Anal. Chem. 2006, 78, 6191-6201
Domain Size-Induced Heterogeneity as Performance Limitation of Small-Domain Monolithic Columns and Other LC Support Types Jeroen Billen,* Piotr Gzil, and Gert Desmet
Department of Chemical Engineering, Vrije Universiteit Brussel, Brussels, Belgium
We have computed the band broadening and the flow resistance in a series of apparently self-similar porous LC support structures, all having the same mean geometric ratios and external porosity, but with a decreasing scale and disturbed by a scale-independent variance on the size and position of the porous solid zone elements. The study shows in a general and qualitative way that each type of LC support that is produced using a manufacturing process displaying a fixed (i.e., domain-size independent) variance on the size and position of the produced solid zone elements will eventually encounter a limit beyond which a further reduction of the domain size can no longer be expected to yield a significant gain in separation speed. This is currently observed in practice for silica monoliths and could also compromise the performance of photolithographically etched columns. In the quest for ever faster and better resolved LC separations, two different approaches to overcome the pressure drop limitation of packed column HPLC are currently heavily pursued. One approach tackles the problem by increasing the volumetric through-pore fraction of the employed supports to reduce their flow resistance (the monolith approach1-5), while the other approach tries to solve the problem by increasing the applied inlet pressure (the UPLC approach6-8). While the UPLC approach certainly struggles with some specific problems such as pressuredependent retention times, viscous friction heating, and column expansion,9 the demonstrated gains in separation speed are clear and significant.10 * Corresponding author. E-mail:
[email protected]. Tel: +3226293617. Fax: +3226293842. (1) Tanaka, N.; Kobayashi, H.; Nakanishi, K.; Minakuchi, H.; Ishizuka, N. Anal. Chem. 1998, 421A-429A. (2) Svec, F. LC‚GC Eur. 2003, 6a, 24-28. (3) Oberarcher, H.; Premstaller, A.; Huber, C. G. J. Chromatogr., A 2004, 1030, 201-208. (4) Ikegami, T.; Dicks, E.; Kobayashi, H.; Morisaka, H.; Tokuda, D.; Cabrera, K.; Hosoya, K.; Tanaka, N. J. Sep. Sci. 2004, 27, 1292-1302. (5) Leinweber, F. C.; Tallarek, U. J. Chromatogr., A 2003, 1006, 207-228. (6) Jerkovich, A. D.; Mellors, J. S.; Jorgenson, J. W. LC‚GC Eur. 2003, 6a 2023 (7) Jerkovich, A. D.; Mellors, J. S.; Jorgenson, J. W.; Thompson, J. W. Anal. Chem. 2005, 77, 6292-6299 (8) Patel, K. D.; Jerkovich, A. D.; Link, J. C.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5777-5786. (9) Martin, M.; Guiochon, G. J. Chromatogr., A 2005, 1090, 16-38. (10) Leandro, C. C.; Hancock, P.; Fussell, R. J.; Keeley, B. J. J. Chromatogr., A 2006, 1103, 94-101. 10.1021/ac060470x CCC: $33.50 Published on Web 07/27/2006
© 2006 American Chemical Society
Similar to the fact that the benefits of UPLC only become apparent if using particles that are significantly smaller than the standard 5- or 3-µm particles,11 it should also be expected that the advantage of the smaller flow resistance of monolithic support columns can only be fully exploited if their domain size would be reduced to or below that of the reference packed bed of spheres. The need for high external porosity columns with a small domain size can also be understood by noting that the currently best performing silica monoliths for which retained component plate height data are available have domain sizes in the order of 3 µm and yield plate heights around 7-8 µm.12 These values are slightly higher than can be obtained for a packed bed of sub 2-µm porous particles.13 To enhance this performance, monoliths with even smaller domain sizes (i.e., of the order of 2 µm and below) need to be manufactured. Such monoliths would have an important economic surplus value, for they would be able to surpass the performance of the packed bed columns over the whole range of plate numbers, without running into the high-pressure problems of UPLC. Despite this clear incentive, well-performing silica monoliths with a domain size in the order of 2 µm or smaller have to the best of our knowledge not yet been reported in the literature. Moreover, making a kinetic plot of all the best performing published silica monolith data reveals the existence of an apparently “forbidden region”.14 This is a region of superior kinetic performances that can never be invaded with packed bed columns, but would be accessible to more open porous structures such as monolithic columns, provided these could be fabricated with the same structural homogeneity but with a smaller domain size than currently possible.15 Noting that the currently best performing silica monoliths yield analysis time (t0) versus N curves that lie in a very narrow band aside of this forbidden region,14 it appears that all attempts that were undertaken to enter this forbidden region by decreasing the domain and skeleton size have been unsuccessful. (11) Desmet, G.; Gzil, P.; Nguyen, D.; Guillarme, D.; Rudaz, S.; Veuthey, J.-L.; Vervoort, N.; Torok, G.; Cabooter, D.; Clicq, D. Anal. Chem. In press (12) Motokawa, M.; Kobayashi, H.; Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Jinnai, H.; Hosoya, K.; Ikegami, T.; Tanaka, N. J. Chromatogr., A 2002, 961, 53-63. (13) Novakova, L.; Solich, P. J. Chromatogr., A 2005, 1088, 24-31. (14) Desmet, G.; Clicq, D.; Gzil, P. Anal. Chem. 2005, 77, 4058-4070. (15) Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. Anal. Chem. 2003, 75, 62446250.
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006 6191
In the present study, we tried to explain this observation making the simplified assumption that the processes occurring during the synthesis of monolithic columns lead to an inherent variance on the size and the position of the formed solid zones that does not scale down in proportion to the domain size. Instead, this manufacturing variance is assumed to remain at a given, nonzero level, independent of the domain size. Since there is currently no real experimental evidence for the existence of such an irreducible manufacturing variance, the present work should mainly be considered as a prospective study and an attempt to visualize the difficulties one could run into if trying to reduce the domain size of monoliths or other LC supports to the extreme. Looking into the future of LC, the assumption of a domain sizeindependent variance on the size and position of the produced solid zones, nevertheless, certainly corresponds to the reality of the photolithographic etching fabrication processes that could be used to produce perfectly ordered pillar array columns as originally proposed by Regnier.15-17 Considering the use of photolithographic manufacturing processes, it should be kept in mind that the variance on the produced feature sizes is determined by the wavelength of the light used to pattern the applied photoresist layer. Even with the most advanced and expensive state-of-the-art deep UV etching process, this is still of the order of 200 nm, independent of the feature size one would try to produce. To make a general investigation of the influence of a domain size-independent solid zone size and position variance that would be inherent to a given existing LC support manufacturing process, two series of hypothetical 2D porous support structures displaying selective retention properties were generated and studied using computational fluid dynamics simulations.15,18,19 In the first series, six perfectly ordered flow domains, with decreasing domain size (12-1.5 µm), but with the same isotropic pattern of circular, porous solid phase zones, were considered. Positioning the circular zones on the points of a perfectly equilateral triangular grid and keeping the same relative dimensions, the six domains could be perfectly overlapped by simply rescaling them and were hence perfectly self-similar.20 In the second series, the same six domain size cases were considered, but now the circular solid zones were subjected to a given absolute variance on their diameter and position. Applying the same absolute variances to all shrinking domain size cases, a limiting case was encountered wherein the size variance of the obtained through pores became of the same order as their mean size. Reaching this limit (ddom ) 2 µm in the present study), the solid zones had a large probability to make contact with each other or even overlap. To pass this limit, the possibility of zone merging was introduced. This is a process wherein two or more zones that would normally overlap or touch are allowed to merge into a single circular zone having the same volume as the sum of the volumes of its original constituents. The latter rule was applied to ensure that all the different domain size cases would have the same external porosity. (16) He, B.; Tait, N.; Regnier, F. Anal. Chem.1998, 70, 3790-3797. (17) Gzil, P.; De Smet, J.; Vervoort, N.; Verelst, H.; Baron, G. V.; Desmet, G. J. Chromatogr., A 2004, 1030, 53-62. (18) Tallarek, U.; Paces, M.; Rapp, E. Electrophoresis 2003, 24, 4241-4253. (19) Hlushkou, D.; Seidel-Morgenstern, A.; Tallarek, U. Langmuir 2005, 21, 6097-6112. (20) Giddings, J. C. Dynamics of Chromatography. Part 1: Principles and Theory; Marcel Dekker: New York, 1965.
6192
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
Considering that the permeability (and hence also the separation impedance number E ) H2/Kv 21) of LC supports is a strong and continuously decreasing function of the external porosity,15,22 the condition of a constant external porosity is a condition sine qua non to make a fair comparison between the different considered domain size cases. To compare different degrees of zone merging, two different zone merging rules were considered. The first one, further on referred to as “contact merging”, only came into effect as soon as two zones touch or overlap after application of the size and position variances. The second one, further on referred to as “proximity merging”, came into effect as soon as the zones were lying within a range of 0.25 µm or smaller from each other after application of the size and position variances. It is assumed that the adopted solid zone merging processes at least partly reflect the real monolith formation process, for it can be inferred that if two solidifying phase zones would only make the slightest contact during the spinodal demixing process, the wetting tendency of the two similar solid zones would cause this pointlike contact zone to grow into a much larger contact zone, providing the basis for the creation of a stable aggregated zone having a volume equal to the volume of the two initial zones. Such merging processes can obviously not occur in a photolithographic etching process, but they still make a good approximation for a geometry consisting of groups of pillars lying so close to each other that they form one large, hardly permeated zone. EXPERIMENTAL SECTION Flow Domain Construction. Figure 1 gives a detailed view of the employed flow domain generation techniques. To construct the perfectly ordered flow domains, serving as the reference against which all flow domain heterogeneity effects can be measured, arrays of 6 by 21 identically sized circular solid regions were drawn using a commercial CAD program (Gambit v2.1.6). In these arrays, each circular region was positioned on the corner points of a perfectly ordered equilateral triangular grid. As can be noted from Figure 1a, the side of the equilateral triangle unit cell of this grid corresponds to the sum of the cylinder diameter and the pore neck size and is hence the obvious measure for the domain size (ddom). In total, six perfectly ordered arrays were constructed, with domain sizes going from 12 to 1.5 µm. The size of the circular regions was always adapted such that the external porosity of all considered structures was exactly equal to ) 0.60. To construct the disordered flow domains, the x- and ycoordinates of the solid zone centers in the ordered arrays were subjected to a random variation, using the evenly distributed random number generator of MS Excel. This is indicated in Figure 1b by the red arrows. Independently of the considered domain size case, the maximal displacement of the cylinder center was always the same. This value was arbitrarily chosen and equal to plus or minus 0.60 µm in both the x-direction and y-direction. A similar size variance algorithm was applied to the diameter of the circular solid zones. In this case, the values were randomly varied in an interval lying between plus or minus 0.26 µm around the particle diameter, also arbitrarily chosen and independent of the domain size. The thus obtained transformation from a uniform to (21) Knox, J. H. J. Chromatogr. Sci. 1980, 18, 454-461. (22) Vervoort, N.; Gzil, P.; Baron, G.V.; Desmet, G. Anal. Chem. 2003, 75, 843850.
Figure 1. Employed solid zone position and size variance algorithm used to construct disordered array cases. (a) Uniform array of equally sized 2D particles. (b) Randomization by employing position and size variance. The randomized array is indicated in red, while the original uniform array is still visible in gray. (c) Proximity (green) and overlap (red) merging procedures employed in the small-domain arrays and (d) the resulting array.
a randomized array is illustrated in Figure 1a,b. Figure 1b is representative for the situation in all cases with ddom g 3 µm, i.e., the cases in which none of the solid zones made contact or overlapped after application of the size and position variances. Panels c and d in Figure 1 illustrate the adopted contact and proximity merging strategies that needed to be used for the smallest domain cases (ddom ) 2 and 1.5 µm). The contact merging strategy proceeded as follows. As soon as two zones touched or overlapped after application of the size and position variances (see example indicated by red arrows in Figure 1c), they were replaced by a new solid zone having a surface area equal to that of the combined surface areas of the original solid zones and with its center positioned at the center of gravity of the two of original zones (cf. the red circular zone in Figure 1d). If this array rearrangement caused the new circular zone to overlap again with a third zone, the whole procedure was repeated until all overlapping zones were removed from the system. For the proximity merging process, a similar strategy was employed, the only difference being that now two zones were already merged into a larger one as soon as they lay within a distance of less than 0.25 µm from each other (see example indicated in green in Figure 1c,d). To implement the proximity merging strategy, every solid zone was virtually expanded with 0.125 µm in every direction (indicated in green in Figure 1c). Overlapping expanded zones were subsequently detected and treated in the same way as for the contact merging. Numerical Methods. Based on the zone size and position data generated in the MS Excel spreadsheet used to apply the size and position variances, the ordered and disordered arrays were generated in Gambit v2.1.6. The same program was also
used to generate the discretization grids needed for the numerical solution of the species and impulse conservation equations determining the flow and dispersion of the initially injected sample species. A commercial computational fluid dynamics (CFD) software package (Fluent v.6.2.16) was subsequently used to solve these conservation equations across the entire flow domain, consisting of one single fluid zone and a large number (maximally 126) of independent porous zones. During the simulations, the concentration of an injected sample species was monitored as a function of time on a series of detection planes equally distributed along the x-axis of the domain. Plate height values were subsequently obtained by calculating the zeroth-, first-, and second-order moments of the recorded breakthrough curves as described earlier.15 For each case, it was ensured that the obtained data were independent of the grid size as well as from the employed time step size. The front and end of the flow domain were treated as a velocity inlet and a pressure outlet, respectively. Along the side walls, a symmetry boundary condition was defined by imposing a zero normal species and velocity (slip flow) gradient condition. In this way, the flow domain acts as if it is imbedded in an infinite wide medium. At the surfaces of the solid zones, a no-slip boundary condition (velocity equals zero at the surface) was imposed to simulate the flow arresting effect. To mimic the diffusion and adsorption processes inside the stationary phase, the porous zones were defined as a zone with an infinitely large flow resistance and the CFD program was extended with two userdefined functions (UDF), as described in an earlier paper.15 The first UDF was used to attribute slow intraparticle diffusivity to the species entering a solid zone. In all cases, the diffusion coefficient in the solid zone was put at Ds ) 5 × 10-10 m2/s. This Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
6193
Figure 2. Overview of obtained flow domains for disordered array cases, rescaled by dividing the actual space coordinates by the domain size. All flow domains have the same external porosity ) 60%. Highest velocities are indicated in red and lowest velocities in blue. The solid zones are in gray and were treated as a microporous zone with selective adsorption properties. For the largest domain cases (a-d) no contact or proximity merging had to be taken into account. For the smaller domain cases, the contact (e, f) and proximity merging effects (g, h) lead to clearly different flow domains. The scale bar is 10 µm long.
is 50% of that of the diffusion coefficient in the mobile zone (Dmol ) 1 × 10-9 m2/s). The liquid-phase viscosity was set at η ) 1 × 10-3 kg‚m-1‚s-1. The second UDF mimics the selective adsorption process by subjecting the species to an adsorptive reaction with equilibrium constant K ) 5, leading to a zone retention factor k′′ ) 2. Size Distribution Measurement. As can be anticipated, a crucial factor in determining the final band broadening in the disordered flow domain cases is the variance of the pore size and the solid zone size distribution. To determine these size distributions, the shortest distances between each solid zone and all of its surrounding solid zones were measured by using a commercial image analysis software package (Imaq Vision v5.0, National 6194
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
Instruments). The distribution of line lengths was subsequently calculated with MS Excel. RESULTS AND DISCUSSION Figure 2 shows the different packing geometries obtained for the disordered domain cases. The ordered domain cases are not represented, because they resemble the only very slightly disordered 12-µm domain case shown in Figure 2a. The reader should note that, to maintain a clear view on the structure of the flow-through pores, the images for the different domain size cases have been scaled according to their nominal domain size (i.e., the domain size of the employed ordered flow domain before the size and position variances were applied). The zoom in effect can
Table 1. Geometrical Values and Fitting Parameters of the Disordered Array Cases ddom (µm) ordered all sizes 12 8 4 3 2 1.5 2 1.5 a
merging no no
contact proximity
ddom (µm) disordereda
no. of solid zones
same as ordered
126
12.00((0.45) 7.96 (( 0.45) 3.96 (( 0.45) 3.00 (( 0.45) 2.17 (( 0.54) 1.81 (( 0.64) 2.73 (( 0.89) 2.31 (( 0.85)
Disordered Flow Domains 126 0.088 2.661 126 0.109 2.653 126 0.229 2.507 126 0.382 2.406 113 0.517 2.270 94 0.612 2.237 68 0.343 2.185 56 0.404 2.361
A
B
Ordered Flow Domains 0.07 2.75
C
Hmin (µm)
u0,opt (10-3 m/s)
φ0 (/)
E0,min (/)
0.036
0.77ddom
0.77Dmol/ddom
125
74
0.037 0.039 0.043 0.043 0.043 0.044 0.085 0.068
9.61 6.84 4.33 4.05 3.36 3.09 3.91 3.46
0.64 0.94 1.54 1.73 2.18 2.45 1.51 1.89
127 126 127 130 155 189 108 110
81 93 152 244 372 548 222 248
The values in parentheses correspond to the measured standard deviation.
be appreciated from the 5-µm scale bar that continuously grows from the 12-µm to the 1.5-µm domain. Figure 2 also displays the velocity fields in the different considered geometries. Although the different presented cases all have the same average fluid velocity, the red-colored zones representing the zones with the highest velocities in each domain do not correspond to the same absolute velocity. Because of their increased heterogeneity, the smallest domain cases were found to have maximal velocities that are roughly twice the maximal velocity in the perfectly ordered cases. This is a consequence of the fact that the majority of the flow is in the most heterogeneous cases transported through only a limited number of preferential flow paths, whereas in the larger domain cases, i.e., those that are the most homogeneous, each flow-through pore draws about the same flow. Figure 2 also shows the clear difference between the cases that did not undergo a proximity or contact merging and those that did. For the cases between ddom ) 12 µm and ddom ) 3 µm, the applied size and position variance did not lead to any zones touching each other or coming within each others proximity limit, hence leaving the appearance of the obtained flow domains still relatively homogeneous. For the ddom ) 2 and 1.5 µm domain size cases, where the contact and proximity merging processes needed to be applied, this view clearly changes, as the obtained flow domains clearly are more heterogeneous. At the same time, the number of individual solid zones also decreased significantly (values in Table 1). This is also reflected in the domain size values calculated using the geometrical visualization method described in the Experimental Section and reported in Table 1. From ddom ) 12 µm to ddom ) 3 µm, the domain sizes of the disordered flow geometries lie very close to the domain sizes of their ordered counterparts. This good agreement follows from the fact that, because of the complete stochastic nature of the adopted flow domain generation process, the number of growing pores and solid zones is nearly exactly compensated by the number of pores and solid zones that become smaller. This is also reflected in the fact that the observed pore size and solid zone size distribution curves of the ddom g 3 µm cases are perfectly symmetrical (Figure 3). For the ddom ) 2 and 1.5 µm cases, the number of solid zones has decreased compared to the perfectly ordered array case because of the adopted solid zone merging processes. It is hence obvious to see in Table 1 that this is also reflected by the fact that the mean
observed domain size is significantly larger than that in the perfectly ordered counterpart with the same size. The solid zone merging obviously also leads to a clear asymmetry of the pore and solid zone size distributions (Figure 3), as it leads to the creation of a number of extraordinarily large solid zones and pores. Since the considered flow domains have been produced so that the position and size of each solid zone is independent of that of its neighbors, the simulated beds only display short-scale randomness effects23 and not any of the large-scale randomness effects discussed by Giddings20 as long-range coupling distance effects. It can however be inferred that this is not detrimental to the general validity of the qualitative results drawn at the end of this paper. Figure 4a shows how the plate heights in the different perfectly ordered domain cases vary with the domain size. A closer inspection of the plate height curves shows that the Hmin values decrease proportionally with ddom and that the slope of the curves in the C-term dominated range decreases according to ddom2, in agreement with Giddings’ theory on the band broadening in selfsimilar systems. Given that the considered ordered domain cases all possess the same ratio between their characteristic sizes and can be perfectly overlapped by simply rescaling them, they indeed satisfy the definition of self-similar systems. Also in agreement with the theory of self-similar structures is the fact that the A-, B-, and C-term constants, as well as the unretained species-based flow resistance were found to be independent of the domain size (φ0 ) 125, Table 1). The A-, B-, and C-term constant data presented in Table 1 have been calculated by first transforming the obtained plate height curves into the domain size-based reduced variables h and ν and subsequently fitting the obtained curves using the Knox empirical,24,25 yet often used, reduced plate height expression:26
h ) Aυ1/3 + B/υ + Cυ
(1)
Comparing Figure 4a with the disordered domain cases in Figure 4b clearly shows the impact of the imposed solid zone size and (23) Billen, J.; Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. J. Chromatogr., A 2005, 1073, 53-61. (24) Tallarek, U.; Bayer, E.; Guiochon, G. J. Am. Chem. Soc. 1998, 120, 14941505. (25) Berdichevsky, A. L.; Neue, U. D. J. Chromatogr. 1990, 535, 189-198. (26) Knox, J. H.; Parcher, J. F. Anal. Chem. 1969, 41, 1599-1606.
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
6195
Figure 3. (a) Pore and (b) particle size distribution curves obtained for the disordered domains cases shown in Figure 2. Domain sizes without merging (red curves) of solid zones are (×) 12, (+) 8, (b) 4, and ([) 3 µm; with contact merging (9) 2 and (2) 1.5 µm; with proximity merging (0) 2 Rνδ (∆) 1.5 µm. The pore and particle sizes are normalized against the domain size.
position variance. The plate height curves now decrease with decreasing ddom, eventually ending up in a band of curves, where the different size and different merging type cases run through each other. As expected, the more pronounced zone merging in the proximity merging cases (open symbols) leads to larger plate heights than in the corresponding contact merging cases with the same domain size (solid symbols). This obviously is a consequence of the increased heterogeneity of the proximity merging flow domains, as can be assessed from their broader pore and solid zone size distribution in Figure 3 and from the flow domain pictures shown in Figure 2. It should be noted that the obtained plate height curves depend strongly on the specificity of the considered flow domain geometry. It is straightforward to assume that other plate height curves would have been obtained if the position of the solid zones would be slightly different. For example, the presence of the strong radial velocity component that can be observed in the ddom ) 1.5 µm proximity merging case (Figure 2h), caused by the 6196
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
downstream presence of a cluster of large solid zones, can be assumed to promote the radial equilibration of migrating species. It can hence be expected that, with a slightly different position of the solid zones, this radial flow would have been much less pronounced, leading to significantly different (i.e., larger) plate heights. The plate height curves shown in Figure 4b should hence not be considered as the single possible outcome for each of the different considered domain size cases. The interaction between the shrinking domain size on one hand (expected to yield smaller plate heights) and the increasing heterogeneity (expected to yield larger plate heights) on the other hand is highly complex and depends on the exact microscopic structure of the generated flow domain. For the proximity merging cases, for example, the ddom ) 2 µm case yields larger plate heights than both the ddom ) 3 µm case (reflecting the larger heterogeneity of the ddom ) 2 µm case) and the ddom ) 1.5 µm case (reflecting the larger domain size and corresponding mass transfer distances in
Figure 4. Obtained plate height curves for the (a) perfectly ordered and (b) disordered (contact merging and proximity merging plotted together) flow domains. Same symbols as in Figure 3. Adopted chromatographic conditions: Dmol ) 1 × 10-9 m2/s, Ds ) 5 × 10-10 m2/s, η ) 1 × 10-3 kg‚m-1‚s-1 and k′′) 2.
the ddom ) 2 µm case). To arrive at more definitive and representative curves, a large number of structural variants should be considered for each domain size and their results should be averaged. This, however, lies far beyond our current calculation possibilities. To quantify the position and course of the plate height curves shown in Figure 4b, their most important characteristics are represented in Figure 5a-c. The main conclusion that can be made from these graphs is that the increased heterogeneity of the smallest domain size cases leads to a very sharp increase of the minimal reduced plate height and of the A-term band broadening as compared to the horizontal line of the ordered cases (dashed line). This is in full agreement the physical meaning of the A-term constant.27 Whereas for the hmin value nearly no difference can be noted between the two investigated merging cases, concerning the A-term dependency there is a clear difference between the proximity merging (open symbols) and for the contact merging (closed symbols) cases. It is assumed that this should not be considered as a reflection of some kind of inherent difference between the two merging types (for there simply is no fundamental difference between them), but simply reflects the
stochastic nature of the constructed flow domains. That is, it is assumed that, with a slightly different outcome of the Excel random number generator, other A-term values would have been obtained, still lying in the vicinity of the values obtained with the current domains, but anyhow clearly different. It is hence assumed that if many more geometries would have been constructed, this would have resulted in a cloud of A-term constant data points, all lying around the A-term values obtained for the presently considered structures. The same can be expected for the C-term constant (Figure 5c). For the domain size reduced flow resistance φ0 (Figure 5d), a relatively unexpected trend is seen, because in the proximity merging case the flow resistance decreases with decreasing domain size, while in the contact merging case the opposite trend is noted. This opposing behavior is partly due to the employed reduction basis. Using the pore size to reduce the bed permeabilities (data not represented here), the two merging types both yielded increasing flow resistances with decreasing domain size (i.e., with increasing heterogeneity). The flow resistances in the (27) Knox, J. H. J. Chromatogr., A 1999, 831, 3-15.
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
6197
Figure 5. (a) Hmin/ddom as a function of ddom, (b) A versus ddom, (c) C versus ddom, and (d) φ0 versus ddom. The represented ddom values correspond to the measured ddom data given in Table 1. (9) Contact and (0) proximity merging. As a reference, the values obtained for the ordered domain case values (2) are given as well.
proximity merging case however remained still smaller than those for the (less heterogeneous) contact merging case. It is nevertheless assumed that no main conclusions should be derived from this observation, as the obtained values can be assumed to be highly sensitive to the local structure of the considered flow domains. With a different outcome of the stochastic cylinder displacement and merging process, significantly different bed structures and hence significantly different flow resistances could have been obtained. As elaborated in refs 14 snf 28, it is very straightforward to combine the effect of flow resistance and band broadening into a single picture that immediately reveals the general kinetic performance potential of any given tested support. All this requires is recombining the (u0,H) data couples from Figure 4 with the (28) Desmet, G.; Gzil, P.; Clicq, D. LC‚GC Eur. 2005, 18, 403-409.
6198
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
bed permeability, the mobile-phase viscosity, and a freely selectable reference pressure gradient (put at 400 bar in the present case) according to the two following very simple expressions, yielding the fastest possible t0 time as a function of the corresponding number of theoretical plates N:
t0 ) (∆P/η)[Kv/u02] exp
(2)
N ) (∆P/η)[Kv/u0H] exp
(3)
As explained in ref 14, the most clear view on the kinetic performance is obtained when plotting the thus obtained values as t0/N2 versus N, rather than as t0 versus N. To remain as close as possible to the familiar view of a traditional van Deemter plot, the direction of the N axis can be reversed, so as to run from
Figure 6. (a) Kinetic plot curves for ordered (gray curves) and disordered arrays (red curves). Same symbols were used as in Figure 3. (b) Comparison with experimental data of silica monoliths. Plate height and bed permeability data sets were taken from Motokawa et al.12 (black 9) MS(50)-A, (black 2) MS(50)-B, (black b) MS(50)-C, (black [) MS(50)-D, (red 9) MS-H(50)-I, and (red 2) MS-H(50)-II; data set from Kobayashi et al.29 (blue 9) MSH-100 I, (blue 2) MSH-100 II, and (blue b) MSH-100 III. The geometrical characteristics of the represented experimental monolithic columns are given in Table 2. Indicated target region with improved kinetic performance is inaccessible due to heterogeneity.
high N values at the left to small N values at the right. Doing so, the obtained curves have their B-term dominated and their C-term dominated regime respectively situated to the left and the right of the curve minimum corresponding to the optimal mobile-phase velocity. As a consequence, the curves allow for the same type of analysis (e.g., comparing the steepness of the C-term regime or comparing the value of the curve minimum to assess the magnitude of the A-term contribution) that is typically carried out in a normal or a reduced van Deemter plot. Although the absolute value of the corresponding t0 time is no longer explicitly available from the y-axis, this problem can easily be solved by adding a grid of constant t0 reference lines to the established graphs, as was done in Figure 6. Comparing the ordered with the disordered domain cases in Figure 6a, the effect of the small domain size-induced heterogeneity governing the disordered domain cases is immediately appar-
ent. The perfectly ordered domain curves (gray curves) clearly show how a continuous reduction of the domain size that would not be accompanied by an increased heterogeneity can be expected to shift the kinetic plot curves from the impractically large N ) 106-107 plates range toward the practically more interesting N ) 5 × 104 plates range without increasing the position of the curve minimum. The red curves for the disordered flow domains on the other hand show that if the domain size reduction is accompanied by a nonshrinking structural variance, the time curves shift less to the right and also display an upward shift of their minimum and of their C-term dominated part. As a consequence, the kinetic performance curves of the smallest domain cases tend to heap up in a relatively narrow band that is difficult to surpass by further reducing the domain size, because this would only further increase the heterogeneity of the obtained structures. Around N ) 5 × 104, a difference of nearly a factor of Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
6199
Table 2. Geometrical Parameters of the Experimental Monolithic Columns column
ddom (µm)
Hmin (µm)
KV0 (×10-14 m2)
MS(50)-A MS(50)-B MS(50)-C MS(50)-D MS-H(50) I MS-H(50) II MSH-100 I MSH-100 II MSH-100 III
10 4.2 3.3 3 6.5 3.5 5.8 3.2 2.3
11.8 10.1 8.9 8.0 11.3 8.5 8.1 6.2 4.9
130 25 15 8 56 19 26 7.8 4.3
10 in analysis time is found between the ordered and the disordered cases. The number of investigated bed morphologies is certainly too limited to draw any quantitative conclusions on the combined effect of a decreasing domain size and a simultaneously increasing heterogeneity (cf. the discussion of Figure 5). There is however no physical reason to doubt that the qualitative picture of Figure 6a is also representative for all other possible flow domain series wherein it would be attempted to decrease the domain size using a manufacturing process displaying a domain-size independent variance on the position and size of the produced solid zone elements. The negative contribution coming from the small domain size-induced heterogeneity can only be expected to grow with decreasing domain size, leading to ever growing deviations from the pure horizontal shift of the t0 time curves that can be expected in the absence of this increased heterogeneity (gray curves in Figure 6a). These ever growing deviations can only result in a heaping up of the t0 time curves. This heaping up of kinetic plot curves of supports with decreasing domain size is also observed experimentally. To illustrate this, three data sets (respectively colored in black, red, and blue in Figure 6b) were taken from Kobayashi et al.29 and Motokawa et al.12 They fabricated and tested silica monoliths with an external porosity of 80-85% in capillaries with a 50-100-µm diameter. In Figure 6b, each different colored data set corresponds to a different type of manufacturing process (see figure captions and cited references for the full details) and consists of a series of monolithic columns with decreasing domain size (ddom values are given in Table 2). The experimental kinetic plot curves shown are obviously in very good qualitative agreement with the currently investigated “synthetic” disordered domain cases (cf. red curves in Figure 6a): the experimental data curves also tend to heap up in their C-domain dominated part and also have their minimum shifted upward with decreasing domain size. From this qualitative agreement, it can only be concluded that, to avoid this heaping up of kinetic performances, solutions need to be found to let the manufacturing tolerance on the position and the size of the produced solid zone elements shrink in proportion to the domain size.
produce small domain size LC supports. If the employed manufacturing method would display a domain size-independent variance on the size and position of the produced solid zone elements, this will inevitably lead to a domain size-induced heterogeneity that from a given domain size on breaks the self-similarity with its larger domain size counterparts, yielding a chaotic cluster of increased reduced plate height or flow resistance data points, or both. Using the kinetic plot method to directly transform the plate height and flow resistance data into a plot of analysis time versus plate number, the effect of the small domain size-induced heterogeneity on the plate heights and the flow resistance of the obtained supports is combined into a single plot. This plot shows that the analysis speed of systems experiencing severe small domain size-induced heterogeneity problems can at least be 10 times lower than what would be possible in the absence of such problems. It also shows that, under the presently adopted assumption, all attempts to reduce the feature or domain size of LC supports will eventually face a lower limit band of kinetic performances that can no longer be surpassed significantly by further decreasing the domain size. This lower limit manifests itself as a relatively narrow band in the kinetic plot of t0 versus N, and this is for example precisely what is observed experimentally when attempting to produce small domain silica monoliths. Similar small domain-induced heterogeneity effects can also be expected to compromise the performance limits of photolithographically etched columns. Due to the heterogeneity of the skeletons and flow-through pores it is up until today impossible to prepare small-domain monolithic silica columns that have the same performance of a packed bed column packed with sub 2-µm silica particles. To find ways to improve the currently employed manufacturing procedures, large systematic experimental studies are needed wherein the pore and skeleton size variability is quantified as a function of the duration and intensity of the reagent mixing and temperature control.
GLOSSARY A, B, C
A-, B-, C-constants in eq 1 (/)
ddom
domain size, ddom ) dpor + ds (m)
dpor
pore size (m)
ds
solid zone size (m)
Dmol
molecular diffusion coefficient in mobile zone (m2/s)
Ds
molecular diffusion coefficient in stationary zone, (m2/s)
f
relative frequency, the frequency divided by number of pores or particles
E0
t0-based separation impedance, E0 )dH2/Kv0 (/)
H
height equivalent of a theoretical plate (m)
h
reduced plate height, h ) H/ddom (/)
CONCLUSIONS The present study provides a qualitative illustration of the inherent difficulties that can be expected if it is attempted to
k′′
zone retention factor, k′′ ) k′′0(1 + K) (/)
k0′′
(29) Kobayashi, H.; Tokuda, D.; Ichimaru, J.; Ikegami, T.; Miyabe, K.; Tanaka, N. J. Chromatogr., A 2006, 1109, 2-9a.
zone retention factor of non retained component, k′′0 ) int[(1 - )/] (/)
K
adsorption equilibrium constant (/)
6200 Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
u0-based column permeability (m2)
ACKNOWLEDGMENT
N
plate number (/)
t0
dead time of a column (s)
u0
mean velocity of permeating, but nonretained solute (m/s)
The authors greatly acknowledge a research grant (FWO KNO 81/00) from the Fund for Scientific ResearchsFlanders (Belgium). J.B. is supported through a specialization grant from the Instituut voor Wetenschap en Technologie (IWT) of the Flanders Region (grant SB/1279/00). P.G. is a Postdoctoral Fellow of the Research FoundationsFlanders (FWO Vlaanderen).
Kv0
Greek Symbols ∆P
pressure drop (Pa)
external or through pore porosity (/)
φ0
u0-based flow resistance (/) (kg‚m-1‚s-1)
η
dynamic viscosity
ν
reduced velocity, n ) u0‚ddom/Dmol (/)
Received for review March 15, 2006. Accepted May 21, 2006. AC060470X
Analytical Chemistry, Vol. 78, No. 17, September 1, 2006
6201
