Anal. Chem. 1997, 69, 361-363
Effect of Electrical Double-Layer Overlap on the Electroosmotic Flow in Packed-Capillary Columns Qian-Hong Wan*
Laboratory for Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands
A plug electroosmotic velocity profile is generally assumed to be characteristic of capillary electrochromatography. However, this ideal plug flow may be illusive in some experiments with packed-capillary columns due to overlap of electrical double layers in flow channels. We report here a theoretical analysis of the double-layer overlap effects in packed-capillary columns, which is based on Rice and Whitehead’s theory of electroosmotic flow combined with a capillary tube model for porous packing. The results show that the electroosmotic velocity under the influence of double-layer overlap depends strongly on the operating parameters, which increases with the column porosity, the particle diameter, and the electrolyte concentration. In recent years, capillary electrochromatography (CEC) has come to be appreciated as a promising separation method.1-7 In this method, neutral solutes are carried through an open-tubular3 or packed-capillary column1,2 by electroosmotic flow and separated by partitioning between the mobile and stationary phases. Owing to the plug nature of this electrically induced flow, CEC offers the potential to attain column efficiency much higher than that possible with parabolic pressure-induced flow. Despite enormous efforts devoted to the improvement of column preparation,1,2,4,6,7 CEC with packed columns is shown to be only marginally superior to pressure-driven system in terms of plate height, and the expected very narrow solute peaks have not been seen on a routine basis. In our study of key factors which limit the performance of CEC, we found electrical double-layer overlap effects to be of particular interest, as they might contribute to excessive peak dispersion in packed-column CEC. The electrical double-layer overlap may occur when the size of the flow channel is not significantly greater in magnitude than the electrical double-layer thickness.8 This double-layer overlap has detrimental effects in CEC not only because it causes a reduction in the electroosmotic velocity, but also because it induces a peak dispersion due to a transition of the velocity profile from flat to parabolic. Although the double-layer overlap effects * Present address: Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J3. Email:
[email protected]. (1) Jorgenson, J. W.; Lukacs, K. D. J. Chromatogr. 1981, 218, 209-216. (2) Knox, J. H.; Grant, I. H. Chromatographia 1987, 24, 135-143. (3) Bruin, G. J. M.; Tock, P. P. H.; Kraak, J. C.; Poppe, H. J. Chromatogr. 1990, 517, 557-572. (4) Knox, J. H.; Grant, I. H. Chromatographia 1991, 32, 317-328. (5) Yamamoto, Y.; Baumann, J.; Erni, F. J. Chromatogr. 1992, 593, 313-319. (6) Yan, C.; Dadoo, R.; Zhao, H.; Zare, R. N.; Rakestraw, D. J. Anal. Chem. 1995, 67, 2026-2029. (7) Liao, J.-L.; Chen. N.; Ericson, C.; Hjerten, S. Anal. Chem. 1996, 68, 34683472. (8) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. S0003-2700(96)00478-7 CCC: $14.00
© 1997 American Chemical Society
on electroosmotic flow in a cylindrical capillary tube were discussed by Rice and Whitehead8 30 years ago, detailed studies in the context of CEC have been limited.2,9 In this work, we investigate the double-layer overlap effects in a packed column, using a theoretical model in which the porous packing structure is viewed as a bundle of capillaries of the size determined by the local packing density so that the electroosmotic flow in the packed column can be described by Rice and Whitehead’s theory. Relative velocity, which is defined as the ratio of the electroosmotic velocity with to that without the double-layer overlap effects, is used to show the variation of electroosmotic flow with packing density, particle size, and electrolyte concentration. RESULTS AND DISCUSSION The expression for the electroosmotic velocity in a cylindrical flow channel has been derived by Rice and Whitehead8 as
u(r) )
[
]
0rζE I0(κr) 1η I0(κa)
(1)
where 0, r, ζ, κ, E, η, r, and a represent respectively the permittivity of a vacuum, the relative permittivity of the medium, the zeta potential of the flow channel, the reciprocal of the electrical double-layer thickness, the applied electric field strength, the viscosity of the medium, the distance from the channel center, and the radius of the flow channel. The cross-sectional average velocity is then given by
u)
[
0rζE 2I1(κa) 1η κaI0(κa)
]
(2)
In eqs 1 and 2, I0 and I1 are respectively the zero-order and firstorder modified Bessel functions of the first kind. Asymptotic expansions show that the functions in brackets approach unity for large values of κa. In these cases, eqs 1 and 2 are equaled, giving
u∞ )
0rζE η
(3)
which is the usual result in the case of no double-layer overlap. The relative velocity µ, by definition, is given by (9) Stevens, T. S.; Coates, H. J. Anal. Chem. 1983, 55, 1365-1370.
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997 361
Figure 1. Variation of relative velocity with electrokinetic radius.
Figure 2. Variation of relative velocity with particle diameter for interparticle porosities of 0.26, 0.40, and 0.48 (from the lower to the upper curve), where κ ) 0.10 nm-1 is assumed.
µ)
[
2I1(κa) u ) 1u∞ κaI0(κa)
]
(4)
The relative velocity has a limiting value of unity in the absence of the double-layer overlap and approaches zero at extremely small values of κa. In Figure 1, the relative velocity µ is plotted against electrokinetic radius κa, which clearly shows that double-layer overlap has a profound effect on electroosmotic velocity only in the low range of κa. For κa ) 10, the electroosmotic flow is reduced by 20%. In order to evaluate the relative velocity using eq 4 in relation to experimental conditions of capillary electrochromatography, one needs to know magnitudes of flow channel diameters in a packed column. As shown elsewhere,10 the mean channel diameter d may be estimated by the following equation:
(5)
Figure 3. Variation of relative velocity with electrolyte concentration for interparticle porosities of 0.26, 0.40, and 0.48 (from the lower to the upper curve). (a) dp ) 1 µm; (b) dp ) 3 µm; (c) dp ) 5 µm.
where dp is the mean particle diameter and n the interparticle
porosity. It appears that the channel diameter is directly proportional to the particle size, with the proportionality mainly deter-
d ) 0.42dp
n 1-n
362 Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
mined by the interparticle porosity. The interparticle porosity of a chromatographic column is known to vary widely, depending on the packing method, particle shape, and size distribution.11 Due to the heterogeneous nature of the random packing, the local values of porosity are even more variable and unpredictable. As pointed out by Giddings,11 the geometrical complication of pore structure has defied all efforts to come to grips with it mathematically, and our practical understanding of porosity must be limited to inexact and intuitive concepts supported by experience. As a starting point, we may consider simple models, such as regular packings of uniform spheres. According to Graton and Fraser,12 the least compact arrangement of such spheres is that of simple cubical packing with a porosity of 0.48, whereas the most compact packing is the rhombohedral one (where each sphere is tangential to 12 neighboring spheres), which has a porosity of 0.26. A random packing of spherical particles in a chromatographic column has an average interparticle porosity around 0.40, a value falling in between these limits. It seems reasonable to assume that the tightly fitted particles in the column will have a porosity approaching the minimum value of 0.26. It is these densely packed regions that become our concern, as the electroosmotic velocities in these regions may be markedly reduced due to double-layer overlap effects. The effect of packing density and particle size on electroosmotic velocity can be shown by substituting 0.5d from the above equation for a in eq 4. The relative velocity is plotted as a function of dp in Figure 2 for three porosity values between 0.26 and 0.48, assuming κ ) 0.10 nm-1. The relative velocity is increased with interparticle porosity in the lower range of dp. With respect to the effect of particle size, all the curves show the same trend, whatever the n values. The relative velocity increases rapidly with dp at low dp values and more gradually for high dp values. From the curves in Figure 2, it is also evident that the transcolumn variation of electroosmotic velocity caused by the variation of the interparticle porosity is less than 30% for dp > 1 µm. This magnitude of variation is, indeed, very small as compared with that in pressure-driven chromatographic systems, where the mean velocity is proportional to the square of the channel diameter.11 However, the velocity inequality increases considerably with submicrometer-sized particles as a result of a substantial doublelayer overlap occurring in narrow channels. The electrolyte concentration affects the relative velocity through the electrical double-layer thickness. The reciprocal of the double-layer thickness is related to the molar concentration
by eq 6 for an aqueous solution of 1:1 electrolyte at 25 °C:13
(10) Wan, Q. H. J. Phys. Chem., submitted. (11) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965; Chapter 5. (12) Graton, L. C.; Fraser, H. J. J. Geol. 1935, 43, 785-909. (13) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: London, 1981; Chapter 2.
Received for review May 16, 1996. Accepted November 7, 1996.X
κ ) 3.29c0.5(nm-1)
(6)
The values of µ as a function of log c were calculated by substituting eq 6 for κ in eq 4, and the results are shown Figure 3 for three values of dp between 1 and 5 µm. These calculations assumed the same values of n as in Figure 2. In general, the relative velocity increases with the electrolyte concentration over a reasonable range covered. As can be seen, electrolyte concentration has a major influence on electroosmotic velocity for low values of dp and n. In these cases, the use of higher electrolyte concentrations is essential to minimize the undesirable effects of double-layer overlap. However, the upper limit of allowable electrolyte concentration is set by other factors such as thermal effects and bubble formation. From these considerations, it is obvious that electrolyte concentration is an important operating parameter that needs to be optimized for best performance in CEC. In conclusion, the present theoretical study predicts that the effect of electrical double-layer overlap on electroosmotic velocity in a packed-capillary column decreases with increasing interparticle porosity, particle diameter, and electrolyte concentration. To our knowledge, there exist no measurements of these effects under the conditions specified here to confirm the predicted trends, but we anticipate that systematic experimental studies aimed at addressing the issues raised in this work will appear in the near future. It is important to note that the theoretical model presented here is highly simplified. No constriction of the flow channel is taken into account, and that particles are of equal size is assumed. Despite these approximations, the model provides a detailed understanding of the dependence of electroosmotic flow on various CEC operating parameters; therefore, we believe is a useful starting point for the more refined analysis of double-layer overlap phenomena. ACKNOWLEDGMENT I thank Professor H. Poppe (University of Amsterdam) for his encouragement and support during the research. The work was supported by a postdoctoral fellowship from the University of Amsterdam. Part of the work was presented at the 5th International Symposium on High Performance Capillary Electrophoresis, Orlando, FL, January 25-28, 1993.
AC960478E X
Abstract published in Advance ACS Abstracts, December 15, 1996.
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
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