, Hamamatsu Photonics, Japan). Injections were carried out manually as described by Clicq et al.23,25
EXPERIMENTAL AND NUMERICAL PROCEDURES Pillar Array Manufacturing. Channels with a depth in the range of a few hundred nanometers were fabricated by depositing
(27) (28)
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(20) (21) (22) (23) (24) (25) (26)
(29)
Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017–4024. Desmet, G.; Baron, G. V. J. Chromatogr. A 1999, 855, 57–70. Desmet, G.; Baron, G. V. Anal. Chem. 2000, 72, 2160–2165. Clicq, D.; Pappaert, K.; Vankrunkelsven, S.; Vervoort, N.; Baron, G. V.; Desmet, G. Anal. Chem. 2004, 76, 430A–438A. Fekete, V.; Clicq, D.; De Malsche, W.; Gardeniers, H.; Desmet, G. J. Chromatogr. A 2006, 1130, 151–157. Clicq, D.; Vervoort, N.; Vounckx, R.; Ottevaere, H.; Gooijer, C.; Ariese, F.; Baron, G. V.; Desmet, G. J. Chromatogr. A 2002, 979, 33–42. Clicq, D.; Vankrunkelsven, S.; Ranson, W.; et al. Anal. Chim. Acta 2004, 507, 79–86. Clicq, D.; Vervoort, N.; Baron, G. V.; et al. LC-GC Eur. 2004, 17, 278. Vankrunkelsven, S.; Clicq, D.; Pappaert, K.; et al. Electrophoresis 2004, 25, 1714–1722. Fekete, V.; Clicq, D.; De Malsche, W. J. Chromatogr. A 2007, 1149, 2–11.
Figure 2. CCD-camera images of a Rhodamine 6G tracer plug moving through different pillar beds at 40× magnification (uwall ) 2 mm/s). (a) Pillar bed of diamond shaped pillars (dlat ) 7 µm). (b) Pillar bed of cylindrical pillars (dp ) 7 µm). (c) Image of a plug on the interface region between an open channel and a pillar bed of cylindrical pillars, showing the retardation of the plug in the pillar bed. The full red line added to Figure 2c is a schematic representation of the depth-averaged velocity profile.
At the start of the experiment, both the moving glass substrate and the silicon channel were always first cleaned with a detergent and then rinsed with the used mobile phase. This way all surfaces are wetted completely, and the influence of the contact angle disappears. The CCD camera could differentiate 256 gray intensity values running between completely black and completely white. All intensity values shown in the present study were obtained by averaging the pixel values in each row within the pillar array. Data processing was done using the “simple PCI”-software (Hamamatsu Photonics) accompanying the CCD camera. In the described experiments, chips with two different layouts have been used. The first design consisted of relatively narrow pillar arrays (w ) 100 µm) separated by 100 µm wide zones devoid of pillars. The second design had wider pillar arrays (w ) 600 µm; h ) 250 nm) while the inter-array distance was still at 100 µm. Computer Simulations. In addition to the experimental work, Computational Fluid Dynamics (CFD) simulations have been performed to obtain a better insight into this new flow type and its characteristics. All simulated flow domains were generated using the Gambit v2.1.6 software. The same software was used to generate the discretization grids needed for the numerical solution of the species and impulse conservation equations. The flow domains with the discretized simulation grids were subsequently fed to a commercial computational fluid dynamics (CFD) solver (Fluent v6.2.16) to solve the conservation equations. During the simulations, the concentration of virtual injected sample species was monitored as a function of time on a series of successive detection planes evenly distributed along the axial length of the channel. Plate height values were subsequently obtained by calculating the zeroth-, first- and second-order moments of the recorded breakthrough curves. It was ensured that the obtained data were independent of the grid size, as well as from the employed time step size, by reducing both values until the influence upon the obtained result dropped below 0.5%. Obviously, different boundary conditions needed to be used to study pressure-, and shear-driven flows. In the pressure-driven flow simulations, the front and end of the channel were treated as a velocity inlet and a pressure outlet (both set at atmospheric pressure), respectively, and all side-walls as stationary walls. In the shear-driven flow simulations, the front and end of the channel
were treated as pressure inlet and pressure outlet (both atmospheric pressure), and the top wall was attributed a certain nonzero velocity to generate the flow. In all cases, a no-slip condition was imposed on all solid walls. Along the fluid boundaries of the defined channel, a symmetry boundary condition was imposed to make the flow domain act as if it were imbedded in an infinite wide medium. SAFETY CONSIDERATIONS Rhodamine 6G, like other rhodamines, is a possible carcinogen and should be handled with caution. Skin contact should be avoided as much as possible. RESULTS AND DISCUSSION In a first series of experiments, a 40× objective was used to monitor the flow of injected tracer plugs of the Rhodamin 6G solution (axial width of about 100 µm, Figure 2) to check whether the moving glass substrate and the substrate carrying the etched pillars were actually touching each other. The strong difference in light intensity between the light tracer plug and the dark pillars in the thus obtained images is an indirect proof of the fact that there is almost no fluorescent tracer flowing between the pillar tops and the glass plate. This implies that the pillars and the moving wall are in intimate contact and that the effective channel depth during the operation is very close to that of the nominal depth measured with the WYKO instrument. Whereas the picture in Figure 2a,b was taken in the central portion of the channel, the picture shown in Figure 2c was taken near the edge of the pillar array. Figures 2a and 2b show that, independently of the shape of the pillars (diamonds or cylinders), the tracer bands remain very sharply delimited during their passage through the pillar arrays. Figure 2c has been added to show that the velocity inside the pillar array is different from that in the open channel areas aside of it, despite the fact that both areas are subjected to the same moving wall velocity (the full red line added to Figure 2c is a schematic representation of the depthaveraged velocity profile). As can be noted, the difference in velocity warps the migrating tracer bands in the region close to the velocity interface. The warped region is broader than the region wherein the velocity gradient exists, in agreement with recent simulations of the side-wall effect in chromatographic Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
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Figure 3. (a, b) Movement of a Rhodamine 6G tracer plug (uwall ) 8 mm/s) through three different lanes (1, 2, and 3, width: 100 µm) of diamond shaped pillars in a 300 nm deep channel. The pillars are arranged in an equilateral triangular pattern with an external porosity of 80%. Ratio of the axial diagonal to radial diagonal is 1.73. Lateral diameters were (1) 5 µm, (2) 3 µm, and (3) 2 µm. (a) picture taken at t ) 0 s and (b) picture taken at 0.28 s.
columns,30 and is a consequence of the diffusional exchange of tracer species moving back and forth between the open and the packed channel regions. This diffusional exchange also causes the width of this warped band region to grow radially with the square root of the axial distance (cf. Einsteins’ law of diffusion saying that diffusional distance δ ) (2Dmolt)1/2). The reader should note that this effect can relatively easily be circumvented, either by arranging a thin, fully solid spacer line between the open and the pillar zone, or by using pillar array channels that are very wide. The main reason why the effect is discussed here is to explain why it is present in the images shown in Figure 3 discussed further on. During the conduct of the experiments, it was also found that the system is extremely sensitive to the alignment of the pillar arrays and the direction along which the moving wall substrate is moved. It was found that a misalignment of only 1° already leads to a dramatic increase of the band broadening. At larger degrees of misalignment, bands that were injected in one of the pillar arrays even easily drifted toward the open channel regions and vice versa, hence completely ruining the ordered flow field needed to obtain good separation performances. Another concern was the breakage of the pillars, caused by the imposed shear stresses. Fortunately, it turned out that only the first row of pillars, that is, the row of pillars closest to the dicing line that was used to define the inlet of the channels, was sensitive to shear stresses. Especially in cases where the dicing line leads to pillars that were cut at some intermediate position, it occurred regularly (typically once every hundred runs) that one of these first row pillars collapsed and had its debris dragged into the following rows of pillars, thereby drawing a line of damaged pillars running through the array in the axial direction. Other than that, we never observed any damage originating from any position other than the first row of pillars. Hence if one would design a wafer with 150 µm pillar-free gaps between the pillar beds of the different desired chips, it would be possible to dice the wafer (30) Broeckhoven, K.; Desmet, G. J. Chromatogr. A 2007, 1172, 25.
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exactly through this pillar-free zone, thus avoiding a first row of damaged, unstable pillars in the obtained chips. In the experiment represented in Figure 3, the arrays with the smallest width (w ) 100 µm) were used because they allow to simultaneously monitor the flow in several parallel pillar arrays if using a 4× zoom objective. Figure 3 shows the movement of a tracer plug (having a velocity of uliq ) 2.6 mm/s) through a series of adjacent pillar beds with a depth of 300 nm. The pillar beds (ε ) 0.8) were all composed of diamond shaped pillars arranged according to an equilateral triangular pattern, but the pillar dimensions varied (dlat varied between 2 and 5 µm, with dax ) 1.73dlat). Although severely influenced by the warped band effect already mentioned when discussing Figure 2, Figure 3 clearly shows how the fluid velocity differs from array to array according to the following simple rule: higher pillar densities lead to smaller velocities. The fact that the velocity in the open regions between the pillar arrays is the highest is obviously also in agreement with this rule. Figure 4a shows the evolution of the mean band position with the product of time and uwall for a range of different velocities in one of the pillar arrays. As can be noted, the effect of the velocity on the slope of the line is negligible. This implies that there is a fixed, linear relation between the mean liquid velocity and the moving wall velocity, which can hence be written as uliq ) Ruwall
(1)
where R is a shape factor that is independent of the mean fluid velocity. Laminar flows between parallel plates and in axisymmetric systems with a sufficiently low Re-number (creeping flows) are known to produce flow fields that have a shape which is independent of the mean velocity (cf. the parabolic flow profile in laminar pipe flow).32 Changing the viscosity of the liquids by mixing pure water into the methanol (allowing to vary the dynamic viscosity of the liquid over roughly a factor of 331) had no effect on the value of R. This observation is in agreement with the theory of hydrodynamics and the laminar nature of the flow and the noslip condition one can expect at the wall surfaces. Since the pressure drop is zero, and since the viscosity only enters the impulse balance in combination with this pressure drop,32 it is obvious to find that the liquid viscosity does not enter the velocity expression. In Desmet and Baron33 it has been rigorously shown (by solving the Navier-Stokes equations with the appropriate boundary conditions) that the mean velocity in a shear-driven flow is related to the moving wall velocity via a shape factor that is independent of uwall and the viscosity of the fluid (see righthand side of eq 17 of ref 33 with R ) 1 - 0.54d/w). This liquid viscosity independence constitutes a fundamental difference with the traditionally employed pressure- and electrically-driven flows, where the established velocity is always inversely proportional to the liquid viscosity. Similar velocity-independent relationships were observed in all other investigated pillar arrays. The only difference between the different arrays is the value of R, which was found to decrease with increasing pillar density, or, more precisely, with increasing outer pillar surface. (31) Colin, H. J.; Diez-Masa, K.; Czaykowska, I.; Miedziak, I.; Guiochon, G. J. Chromatogr. 1978, 167, 41. (32) Schlichting, H. Boundary-Layer Theory; Mc-Graw Hill: London, 1958. (33) Desmet, G.; Baron, G. V. J. Chromatogr. A 2002, 946, 51–58.
Figure 4. (a) Plot of the mean traveled distance of the moving bands versus uwall · t for the different wall velocities (uwall ) 2 mm/s ([), 4 mm/s (9), 6 mm/s (2), 8 mm/s (×), 10 mm/s (b), 12 mm/s (0). The straight line with slope 0.5 represents the open channel case. (b) Plot of the apparent flow resistance φ as a function of Sstat/Smov for various pillar dimensions and shapes. The straight line with slope ) 1 represents eq 4.
This dependency on the outer pillar surface is not surprising since the analytical expression for the velocity in an open-tubular shear-driven channel (cf. eq 13 of ref 33) can be used to show that the proportionality constant R that would be obtained in an open channel varies more or less proportionally with the ratio of the moving wall surface over the total wall surface: R=
Smov Stot
(2)
Although the equality in eq 2 is not exact (the shape of the flow field also has a small effect on the mean liquid velocity and this shape obviously also depends on the channel width to depth ratio), it allows to understand that the amount of moving surface plays a decisive role in the velocity that can be achieved with shear driven flows. In pillar arrays that would be very deep, and wherein only the cover wall would be moving, only a small fraction of the total surface would be moving. As a consequence, only a very small impulse source is available, implying that only a very low mean velocity can be expected in such channels, perhaps only one tenth of the moving wall velocity or even less. On the other hand, in very shallow open channels with a flat-rectangular cross section (w . d),with a moving top or bottom wall, a much larger moving surface is available, leading to an Smov/Stot ratio approaching 0.5. According to eq 2, this then leads to a value of R ) 1/2. This value was indeed consistently observed in all experimental
studies on open-tubular shear-driven chromatography performed in our group, all conducted in channels that were orders of magnitude wider than deep.23,34 As a line of reference, the R ) 1/2-case of eq 1, representing the case of an open-tubular channel with w . d, has been added. The proportionality constant between uliq and uwall can also be used to define an equivalent flow resistance (φ). Although the concept of a flow resistance has been developed to describe the pressure drop experienced in pressure-driven flow systems, and although the pressure-drop in shear-driven flow systems is strictly zero, it nevertheless remains possible to express how strongly the fluid is arrested by the stationary surfaces as compared to the mechanical force it is subjected to by the moving wall surface. Given this physical interpretation, it is proposed to define the flow resistance of a shear-driven flow system as
φ)
uwall -1 uliq
(3)
which according to the approximate relationship in eq 2, and considering that Stot ) Sstat + Smov, can also be rewritten as (34) Vankrunkelsven, S.; Clicq, D.; Cabooter, D.; et al. J. Chromatogr. A 2006, 1102 (1-2), 96–103.
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φ=
Sstat Smov
(4)
The flow resistance definition adopted in eq 3 is in agreement with one’s intuitive understanding of the notion of a flow resistance. Considering, for example, the (hypothetical) case of an open-tubular channel or a pillar array having a slip-wall velocity condition on all its stationary surfaces, the flow would not experience any braking effect coming from the stationary walls, and uliq would be equal to uwall. In this case, the flow resistance factor would be equal to zero, in agreement with the physical meaning of the concept of a flow resistance. On the other hand, considering a pillar array displaying the usual no-slip wall condition at its surfaces and being filled with very high pillars, hence having a total stagnant surface that is orders of magnitude larger than the surface of the moving flat wall, it can be inferred that uliq will be much lower than uwall and approach zero. As a consequence, this system would, according to the definition in eq 3, yield a flow resistance φ approaching infinity. This is again in agreement with one’s intuitive understanding of the concept of a flow resistance. For the case of an open tubular channel with a flat-rectangular cross section, a flow resistance of exactly 1 is obtained. For the pillar arrays considered in the present study, flow resistances ranging between 1.13 and 1.47 were obtained. According to eq 4, there should be a clear relation between φ and Smov/Stot. In Figure 4b, this relation can indeed be roughly discerned, but the agreement is far from perfect, that is, there is no one-to-one relation as predicted by eq 4. The latter is not unexpected, since it can be inferred that the influence of the velocity profile on the uliq/uwall ratio will be even more shape dependent in a pillar array than it already is in an open channel (cf. the discussion of eq 2). Given that a blunt object will inevitably yield a higher flow resistance than a sharp object, and since the information about the “sharpness” of the objects is not contained in the Smov/Stot ratio, it is in fact obvious to find that the latter does not perfectly predict the uliq/uwall ratio and φ. However, considering that it is very difficult to establish a perfectly accurate expression, accounting for all geometrical subtleties of the flow field, eqs 3 and 4 certainly are very convenient as good first estimates of the uliq/uwall ratio and φ. Figure 4b shows how the obtained φ data are plotted versus Sstat/Smov. The figure shows that the flow resistances of the differently considered pillar arrays follow a linearly increasing relation with the primary design parameter Sstat/Smov, similar to eq 4, but shifted upward with respect to it. The upward shift is not unexpected, since in the open channel all surfaces are oriented parallel to the flow field whereas the pillar arrays also have a significant amount of their stationary surface oriented normal to the mean flow direction. Since perpendicularly oriented surfaces have a much larger flow braking effect than parallel oriented surfaces, and since the difference between parallel and perpendicularly oriented surfaces is not contained in the Sstat/Stot ratio, it is obvious to find that the pillar array data lie above the trend predicted by eqs 2 and 3. The mutual order of the diamond pillar array data is consistent with what already was established in Figure 3. The more dense ε ) 0.6 arrays have a higher flow resistance than the less dense ε ) 0.8 arrays. The higher flow 948
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resistance of the cylindrical pillar array as compared to the diamond pillar array with the same external porosity and the same pillar size can be explained by the fact that the pillars in the former are blunt-shaped, whereas they are sharper in the latter. Apart from the achievable liquid velocity, another important feature of a well-performing separation device is its ability to transport sample species and particles with a minimum of axial dispersion. The latter is traditionally quantified by measuring the variance of an injected narrow band of tracer species as a function of the time. As already remarked, the band broadening can best be studied in wide arrays. Partly because this circumvents the problem of the band warping effect originating from the diffusional crosstalk between neighboring channels with a different velocity, and partly because they allow to average the pixel intensities of the migrating band over a much larger transversal distance, hence leading to enhanced signal-to-noise ratios. The excellent signalto-noise ratio that can be obtained by averaging the pixel intensity in the y-direction along each row of pixels is clearly illustrated by the intensity profile added in overlap to Figure 5, showing the migration of an injected band (initially injected width ) 100 µm) in the first 2 mm of one of the 600 µm wide arrays. As can be observed, the band broadening in the pillar array is extremely small. To quantify this, the spatial variance σ2 of the intensity profiles has been calculated using the zeroth, first, and second moment (M0, M1, and M2). The expression used for the nthorder moment40,41 Mn )
∫
∞
o
c(x)xn dx
(5)
The peak variance can then be calculated according to the following well-established expression:
σ2 )
( )
M1 M2 M0 M0
2
(6)
Figure 6a shows how the spatial variance values that are obtained for a given run with given mean velocity vary in a linearly proportional way with the time. This is in full agreement with the axial dispersion variant of Einstein’s law of diffusion: σ2 ) 2Daxt
(7)
The slope of the straight line relationships observed in Figure 6a depends upon the mean velocity, reflecting the fact that the degree of axial dispersion varies with the velocity. To investigate this dependency in more detail, we found it customary to transform the Dax values into plate height values (H), for the latter are traditionally employed to analyze the band broadening in chromatographic columns. As is well-established, the relation between both quantities is given by35
H)
2Dax uliq
(8)
(35) Gidding, C. Dynamics of Chromatography; Marcel Dekker Ltd: New York, 1965.
Figure 5. Movement of a band of a fluorescent tracer (Rhodamine 6G) through a pillar bed consisting of cylindrical pillars (dp ) 7 µm) at a velocity of uliq 2.6 mm/s. The plug was followed for 1 mm.
Subsequently plotting H versus uliq yields a so-called Van Deemter curve (see Figure 6b), originally established for the dispersion in packed sphere beds, but in its general form also applicable in any type of flow structure. Traditionally, the thus obtained curves are fitted using the so-called Van Deemter equation, which can be written as35
H)A+
B + Culiq uliq
(9)
In eq 9, the A-term part represents the contribution to the band broadening stemming from the tortuosity and the different lengths of the streamlines penetrating through the column, the B-term part represents the influence of the axial component of the molecular diffusion (at u ) 0 this is the single source of band broadening), while the C-term part reflects the band broadening originating from the fact that the molecules continuously diffuse between low and high velocity zones. Figure 6b shows the experimental H-values for the different considered velocities. Each velocity was repeated five times, and all different obtained values are represented to allow assessing the scatter on the data. The scatter became very large at mean liquid velocities of uliq ) 5 mm/s and was caused by the recording limitations of the employed CCD camera. From a given velocity on, the plug travels a non-negligible distance during one image frame. The minimally used exposure time for a frame is 0.01 s. A plug traveling at 5 mm/s traverses about 50 µm during this time, which makes the plug seem wider than it actually is. The reader should note that the pillar matrix acts as a stationary phase that promotes longitudinal diffusion across
the boundary layer lining the pillars. Making reproducible measurements at liquid velocities larger than 5 mm/s was impossible with the current setup. From the fact that the plate heights related to the largest fraction of the explored velocities decrease with increasing uliq, it can be concluded that the plate height values over the range up to 5 mm/s are essentially dominated by the B-term part of eq 9. The shape of the curve suggests that the minimum of the curve would be situated around 4.5 to 5 mm/s, although the scatter prevents any firm conclusion to be established. In any case, the fact that the plate heights do not increase with increasing velocity in the range up to 5 mm/s is highly uncommon for a chromatographic system, since conventional packed bed columns reach a minimal plate height between 0.5 and 3 mm/s36 and, at higher velocities, enter a regime wherein the plate heights increase with the velocity (hence reflecting the C-term part of eq 9). Having plate height values that increase with increasing velocity prevent the achievement of high separation speeds, for this behavior implies that if one tries to raise the speed of the separating bands, one also makes them broader, which counteracts the separation. Hence, having a system that has plate heights that decrease or are independent of the applied velocity up to a value of 5 mm/s is very beneficial from the perspective of achieving a very high separation speed. The possibility to achieve low plate heights at high velocities has already been abundantly demonstrated for shear-driven chromatographic separations con(36) Cabooter, D.; de Villiers, A.; Szucs, R.; Sandra, P.; Desmet, G. J. Chromatogr. A 2007, 1147, 183–191.
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Figure 6. (a) Typical peak variance versus traveled distance for a pillar array with dp )7 µm and an external porosity of 60% and for different values of uliq (working fluid ) methanol). (uwall ) 2 mm/s ([), 4 mm/s (9), 6 mm/s (2), 8 mm/s (b), 10 mm/s (4), 12 mm/s (0). (b) Experimental Van Deemter plot (plate height versus uliq), same geometry and working fluid as in (a) ([, full line). Simulated Van Deemter curve for pressure driven (b, dotted line) and shear driven (2, dashed line) flow through a cylindrical pillar array (same geometry as in (a), Dmol ) 10-9 m2/s, η ) 10-3 kg/(m s)). All data sets have been fitted by the Van Deemter model (eq a7).
ducted in open-tubular channels.29,37 The present study now reveals that these high separation speeds remain possible if the channels are filled with micropillars. The actual minimal plate height value that is reached (Hmin = 0.25 µm) is also much smaller than in conventional chromatographic systems, where plate heights are usually in the 5 to 10 µm-range. If we take the limited column length L (in the present setup, L ) 2 cm) into consideration, according to N)
L H
(10)
a total plate number N between 20000 and 30000 would be possible in retained conditions at the optimal liquid velocity. Similar plate height curves were obtained for all other investigated channels, but the difference between the different pillar sizes and shapes was too small to be significant over the scatter on the data. The small impact of the pillar shape on the observed plate height values has two reasons. The first reason is that most data points are most strongly affected by the B-term (37) Fekete, V.; Clicq, D.; De Malsche, W. J. Chromatogr. A 2008, 1189, 2–9; use of 120 nm deep channels for liquid chromatographic separations.
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part of eq 9, relating to the axial component of the effective molecular diffusion, which in turn is nearly completely independent of the pillar shape for porosities of ε ) 60% and larger.38 The second reason is that the contribution originating from the other two terms (A- and C-term), making up about one half of the total plate height value when the Van Deemter curve reaches its minimum, is essentially dominated by the depth of the channels and not by the shape or the size of the pillars. This is because the considered channels are very shallow, so that the majority of the liquid elements are always closer to the top or the bottom wall than to one of the pillar side walls. A series of CFD-simulations were conducted to better understand the nature of the observed shear-driven (SD) pillar array flows and the reasons underlying the unexpectedly small additional band broadening as compared to a pure open channel flow. Although pressure-driven (PD) flows are hardly feasible in channels having the dimensions used in the present study (they would cause a pressure drop of more than 500 bar per cm column length for the highest considered velocities), we nevertheless (38) Desmet, G.; Broeckhoven, K.; De Smet, J.; Baron, G. V.; Gzil, P. J. Chromatogr. A 2008, in press.
Figure 7. (a,b) Velocity field lines at half-channel height in, respectively, a pressure driven and a shear-driven flow through a cylindrical pillar array channel with dp ) 3 µm, external porosity 60%, and channel height ) 300 nm. Working fluid ) water (η ) 10-3 kg/(m s)). Added white ellipsoidal regions denote regions with zero velocity and added white arrows denote backflow regions. (c,d) Corresponding pressure fields (black arrows denote direction of flow).
included them in this theoretical part for most people active in the field are very well acquainted with the typical velocity profiles that are obtained under PD flow conditions. The PD flow variant is therefore ideally suited to serve as a reference case. In Figure 7 the velocity vector fields and the static pressure field (both averaged across the channel depth) of a SD flow are compared to the fields that would be obtained in the case of a PD flow through the same circular pillar bed. Whereas the PD flow case exhibits the typically expected patterns (pressures go from high to low and all velocity vectors run forward and follow a smoothly running tortuous path around the pillars), the velocity and pressure profiles in the SD case display a number of clearly different features. Perhaps the most striking difference is that for the pressure field, which no longer decreases gradually as it does in the PD case but oscillates above
and below the mean atmospheric pressure governing the inlet and outlet section of the channels. The absence of a gradual decrease of course reflects the fact that the SD flow does not lead to any pressure drop. The origin of the regions with over- and under-pressure in turn can be best understood from the corresponding velocity vector plot. Since the considered channels are very shallow, the majority of the fluid elements are determined by the velocity field of the nearest wall. Since the moving wall purely moves in the axial direction, the fluid elements approaching the pillars experience a strong forward-driving force nearly all the way up to the surface of the pillars. Since the fluid cannot penetrate the pillars, a local excess pressure is created. A similar effect occurs at the back side of the pillars. Here the moving wall exerts a forward moving force on the liquid, but since the impermeable Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
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pillar wall prevents a swift continuous supply of fluid elements, an under-pressure region is created. The existence of these excess- and under-pressure regions also induces a number of important differences with the velocity vector field of the PD case. A first consequence is that they induce small regions (cf. added ellipsoids) where the flow is directed opposite to the flow in the corresponding region in the PD-case. These backflow regions induce a prolonging of the tortuous path of the fluid elements moving through the channel. The tortuosity of the flow can be exactly calculated by taking the ratio of the average liquid velocity magnitude to its axial component. As can be expected, this tortuosity was found to be velocity independent because the shape of the flow field is velocity independent. For every tested velocity an 11% increase in tortuosity was observed in the shear driven cases. A second consequence of the existence of local excess- and under-pressure regions is that the altered velocity vectors combine into a pattern containing a number of small regions where the net velocity is nearly identically zero (cf. added circles). Nevertheless, the regions wherein these differences occur remain relatively small. To investigate the influence of these regions on the band broadening, we also conducted an extensive series of species dispersion simulations. The results were again quantified in terms of plate heights H and are represented in Figure 6b (see dashed and dotted lines). As can be noted, the difference between the PD- and the SD-case is small. At the highest considered velocity (uliq)20 mm/s, not shown), the difference is not larger than 0.15 µm. Near uliq ) 5 mm/s, the difference is only about 5%. The presence of the back-flow and zero-velocity regions is hence clearly far from devastating. This is mainly because of two reasons. The first one is that the regions remain relatively small, and the second one is that the diffusion across the depth of the channels occurs so extremely fast (because of the nanosized diffusion distances) that difference between the parabolic velocity profile of the PD-case and the linear flow profile of the SD-case are very effectively eliminated. Figure 6b shows a good quantitative agreement between simulations and experiments. Differences may be due to differences between the Dm values and/or measurement errors. CONCLUSION The present study discussed the hydrodynamic properties of a novel flow type, originating from the application of the sheardriven flow principle in a pillar array channel. New to this type of shear-driven flow is that it contains stable back-flow regions caused by a stable pattern of local pressure gradients (with local minima and maxima). Being based on the shear-driven flow principle, very high fluid velocities can be achieved without generating a pressure-drop. This implies that the achievable velocity is not affected by a decrease in channel depth. Unlike electrically driven flows, the (39) De Smet, J.; Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. Anal. Chem. 76 2004, 3716–3726. (40) Berdichevsky, A. L.; Neue, U. D. J. Chromatogr. 1990, 535, 189–198. (41) Aris, R. Proc. R. Soc. 1959, A252, 538–550.
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achievable velocity is completely independent of the nature of the fluid. The apparent flow resistance of the channels can to a first approximation be calculated as the ratio of stationary to moving surfaces. If applied in shallow channels, such as the currently employed 1-D nanochannel, the degree of axial dispersion or band broadening remains remarkably low. The presence of the pillars opens new possibilities over the previously studied shear-driven flows through open-tubular channels. For reversed-phase chromatographic separations, the presence of the pillars obviously increases the mass-loadability of the column. A higher mass-loadability implies that more concentrated sample plugs can be injected before overloading occurs. Furthermore, and more appealing, the presence of the pillars might open the road to a series of novel separation methods, using the pillars as a sieving matrix or as obstacles in an obstacle course13,16 to induce size-dependent separations of macromolecules. The possibility to combine very high velocities with very low channel depths (in the nanometer range) might open a new range of separation speeds in this field. ACKNOWLEDGMENT J. Vangelooven is supported through a specialization grant from the instituut voor Wetenschap en Technologie (IWT) from the Flanders region. F. Detobel is supported through a specialization grant from the Flemish Fund for Scientific Research. APPENDIX R) δ) ddom ) dp ) dlat ) Dax ) ε) φ) H) L) M0, M1, M2 ) N) Smov ) Stot ) σ2 ) t) uliq ) uwall )
SYMBOL LIST Proportionality constant between uliq and uwall in eq (a1) Diffusional distance in Einstein’s law (m) Domain size (m) Pillar diameter (m) Lateral diameter of a diamond shaped pillar (m) Axial dispersion coefficient (m2/s) External bed porosity Flow resistance Plate height value (m) Column length (m) Zeroth, first, and second moment (m2, m2 t, m2 t2) Number of theoretical plates Total moving surface in a shear driven flow channel (m2) Total sum of wall surface in a shear driven flow channel (m2) Peak variance (m2) Time (s) Liquid velocity (m/s) Moving wall velocity (m/s)
Received for review August 12, 2008. Accepted November 21, 2008. AC801691E
