Article pubs.acs.org/ac
In Situ Measurement of the Transversal Dispersion in Ordered and Disordered Two-Dimensional Pillar Beds for Liquid Chromatography Selm De Bruyne,†,‡ Wim De Malsche,†,‡ Sander Deridder,† Han Gardeniers,‡ and Gert Desmet*,† †
Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium MESA+ Research Institute, University of Twente, Enschede, The Netherlands
‡
S Supporting Information *
ABSTRACT: Using a fully transparent micropillar array chip and an optical “injection” setup capable of writing pulsed and continuous patterns into the flow by uncaging a fluorescent dye, highly detailed measurements of the transversal dispersion process in twodimensional (2D) chromatographic beds could be made. With the use of water-glycerol mobile phase spanning a wide range of viscosities, the obtained data cover a broad range of reduced velocities (0 < ν < 3000) and show a clear leveling-off of the transversal dispersion coefficient at large reduced velocities. With dependence on the packing density, this leveling-off occurs at a value of about Dtrans/Dmol = 10 (ε = 0.4), Dtrans/Dmol = 4 (ε = 0.6) and Dtrans/Dmol = 2.5 (ε = 0.8). Another interesting observation that could be made is that the effect of the bed order on the observed transversal dispersion process is relatively small. The observed leveling-off in the relation between the measured Dtrans values and the reduced liquid velocity furthermore clearly invalidates the classical Galton-board model, predicting a linear increase of Dtrans with the reduced velocity. On the other hand, it corroborates a recently proposed series-connection transport model for Dtrans in 2D porous media. Dmol (where u is the fluid velocity in meters per second and Dmol is the molecular diffusion coefficient in meters squared per second) and represents the ratio of the time needed to cover a certain distance by diffusion velocity to the time needed to cover the same distance by advection and in engineering literature is also known as the Peclet number (Pe). The form of eq 1 originates from the assumption that the diffusive and the advective contributions to the transversal transport occur purely in parallel and can hence simply be added. This is an assumption made in many classical studies15,18 and is certainly valid under turbulent flow conditions. In 2011, a series of detailed numerical flow simulations conducted in our group however showed the inaccuracy and physical inconsistency of this model for the laminar flows encountered in 2D flow systems for liquid chromatography.19 The inaccuracy of the model manifested itself by the fact that eq 1 predicts a linearly proportional increase with ν at sufficiently large ν, whereas the simulations showed a nearly complete flattening of the transversal diffusion coefficient (Dtrans) versus ν curve at high ν. The same flattening trend was observed by Daneyko et al. using Lattice−Boltzman simulations.20 Doubts on the validity of eq 1 were also already cast in the classical simulation study of Eisdath et al.21 The physical inconsistency of the Galton-board
S
ince the band broadening in chromatographic systems is determined by the occurrence of transversal velocity differences, and the extent to which these differences can be countered by the transversal diffusion and dispersion processes, it is evident that a detailed insight of these transversal dispersion processes is of primary importance in the understanding of the different band broadening processes in chromatographic systems.1,2 For instance, when applying the Taylor−Aris expression (originally developed for use in an open tube) to express the contribution of trans-column velocity biases in a packed bed medium, the molecular diffusion coefficient appearing in the denominator of this expression should be replaced by the more general radial dispersion coefficient, to express that in a packed medium the transversal transport occurs by a mix of diffusive and advective steps, whereas in an open-tube the transversal dispersion only occurs by molecular diffusion.3−5 Traditionally, the transversal dispersion in packed columns is represented by the so-called Galton-board model:6 Dtrans = Dmol (γeff + βν)
(1)
In eq 1, γeff represents the geometrical obstruction to diffusion of the porous medium and can be calculated from the effective medium theory.7−12 Further, β is another geometrical constant, for which different values have been proposed in literature (ranging from 0.1,13 over 1/1214,15 and 3/16,16 up to 7/1617), and ν is the reduced velocity. The latter is calculated as ν = udp/ © 2014 American Chemical Society
Received: October 1, 2013 Accepted: February 18, 2014 Published: February 18, 2014 2947
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dispersion but will at most only lower the value of the velocity-dependent term constant. Since eqs 1 and 3 essentially differ in the large ν range, special efforts were made to cover the broadest possible range of reduced velocities, using a broad range of micropillar diameters (up to 100 μm) and by increasing the viscosity of the mobile phase by mixing water with up to 90 vol % of glycerol.
model underlying eq 1 follows from the fact that this model assumes that a molecule moving along a streamline can move to another streamline after each particle. In turbulent flow conditions, this is indeed a good representation of the reality. In laminar flows, however, molecules can only move from one streamline to another by diffusion, which, at high ν, is a slow process. To account for this slow intermediate step, it makes sense to assume, as represented in Figure S-1 of the Supporting Information, that the transversal dispersion occurs according to a series connection process, wherein the normal convective transversal dispersion (occurring across a single through-pore with dispersion coefficient Dtrans1 = γeff,1Dmol + β1νDmol) is interrupted at regular intervals by a purely diffusion-based transport step (occurring across the interface between two adjacent boundaries with dispersion coefficient Dtrans2 = Dmol). With consideration that the overall resistance to mass transfer of a series connection of different resistance is equal to the sum of the individual resistances and that the resistance is inversely proportional to the dispersion coefficient, the above can be represented mathematically in the following way:19 ϕ 1−ϕ 1 = + Dtrans DT1 DT2
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EXPERIMENTAL SECTION Chemicals. The uncaging dye solutions were always made starting from small volumes (ca. 10 mL) with a concentration of 1 mM by dissolving CMNB-caged fluorescein (Invitrogen, CA) in ultrapure water (Millipore Synergy UV, Merck Millipore). These concentrated solutions were filtered (Chromafil Xtra, PTFE-20/25, 0.20 μm, Machery-Nagel, Germany) and used as a “stock solution”, which were subsequently mixed with different solvents like glycerol when a different viscosity was required. Spectro-photometric grade glycerol was purchased from Sigma Aldrich (Fluka, Belgium). Optical Set-up & Flow Control. An epi-fluorescence microscope (Olympus IX71), equipped with a mercury vapor lamp and a camera, was used for the visualization of the fluorescence signal. The employed filter setup consists of the typical combination of an excitation (D460/50x, AF Analysentechnik, Germany) and emission (HQ 525/50m, AF Analysentechnik) filter in conjunction with a dichroic mirror (500dxcru, AF Analysentechnik) and is placed in a filter cube inside the microscope. The visualization and detection of the fluorescence signal of the released fluorescein was done using a CCD camera (Hamamatsu digital CCD camera C4742-95). For the activation of the dye, the λ = 351.4 nm line of an argon laser (Coherent Innova 90-6 Argon Ion Laser) was used. In its standard use, this laser has 10 different wavelengths in the spectrum between 454.4 and 528.7 nm. By placing another prism, wavelengths from the UV spectrum (334, 351.1, 351.4, and 363.8 nm) can also be reached. The light exiting the laser was coupled into a multi mode optical fiber (QMJ-33-UVVIS-400/440-3-2, OZ Optics) using a laser coupler (HPUC-23-330-M-10BQ-1, OZ Optics) and was led to the collimation lens (HPUC-23−330-M-10BQ, OZ Optics) of a lightweight home-built optical setup on top of the microscope table. Inside this home-built setup, the collimated light was directed to a holder for a pinhole or a rectangular slit and the shape of this pinhole or rectangular slit was focused inside the channel using a lens (f = 75 mm, LA4725, Thorlabs, Applied Laser Technology, The Netherlands) and an objective. The final shape of the activation zone inside the channel could be altered by changing the shape of the pinhole or slit, and its size can be manipulated by using a different objective or changing the size of the pinhole or slit. The focused laser beam could be rapidly and accurately positioned in the plane of the microfluidic chip using an X−Y translation stage positioned underneath the lightweight home-built setup. The employed pinholes and slits were from the Thorlabs series and the OZ optics series. The objectives used for 5×, 10×, and 20× magnification were LMU-5x/0.13-266, LMU10x/0.25-266, and LMU-20x/0.4-266, respectively, from OFR (Applied Laser Technology, The Netherlands). For the 40× magnification, the Achrostigmat 40×/0.55 objective from Zeiss (Carl Zeiss NV-SA, Belgium) was used. When using a 5× objective for detection, the activation spot was made using a 20× objective (OFR LMU-20x/0.4) in combination with a 50
(2)
wherein ϕ and (1 − ϕ) are the fractions of the time during which the species are subjected to the transversal dispersion process within the individual through-pores and between two adjacent through-pores, respectively, and DT1 and DT2 are the dispersion coefficients occuring across a single through-pore and the interface with two adjacent boundaries, respectively. After recombination of the different terms and factors into the new geometrical constants γeff, β, and δ (see Quantitative Measurements of Dtrans for a more detailed discussion of the physical meaning of these constants), this expression can be rewritten into the following form: ⎛ γ + βν ⎞ Dtrans = Dmol ⎜ eff ⎟ ⎝ 1 + δν ⎠
(3)
Whereas eq 1 predicts a purely linear increase of Dtrans with the fluid velocity at high ν, eq 3 predicts that Dtrans will tend to a constant value in the ν → ∞ limit. Equation 3 certainly still also only represents a crude approximation of the true dispersion process; more complex models can certainly be built, but it has been found to provide a much better approximation to the simulated transversal dispersion data obtained in two-dimensional (2D) systems than eq 1.19 In the present contribution, we report on a series of dedicated experiments conducted to investigate the accuracy of eq 3. This was done using fully transparent nonporous micropillar array columns22−27 with different ordered and disordered pillar arrangements, with different degrees of packing density, and by measuring the radial dispersion in these columns using a high speed CCD camera in combination with the uncaging dye technique pioneered by Paul et al. and others for the visualization of flow profiles in microfluidic devices.28−32 Because of their rectangular shape, the flow in the pillar arrays only has an x,y-component and can therefore still be considered as a 2D system. The presence of the top and bottom wall only locally slows down the velocity but does not induce any additional velocity flow component. It can hence be inferred that the presence of the top and bottom walls will not affect the general velocity dependency of the transversal 2948
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Figure 1. (a) CCD camera image of the dispersion plume for ν = 20 in an ordered array with the pillar diameter (dp) = 60 μm, ε = 0.6 (5× magnification) and (b) corresponding transversal read-out (pink symbols) and best-fit Gaussian curve (full pink line) recorded at monitor line indicated on (a); white dashed line. (c1 and c2) CCD camera images of the dispersion plume for ν = 1800 at the (c1) entrance and (c2) exit of an ordered array with dp = 60 μm, ε = 0.6 (10× magnification, distance between both monitor lines = 3000 μm) and (d) corresponding transversal readouts (pink and blue symbols) and best-fit Gaussian curves (full pink and blue lines) recorded at monitor lines indicated on (c1 and c2); white dashed lines.
μm pinhole. When higher magnification objectives were used for detection (10-20-40X), smaller pinholes were used. The flow of fluid through the channels was controlled using low hysteresis pressure regulators (LRP-1/4-25 up to 2.5 bar and LRP-1/4-10 up to 10 bar, Festo NV, Belgium), and a home-built high-pressure vessel was used to transfer the pressure of the dry nitrogen to the liquid. The gas line contained two valves, allowing for the system to be depressurized without changing the value of the pressure regulator. No valves are used in the liquid line since the presence of extra connections increases the probability of air bubble formation and creating leakages. Microfluidic Chip Fabrication. Five-hundred micrometer thick fused silica wafers containing a number of different nonporous micropillar array channels (10 μm deep and 3 mm wide) with different pillar dimensions, pillar positioning patterns, and void fractions (ε) were produced in the cleanroom of the MESA+ Institute for Nanotechnology (Enschede, The Netherlands) using the procedures already described extensively in previous publications.22,32,33 Through-holes for the in- and outlet connections were powder blasted with an inhouse built setup. The processed wafers were subsequently sealed with a bare 500 μm thick fused silica wafer by thermal fusion bonding. The disordered pillar arrays were created by starting from the ordered configuration (equilateral triangular arrangement) and randomly changing the diameter as well as moving the center points of the individual pillars without changing the porosity.
The random diameter and particle displacement distances for each individual particle were generated using the evenly distributed random number generator of MS Excel. The maximal variation of the particle size and displacement was taken as 10% of the mean particle size. After taking into account the modified particle size, the maximal shift of the particle centers was equal to 90% of the pore size. Since each particle displacement occurs independently of its neighboring particles, this results in a heterogeneous packing with shortscale randomness.2 To quantify the degree of randomness, Figure S-2 of the Supporting Information shows the pair correlation function for the ordered (red) and the disordered (black) ε = 0.4 beds used in the present study. Measurement Procedures. Fluid velocities were measured by first locally uncaging the CMNB-caged fluorescein dye along a narrow transversal line by rapidly moving the focused laser beam, using the X-Y translation stage back and forth, and subsequently recording the displacement of this line along the axial axis. The position of the line center was determined by fitting a Gaussian curve over the line profiles observed in each of the subsequently captured video frames. Plotting this position versus the time, linear plots with an r2 ≥ 0.9999 were obtained. The measurement of the molecular diffusion coefficient (Dmol) was also conducted by locally creating a transversal line of uncaging dye, but now in the absence of a flow and in a open channel (i.e., a channel containing no pillars). Subsequently recording the slow axial spread of the tracer line, an accurate 2949
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Figure 2. (a) Series of CCD camera images visualizing the temporal radial dispersion process at a high reduced velocity in an (a) ordered and a (b) disordered bed. Parameters for ordered bed: ν = 2500, average dp = 73 μm, ε = 0.4 (40× magnification). Parameters for disordered bed: ν = 2500, dp = 81 μm, ε = 0.4 (20× magnification). The bright dot inside the pillar in the lower left corner is the rest position of the activation light spot (which is rapidly switched from a position in the channel to its rest position). White arrows: see text.
Safety Considerations. CMNB-caged fluorescein should be handled with care and contact with skin or eyes, as well as digestion or inhalation, should be avoided. Laser glasses (OD > 6 at 350 nm) should be worn during experiments on the optical setup to avoid eye damage from the UV laser. The chip manufacturing in the clean room should be carried out by trained individuals familiar with the operation and safety guidelines of the processes and equipment due to the many highly toxic and corrosive gases and solutions involved in the processes.
measurement of Dmol could be made from the straight-line slope of the plot of the spatial variance of the tracer line with the time, following the principles of the so-called peak parking method.34,35 Doing so, the diffusion coefficient of uncaged fluorescein in water was measured as Dmol = (3.90 ± 0.05) × 10−10 m2/s at 20 °C). This value is in good agreement with the value, Dmol = (4.15 ± 0.05) × 10−10 m2/s at 25 °C) found in literature.35 The actual measurements of the transversal dispersion coefficient Dtrans were carried out by continuously uncaging the fluorescent dye at a single point and by subsequently measuring the spatial variance of the ensuing dispersion plume (cf. Figure 1a) at two subsequent monitor lines, using:36 Dtrans =
2 Δσtrans 2Δt
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RESULTS AND DISCUSSION Qualitative Measurements. Using very broad pillars (order 50−100 μm) and very high viscosity water/glycerol mixtures as the mobile phase, in combination with the highly localized optical injection method, very detailed studies of the transversal dispersion process in (planar) packed bed systems could be made. Both pulsed (Pulsed Point Injection Experiments) and continuous (Continuous Point Injection Experiments) point injections were applied to study the transient as well as the steady-state behavior of the dispersion process. To prevent the formation of ill-resolved pillars during the lithography and etching steps, the pillars in the disordered channels were consistently taken larger than in the ordered array to prevent the occurrence of through-pores that would be locally too narrow after etching. As a consequence, most of the images recorded in the disordered arrays are recorded at a 2× smaller magnification than in the ordered arrays, after which they have been blown-up to cover the same number of pillars. The images presented for the disordered arrays are therefore slightly less sharp than those for the ordered arrays. 3.1.1. Pulsed Point Injection Experiments. Figure 2 shows the time-lapsed recording of the response to a pulsed point injection in an ordered (Figure 2a) and a disordered (Figure 2b) pillar array at large reduced velocity (ν = 2500). Because the contribution of the molecular diffusion is very low in this case, the transversal dispersion events are slowed down to such an extent that they can be followed in a highly detailed manner using a high-speed CCD camera. At this high reduced velocity, the flow of the injected tracer dye species clearly remains restricted to the flow-through pores they were initially injected
(4)
In this expression, the time interval Δt is the time needed for the analytes to flow from one monitor line to the other and Δσ2trans is the observed difference in transversal variance. The variance values were directly taken from the variance of the best-fit Gaussian, of which examples are shown in Figure 1, panels b−d. At low to moderate reduced velocities (ν ≤ 50), the dispersion plume spread over such a wide lateral distance that the width at the origin of the plume could be neglected compared to the width of the plume at a monitor line positioned 1.5 mm downstream of the plume source (see Figure 1, panels a and b). The latter position was selected because it fell in the same field of view as the injection source at the adopted 5× magnification. At high reduced velocities (Figures 1, panels c and d), the transversal dispersion is so weak that a significant difference in lateral plume width could only be achieved over a distance of several millimeters (cf. the fact that the monitor lines shown Figure 1, panels c1 and c2, are 3000 μm apart). The recorded transversal concentration profiles shown in Figure 1 (panels b and d) and recorded at the dashed white monitor lines indicated on Figure 1 (panels a and c) show the good quality of the recorded signal, as well as the fact that the obtained profiles can be very well-approximated using a Gaussian distribution curve. 2950
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Figure 3. CCD camera image of the dispersion plume at a relatively low reduced velocity obtained by making a continuous optical injection in an (a) ordered and a (b) disordered bed. Parameters for ordered bed: ν = 17, dp = 73 μm, ε = 0.4 (5× magnification). Parameters for diordered bed: ν = 15, dp = 41 μm, ε = 0.4 (5× magnification).
Figure 4. Stitched CCD camera images of the dispersion plume obtained at a very high reduced velocity in (a) an ordered bed (dp = 73 μm, ε = 0.4, ν = 2500, 40 × magnification), and (b) a disordered bed (dp = 81 μm, ε = 0.4, ν = 1400, 20× magnification).
in. Because of the high ratio of advective to diffusive transport, the residence time at the interface with the adjacent flowthrough pore is too short to allow an appreciable amount of the species to diffuse across the interface with the adjacent flowthrough pore. Very importantly, this transversal dispersion behavior does not significantly change when shifting from the ordered array (Figure 2a) to the disordered pillar array (Figure 2b), as in this case the injected species also remain inside the flow-through pore they were injected in and do not move to an adjacent pore. They can only do this by molecular diffusion, but this process is so slow that this can only occur to an appreciable extent over a distance of several hundreds of particles at this very high reduced velocity. The ordered and disordered case also clearly display a very similar pattern of dead-zone regions where the fluid velocity is very small (cf. the white arrows added in the 4 most rightward panels of Figure 2). These zones are filled and emptied later than the other regions because of
the slow diffusive equilibration between the regions of high and low velocity. The only difference between the ordered and disordered cases is that the interfaces between the adjacent though-pores run perfectly parallel (horizontal in the orientation of Figure 2a), whereas they are running skew in the disordered pillar case (cf. Figure 2b). Continuous Point Injection Experiments. As described in many classical studies,36 the most accurate and simple measurement of the transversal dispersion coefficient can be obtained by measuring the transversal variance of the steadystate dispersion plume formed in the long time limit response to a continuous point injection. Figure 3 shows an example of such plumes in ordered and a disordered pillar arrays recorded at a relatively small reduced velocity ν (ν ≅ 15). At this relatively small reduced velocity, the tracer obviously rapidly spreads over a more or less triangular region with an opening angle in the order of 45°. This is due to the fact that, 2951
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because of the low reduced velocity, there is ample time for a significant fraction of the species to diffuse into the neighboring flow-through pore after each particle. Comparing Figure 3 (panels a and b), subsequently, shows that the dispersion plume is very similar to that observed in the perfectly ordered array, again pointing at the relatively small effect of disorder on the transversal dispersion. At very high reduced velocities ν, the local diffusion rate across the interface between two adjacent streamline bundles is so slow that the injected species remain on their initial streamline within the flow-through pore for a very long time (Figure 4). As a consequence, the dispersion plume remains very narrow and has an opening angle close to zero. Interestingly, and in agreement with the observations already made in Figure 2, this behavior is again very similar in the disordered bed, where the tracer species also clearly remains confined to the very narrow bundle of streamlines they were initially released in (Figure 4b). Figure 4, panels a and b, in fact clearly show that the species rapidly disperse over the throughpore they were initially released in, whereas moving over to an adjacent through-pore requires an additional diffusion step which, at high ν, becomes rate limiting. Also noting that the width at the right-hand side of the images shown in Figure 4, panels a and b, is very similar in the ordered and disordered array, the only difference between both cases is that the width of the region where the streamlines are passing through is much more fluctuating in the disordered case compared to the ordered bed case, as a consequence of the (disordered) disturbance caused by the pillars in the former case. Despite this disturbance, the average width of the dispersion plume is essentially determined by the width of the uncaging laser spot (because of the quasi-negligible molecular diffusion). Figure 4 (panels a and b) hence clearly refute the classical image of streamlines splitting after each particle, which is underlying the Galton-board model, leading to eq 1. This is also in agreement with the theory of hydrodynamics, according to which the streamlines in a laminar flow can never intersect,16,37 irrespective of whether the fluid flows through an ordered or a disordered system. The transversal transport in both cases hence eventually always becomes rate limited by the exchange process between adjacent streamlines, which, in a laminar flow, is always purely diffusionbased. Quantitave Measurement of Dtrans. Figure 5 shows the evolution of the all measured Dtrans values calculated according to the procedure described by Figure 1 for a wide range of different reduced velocities ν, for different values of the external porosity, and for ordered as well as a disordered case. As can be noted, the data curves all level off at high ν and can be fitted very well with the series transport model proposed by Deridder19 and represented by eq 3. The fact that each of these data sets nicely conform to a single curve (one per data set) is not obvious at all, considering that the data points for each different case have been obtained from two or three different channels (containing pillars with different diameter) and using two or three different mobile phase compositions to cover the different ν ranges. The average fitting error per data point is 5%, 15%, 12%, 15% for the ε = 0,4-disordered, ε = 0,4ordered, ε = 0,6- and ε = 0,8-cases, respectively, whereas these fitting errors amount up to 52%, 71%, 46%, and 30%, respectively, when using eq 1 instead of eq 3. Also interesting to note is the clear effect of the packing density: the smaller ε, the higher Dtrans. This observation is in
Figure 5. Graphical representation of the acquired data and the model (full lines). (blue +) ε = 0.4 and disordered, (red ◇) ε = 0.4 and ordered, (green □) ε = 0.6 and ordered, and (brown ×) ε = 0.8 and ordered. See Table 1 for the parameters of each fitting.
full agreement with the larger tortuosity and the concomitant higher transversal velocity components that are induced in the low ε cases (the closer the pillars are spaced, the more the liquid is forced into a transversal motion). At high ν, the observed Dtrans values tend to decrease from a value of about Dtrans/Dmol = 10 (ε = 0.4) over Dtrans/Dmol = 4 (ε = 0.6) to Dtrans/Dmol = 2.5 (ε = 0.8). The Dtrans values of 10 to 12 at high n are similar to those predicted via our numerical simulations in 2D arrays, showing that the presence of the top and bottom wall only has a minimal impact on the observed transversal dispersion. Interestingly, the difference in Dtrans between the ordered and disordered bed case is relatively small, as both cases level off to a value that only differs by some 10 to 20%. This (perhaps surprisingly) small difference between the ordered and the disordered case can be considered as the quantitative confirmation of the qualitative observations made in Figures 2−4. Despite the relatively low number of data points in the low ν range (caused by the poor flow rate stability of the employed setup), the fitted γeff coefficients (see Table 1), which Table 1. Table of the Fitted Values for the Parameters Appearing in eq 3 Based on the Data Sets Shown in Figure 5 ε
γeff
β
δ
disorder 0.4 order 0.4 order 0.6 order 0.8
0.75 0.74 0.77 0.86
0.38 0.37 0.33 0.26
0.037 0.037 0.078 0.105
correspond to the obstruction factors for the pure diffusive transport (ν = 0), agree very well with the values predicted by the effective medium theory.7 As was shown by Deridder et al.,10,11 the theoretical γeff values that can be expected from this theory in an ordered array with ε = 0.4, 0.6, and 0.8 are given by γeff = 0.62, γeff = 0.71, and γeff = 0.83, respectively. As a sidenote, it can also be remarked that these γeff values are also directly proportional to the B-term coefficient appearing in the reduced van Deemter equation (with B = 2·γeff·Dmol). With remembrance of the main goal of the present study (i.e., differentiating between the parallel connection (eq 1) and the series connection model (eq 3) for the transversal dispersion process), it is especially the high ν end of Figure 5 2952
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needed to discriminate them from the presently considered simple series-connection model. Another major question is whether the same qualitative conclusions as made in the present study for 2D flows can also be made for 3D flows is difficult to answer in a direct way. Whereas the present findings are clearly due to the fact that streamlines in a 2D laminar flow cannot intersect, this no longer holds in the 3D case where the streamlines can twist around each other16 (although this seldom occurs). Hence, whereas in the 2D, the region of streamlines marked by a tracer at infinitely large ν (i.e., by assuming Dmol = 0) should remain constant (neglecting variations caused by the presence of the particles), this should not necessarily be the case in 3D, where streamlines starting on the same spot can diverge to cover a broader radial zone (although the area of the cross-section marked by the species should also remain constant). Such behavior would indeed give rise to a linear increase of Dtrans with ν (as predicted by eq 1). However, given that the space left by the diverging streamlines should be filled with streamlines coming from another starting point, this also implies that there should be streamlines that are contracting (coming from the outside to the inside of a given bundle). These streamlines contribute negatively to Dtrans, such that one can expect that, on the average, the net increase in radial position of a bundle of tracer streamlines at infinitely large ν would be zero, hence giving rise to the same type of leveling-off, as observed in the present study.
that is of interest. Despite the relatively large scatter, essentially caused by the very small differences in transversal variances from which the Dtrans value needs to be determined, it is clear that Dtrans does not increase linearly with ν at high ν as predicted by eq 1, but instead clearly levels off to a constant or near-constant value, as predicted by eq 3. The β values in Table 1 are of the same order as those predicted by Saffman16 and Prausnitz17 (cf. Introduction). The values of the δ parameter listed in Table 1 are such that they allow for representation of the leveling-off trend, starting around ν = 100. The observed leveling off at high ν can be explained by the fact that the slow diffusion step becomes rate limiting again at these high ν values, as the time available for the diffusive mass transfer that needs to be made between adjacent streamlines decreases inversely proportional with ν. Despite the fact that the diffusion process again becomes the rate-limiting factor in this high ν limit, the Dtrans values are substantially larger than in the low ν limit. This is due to the fact that the distances that need to be covered by diffusion are much smaller in the high ν limit than in the low ν limit, as part of the transversal distance is (very rapidly) covered by advection. With reference to the newly added Figure S-1 in the Supporting Information, there is, at high ν, only a small part of the through-pore where the transport still has to occur via diffusion. This is the region near the interface with the adjacent through-pore, where the streamlines run parallel to each other. As a consequence, the net transversal velocity across the interface is equal to zero, and the prevailing dispersion coefficient in this region equals DT2 = Dmol. In the central part of the tube, the streamlines have a net positive and negative displacement, such that the dominant mode of transport at high ν in this part of the through-pore is advection. The time needed to cross this part of the throughpore becomes hence negligibly small at high ν, whereas the time needed to cross the interface zone remains finite (and independent of ν).
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ASSOCIATED CONTENT
S Supporting Information *
Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
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*E-mail:
[email protected]. Tel: (+)32-(0)2-629-32-51. Fax: (+)32-(0)2-629-32-48.
CONCLUSIONS Using a dedicated optical setup to locally and temporarily uncage a laser dye and a series of different glass microchannels containing pillar beds with different diameter and packing density, and using different water/glycerol mixtures, detailed measurements of the transversal dispersion process in the 2D chromatographic systems could be made over a very broad range of reduced velocities. The measurements show that the species rapidly disperse over the through-pore they were initially released in, while crossing over to an adjacent set of streamlines requires an additional diffusion step which, at high ν, becomes rate limiting. As a consequence, the relation between Dtrans and ν tends to level off to a constant value at high ν. With dependence on the packing density, this leveling off occurs at a value of about Dtrans/Dmol = 10 (ε = 0.4), Dtrans/ Dmol = 4 (ε = 0.6), and Dtrans/Dmol = 2.5 (ε = 0.8). Interestingly, the effect of the bed order on the observed transversal dispersion process is relatively small, as the investigated 2D ordered and disordered beds display very similar qualitative and quantitative behavior. All these observations can be represented very well using the series-connection model proposed by Deridder.19 As also shown by Deridder, even more sophisticated Dtrans models can be conceived, based on similar but more complex considerations. Their use however falls outside the scope of the present study, as more accurate measurements would be
Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS G.D. greatly acknowledges the stimulating discussions with Konstantin Smirnov (Lomonosov Moscow State University, Moscow, Russia).
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