Anal. Chem. 2011, 83, 460–461
Comment on “Submicrometer Plate Heights for Capillaries Packed with Silica Colloidal Crystals” Uwe D. Neue Waters Corporation, 34 Maple Street, Milford, Massachusetts 01757, United States In a recent paper, Wirth and co-workers1 discussed the interstitial fraction of capillaries packed with uniform sub-1 µm particles. The authors concluded that packed beds with an interstitial fraction around 25% had been achieved. However, the information provided in the paper does not support such a conclusion. Many studies in the chromatography literature alone have looked at interstitial fractions of packed beds. None of them supports the existence of packed beds with an interstitial fraction of 25%. As an example, Ugelstadt2,3 and co-workers created uniformly sized porous polymer particles and examined columns packed with these beads. They reported interstitial fractions around 35%, quite in line for other polymeric porous particles used in size-exclusion separations. Tallarek4 recently studied isotropic random monosized hard-sphere packing structures from a theoretical point of view and reported interstitial fractions between 36.6% and 46% for random-close to the random-loose packings. The paper by Wirth and co-workers1 contains sufficient information to verify the interstitial fraction of the packed capillaries. In the Results and Discussion section as well as in Figure S-2 of the Supporting Information, the relationship between pressure and flow rate is presented, while the information on the mobile phase composition is given in the Experimental Section. This provides sufficient information to calculate the interstitial porosity εi using the Kozeny-Carman equation, which is widely accepted in the chemical engineering literature and in chromatography.5 For the purpose of this discussion, it is best written as follows: (1 - εi)2 πrc2Pdp2 ) 180 FηL εi3
(1)
where rc is the capillary radius, P the pressure, dp the particle size, F the flow rate, η the viscosity, and L is the length of the packed bed. All readily measurable or accessible parameters are found on the left-hand side of the equation. The function in the interstitial porosity εi found on the right-hand side of eq (1) Malkin, D. S.; Wei, B.; Fogiel, A. J.; Staats, S. L.; Wirth, M. J. Anal. Chem. 2010, 82, 2175–2177. (2) Kulin, L.-I.; Flodin, P.; Ellingsen, T.; Ugelstad, J. J. Chromatogr., A 1990, 514, 1–9. (3) Ellingsen, T.; Aune, O.; Ugelstad, J.; Hagen, S. J. Chromatogr., A 1990, 535, 147–161. (4) Khirevich, S.; Daneyko, A.; Ho¨ltzel, A.; Seidel-Morgenstern, A.; Tallarek, U. J. Chromatogr., A 2010, 1217, 4713–4722. (5) Cabooter, D.; Billen, J.; Terryn, H.; Lynen, F.; Sandra, P.; Desmet, G. J. Chromatogr., A 2008, 1178, 108–117.
460
Analytical Chemistry, Vol. 83, No. 1, January 1, 2011
Figure 1. Porosity function according to Kozeny-Carman.
1 is shown in Figure 1. We see that for a standard packed bed with an interstitial porosity of 0.4 () 40%), the value of the function is about 1000. For the densest packing of uniform spheres, which has an interstitial fraction of 0.2595 (∼26%), the function has a value of about 6000. In other words, a packed bed with an interstitial fraction of about 25% would have a roughly 6-fold higher backpressure than the same bed with a classical and normal interstitial fraction of 40%. The column radius was 50 µm, the pressure was 12 400 psi at the highest flow rate of 200 nL/min, and the particle size was 330 nm. The 90/10 methanol/water mixture has a viscosity of 0.9 cP at 25 °C. The length of the packed bed is found in Figure S-2 to be 2.31 cm for the pressure and flow rate given in the text. Using these data, we obtain a value of 1055. This translates to an interstitial fraction of 0.396, or about 40%, quite in line with expectation for a standard packed bed. This clear and unequivocal value stands in stark contrast with the volumetric displacement reported in the paper. However, it is not clear that the liquid volume measured would penetrate the entire interstitial fraction. We note that water was used as the “wicking” liquid into a packed bed with a hydrophobic C18/C1 coating. The wicking of water into a hydrophobic structure is not possible (see below). If the structure was indeed hydrophobic, then some assistance with a mild pressure on the syringe was necessary. It is quite possible that the wetting angle of the hydrophobic coating together with the small pressure from the syringe was insufficient for a full penetration of all the crevices in the packed bed. Such a phenomenon would exclude the smaller spaces where the particles touch each other but still provide an easy penetration into the more open spaces. 10.1021/ac101900e 2011 American Chemical Society Published on Web 10/05/2010
The Washburn equation explains the pressure P needed for the penetration of a nonwetting liquid into a pore space with radius r:6
P)-
2γ cos(θ) r
(2)
γ is the surface tension (of water) and θ is the wetting angle. Unfortunately, the wetting angle of water on C18 coated beads is unknown, so one cannot pursue this calculation much further. However, for a wetting angle of 100°, a pressure of only 0.2 bar is sufficient to penetrate the largest interparticle pores. Thus a wetting of the bed with only a slight assistance is possible. We note on the side that the measured values for the interstitial fraction reported in Figure S-3 go down to a (6) Washburn, E. W. Phys. Rev. 1921, 17, 273–283.
porosity of 0.225, clearly a physical impossibility for the true interstitial fraction of a bed of uniform spheres but not impossible for a bed that is only partially penetrated by a marginally wetting liquid. Erroneous values for the wetting of the bed can also be caused by other factors, such as incomplete drying, residues from the treatment, etc. The SEM image in Figure 1 of ref 1 only depicts the end of the packed capillary. It is not uncommon to find a different structure at the end of a packed bed, and therefore the inference of a similarity of the entire structure of the packed bed is invalid. I conclude that that there is no valid evidence for a packed bed structure with an interstitial fraction of 25%. Received for review July 16, 2010. Accepted September 24, 2010. AC101900E
Analytical Chemistry, Vol. 83, No. 1, January 1, 2011
461
