Anal. Chem. 1998, 70, 3069-3077
Axial Nonuniformities and Flow in Columns for Capillary Electrochromatography A. S. Rathore and Cs. Horva´th*
Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286
Unlike in capillary zone electrophoresis and microscale high-performance liquid chromatography, columns in capillary electrochromatography (CEC) have discontinuities of the electric field strength and the flow velocity at the interface of the packed and open segments of the column. The goal of the present work is to offer a framework for measuring and interpreting the relevant electrochromatographic parameters such as the electric field strength, the potential drop, and the flow velocity in the packed segment of a CEC column. This would help us gain further insight in the electrochromatographic process and facilitate the design of CEC separation technology and comparison of data from different sources. First, the flow of ions that is governed by the conservation of current is analyzed and the potential drop across the packed and open segments of the column calculated. Then, conservation of volumetric flow rate is used to calculate the flow velocities through the two segments and, further, to estimate the net flow velocity through such a column. To satisfy the mass conservation law, in most cases a “flow-equalizing intersegmental pressure”, which is different from the pressures at the two ends of the column, develops at the interface of the packed and the open segments. The intersegmental pressure, induced to equalize the flow rates in the two segments, has been shown to have a significant effect on the magnitude as well as the radial distribution of the flow velocity in the open segment, where the net flow becomes a mixture of electroosmotic and pressure-driven flows. Capillary electrochromatography (CEC) is a bona fide liquid chromatographic technique that is carried out with packed capillary columns by electroosmotically driven mobile phases at high electric field strengths in an apparatus similar to that used in capillary zone electrophoresis (CZE). The use of electroosmotic flow (EOF) as a “pumping mechanism” in liquid column chromatography was first suggested in 1974,1 and the viability of this approach with packed capillary columns was demonstrated in 1981.2 Later, the salient features of CEC were examined,3,4 and since then the potential of this technique has been demonstrated * Address correspondence to this author at Department of Chemical Engineering, Yale University, P.O. Box 208286, New Haven, CT 06520-8286. Tel.: (203) 432-4357. Fax: (203) 432-4360. E-mail:
[email protected]. (1) Pretorius, V.; Hopkins, B. J.; Schieke, J. D. J. Chromatogr. 1974, 99, 2330. (2) Jorgenson, J. W.; Lukacs, K. D. J. Chromatogr. 1981, 218, 209-216. (3) Knox, J. H.; Grant, I. H. Chromatographia 1987, 24, 135-143. S0003-2700(97)01260-2 CCC: $15.00 Published on Web 05/30/1998
© 1998 American Chemical Society
by several research groups.5-15 Yet, the understanding of the fundamental physicochemical phenomena underlying CEC has received limited attention.15-21 At present, it is believed22 that CEC has the potential to become a high-performance liquid chromatographic separation method on an electrophoretic platform offering high peak capacity and compatibility with mass spectroscopy (MS) for the analysis of a wide variety of sample components. For attaining rapid, accurate, and reproducible separations, it is imperative that the generation and control of the electroosmotic flow in CEC columns packed with suitable stationary phases is well understood. Figure 1 shows a typical CEC column consisting of a packed and an open segment that are uniform both radially and axially and are separated by the retaining frit which is adjacent to the detector window in the most frequently used configuration.21 At the interface of the packed and open segments of the column, there is a discontinuity of almost all electrochromatographic parameters, including the conductivity, electric field strength, and the flow velocity. The presence of such discontinuities in CEC complicates the electrochromatographic process and makes the exact calculation of the above-mentioned electrochromatographic parameters in the packed and the open segments more difficult. The goal of the present work is to examine and characterize the inherent peculiarities of CEC columns and provide a detailed description of the calculation procedures required to evaluate the pertinent electrochromatographic parameters in the packed and open (4) Knox, J. H.; Grant, I. H. Chromatographia 1991, 32, 317-328. (5) Yan, C.; Schaufelberger, D.; Erni, F. J. Chromatogr. 1994, 670, 15-23. (6) Smith, N. W.; Evans, M. B. Chromatographia 1994, 38, 649-657. (7) Rebscher, H.; Pyell, U. Chromatographia 1994, 38, 737-743. (8) Behnke, B.; Bayer, E. J. Chromatogr. 1994, 680, 93-98. (9) Colon, L. A.; Guo, Y.; Fermier, A. Anal. Chem. 1997, 69, 461A-467A. (10) Boughtflower, R. J.; Underwood, T.; Paterson, C. J. Chromatographia 1995, 40, 329-335. (11) Yan, C.; Dadoo, R.; Zare, R. N.; Rakestraw, D. J.; Anex, D. S. Anal. Chem. 1996, 68, 2726-2730. (12) van de Bosch, S. E.; Heemstra, S.; Kraak, J. C.; Poppe, H. J. Chromatogr. A 1996, 755, 165-177. (13) Dittmann, M. M.; Rozing, G. P. J. Chromatogr. A 1996, 744, 63-74. (14) Huber, C. G.; Choudhary, G.; Horva´th, Cs. Anal. Chem. 1997, 69, 44294436. (15) Choudhary, G.; Horva´th, Cs. J. Chromatogr. A 1997, 781, 161-183. (16) Dittmann, M. M.; Wienand, K.; Bek, F.; Rozing, G. P. LC-GC 1995, 13, 800-814. (17) Rathore, A. S.; Horva´th, Cs. J. Chromatogr. A 1996, 743, 231-246. (18) Ståhlberg, J. Anal. Chem. 1997, 69, 3812-3821. (19) Iwata, M. In Electric Field Applications in Chromatography, Industrial and Chemical Processes; Tsuda, T., Ed.; VCH: New York, 1995; pp 133-151. (20) Rathore, A. S.; Horva´th, Cs. J. Chromatogr. A 1997, 781, 185-195. (21) Rathore, A. S.; Horva´th, Cs. Anal. Chem., in press. (22) Majors, R. E. LC-GC 1998, 16, 96-110.
Analytical Chemistry, Vol. 70, No. 14, July 15, 1998 3069
Fopen )
Figure 1. Schematic illustration of a CEC column consisting of a packed and an open segment. The detection window is located immediately after the retaining frit at the outlet of the packed segment.
segments from readily obtainable experimental data. Our objective is to develop a framework that would not only allow us to conveniently measure the various CEC parameters but also make it easy to compare electrochromatographic data obtained in different laboratories using a variety of mobile- and stationaryphase conditions. ELECTRIC FIELD STRENGTH IN THE PACKED AND OPEN SEGMENTS Conservation of Electric Charge. In the absence of any “source” or “sink” for the conducting ions inside the capillary tube, as in our case, the conservation of current and Ohm’s law yield the following expression for the current through the column, i:
i)
)
Vpacked Vopen V ) ) Rpacked Ropen Rpacked + Ropen VAopen FpackedLpacked + FopenLopen
(1)
where Rpacked and Ropen are the electrical resistances of the packed and open segments having lengths Lpacked and Lopen. Aopen is the area of the capillary lumen, Fpacked and Fopen are the resistivities of the respective segments, V is the total potential difference across the column, and Vpacked and Vopen are the potential differences across the packed and open segments. Evidently, FpackedLpacked/ Aopen and FopenLopen/Aopen in eq 1 represent the respective electrical resistances that are dependent on the length and cross-sectional area of the segment. It follows from eq 1 that, once the resistivities of the two segments are known, the respective potential drops and field strengths can be conveniently evaluated as in the ensuing discussion. Evaluation of Resistivity of the Packed and Open Segments. Evaluation of Fpacked and Fopen requires currents measured in two columns that have different fractional packed lengths but otherwise identical properties. The parameters of two such columns are denoted by the superscripts I and II, using which the resistivities in the packed and open segments, Fpacked and Fopen, are evaluated using eq 1 as
Fpacked )
1 σpacked
)
V[iIILIIopen - iILIopen]Aopen LiIiII(LIpacked - LIIpacked)
and 3070 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
(2)
1 σopen
V[iILIpacked - iIILIIpacked]Aopen
)
(3)
LiIiII(LIpacked - LIIpacked)
where L is the total length of the capillary and σ is the conductivity of the respective segment, given by the reciprocal of the resistivity. For the sake of simplicity, we may choose one of the columns to be an open capillary. Further, it follows from the above discussion that resistivity of the packed segment, Fpacked, is independent of its length and the capillary radius and so can be used as a measure of the quality of packing. Evaluation of the Potential Drop across the Packed and Open Segments. Once Fpacked and Fopen are known, the potential drops across the two segments of either column with packed length, Lpacked, are evaluated by using the following expressions:
Vpacked ) V
( ) (
)
Rpacked FpackedLpacked )V R FpackedLpacked + FopenLopen
Vopen ) V - Vpacked ) V
(
)V
(4)
( ) Ropen R
)
FopenLopen FpackedLpacked + FopenLopen
(5)
where R is the total resistance of the column. Equations 4 and 5 express that the total potential drop, V, is distributed across the packed and open segments of the column according to the relative magnitude of their resistances. Figure 2 shows that, with increasing dimensionless packed length, λ ) Lpacked/L, the total resistance of the column and, hence, the potential drops across the two segments vary nonlinearly according to eqs 4 and 5. As λ goes from 0 to 1, the two potential drops, Vpacked and Vopen, monotonicaly increase and decrease from zero to V and V to zero, respectively. Further, a higher conductivity ratio, σopen/σpacked, means a higher resistivity of the packed segment. Thus, a higher potential drop across the packed segment is required to generate current equal to that in the open segment, causing an increase in the nonlinearity of V vs λ plots in Figure 2. Since the total potential drop is kept constant, the plots of Vpacked and Vopen against λ are mirror images of each other in Figure 2. The data plotted in Figure 2 were obtained by measuring the current in columns differing only in the length of packed segment, Lpacked, under otherwise identical conditions.15 The respective values of Fpacked and Fopen were estimated by using eqs 2 and 3, and the average values were calculated as 123.8 and 40.2 Ω m. Thus, the conductivity ratio, σopen/σpacked, was 3.1, and the potential drops across the two segments, Vpacked and Vopen, were evaluated using eqs 4 and 5 and shown in Figure 2. Evaluation of the Equivalent Length of the Packed Segment. Recently, prior works23,24 on EOF through porous media in CEC were briefly reviewed20 without drawing a conclusive picture of the actual flow field in CEC columns. In the present (23) Overbeek, J. T. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: New York, 1952; pp 194-244. (24) Masliyah, J. H. Electrokinetic Transport Phenomena; Aostra: Alberta, Canada, 1994; pp 103-123.
Figure 2. Plots of the potential drop across the packed and the open segments of a CEC column against the dimensionless packed length, λ, with conductivity ratio, σopen/σpacked, as the parameter. Experimental data used in the calculations are from the literature.15 The respective currents with columns having packed segments of 0, 10, and 20 cm were reported as 6.6, 3.9, and 2.6 µA, respectively. The conditions were fused silica capillary, 50 µm × 30 cm; applied voltage, 18 kV; column packing, 3.5 µm Zorbax ODS, 80 Å; mobile phase, 50% (v/v) ACN in 10 mM sodium borate, pH 8.0.
case, we are interested only in the estimation of the volumetric flow rate in the packed segment. For the sake of simplicity, we may replace it by a hypothetical open tube of length, Le, that is the same as the distance traveled by the neutral and inert tracer in the packed segment. The lumen of this hypothetical tube, Apacked, is assumed to be the same as the free cross-sectional area of the packed column, so that Apacked ) πae2, where ae is the radius of the hypothetical tube. The equivalent length, Le, is determined from the ratio of the conductivities of the packed and the open segments of the column, as20,23
x
Le L ) Lpacked Lpacked
1+
(
) ( )
Lpacked σopen Lopen -1 L σpacked Lpacked
(6a)
Alternatively, the equivalent length can be evaluated directly from the current through the CEC column in the absence and presence of the packing, i* open and i* packed, respectively.
x
Le L ) Lpacked Lpacked
i*open
i*packed
-
( ) Lopen Lpacked
(6b)
According to eq 6a, Le depends on the length of the packed segment, Lpacked, as well as the conductivities of both the mobile and stationary phases. The ratio of lengths, Le/Lpacked, as a function of λ is plotted in Figure 3 with the conductivity ratio, σopen/σpacked, as the parameter. A higher σopen/σpacked ratio signifies an increase in the resistivity of the packed segment, which is equivalent to an increase in Le/Lpacked. Further, if the resistivities of the two segments are close, as for σopen/σpacked ) 1.5 or 3.1, Le/Lpacked is nearly a constant, as assumed in the following discussion. Evaluation of the Electric Field Strengths in the Packed and Open Segments. As discussed earlier, the potential drop across both the packed and the open segments of the column
Figure 3. Plots of the ratio of the equivalent and the actual lengths of the packed segment, Le/Lpacked, as calculated according to eq 6a, against λ with conductivity ratio, σopen/σpacked, as the parameter. Conditions are the same as in Figure 2.
can be calculated from the properties of the two segments and the applied voltage. The actual electric field (under which the migrants travel) in the packed segment, Epacked, can be evaluated from the pertinent potential drop, Vpacked, and the equivalent length, Le, as follows:
Epacked ) Vpacked/Le
(7)
whereas the electric field in the open segment is obtained by the relationship
Eopen ) Vopen/Lopen
(8)
Evidently, both Epacked and Eopen are quite different from the “fictitious electric field”, E*, that is obtained by dividing the total potential drop across the column, V, by the total column length, L.
E* ) V/L
(9)
Clear distinction between Epacked, Eopen, and E* is of particular importance when accurate knowledge of the electric field strength is required for evaluation of the electroosmotic and electrophoretic mobilities from the observed migration times. Similar situation arises in other systems that are subject to similar conservation principles, e.g., displacement electrophoresis. Figure 4 illustrates plots of the above three electric field strengths against the dimensionless packed length λ, according to eqs 7-9 with σopen/σpacked ) 3.1. It is seen that the field strength is higher in the packed than in the open segment, i.e., Epacked > Eopen. Both field strengths Epacked and Eopen increase with decreasing λ, and concomitantly the mobile-phase velocity increases in both segments. Figure 4 also illustrates the inequality of the actual electric fields in the two segments, which serves as a caveat that, instead of the fictitious electric field strength, E*, the actual electric field strengths, Epacked and Eopen, should be evaluated and reported together with other chromatographic conditions in CEC. FLOW VELOCITIES IN THE PACKED AND OPEN SEGMENTS Conservation of Volumetric Flow Rate. The packed and open segments of the CEC column being connected in series, Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
3071
Figure 4. Plots of the actual, Epacked and Eopen, and the fictitious, E*, electric field strengths in the packed and the open segments, as calculated from eqs 7-9, against λ for case I with σopen/σpacked ) 3.1. At the top, the discontinuity in the electric field strength at the interface of the two column segments is illustrated. The rest of the conditions are as in Figure 2.
the mass conservation law requires that the volumetric flow rate of the mobile phase be the same in the two segments. The actual flow velocities in the open and packed segments, denoted by a a ueo,packed and ueo,open and taken as the average velocities over the total free cross-sectional area, can be related as follows:
uaeo,packedApacked ) uaeo,openAopen
(10)
However, if the two segments were operated individually under conditions prevalent in the CEC column, the “virtual” flow velocities would be given by the following expressions:
( )
ueo,packed ) µeo,packed
Vpacked Le
(11)
and
( )
ueo,open ) µeo,open
Vopen Lopen
(12)
the open segment due to the tortuosity effects, e.g., raw fused silica capillary is packed with raw or octadecylated silica. Case II: ueo,packedApacked ) ueo,openAopen. Such an equality may occur in practice when the conductivites and relative lengths of the packed and the open segments are such that the velocities in the two segments, as given by eqs 11 and 12, fulfill the conservation of volumetric flow rate given by eq 10. Case III: ueo,packedApacked > ueo,openAopen. The flow rate generated by the packed segment in this case is higher than that generated by the open segment, e.g., the capillary inner wall is neutral and the stationary-phase surface is charged, so that flow is generated mainly in the packed segment. Flow-Equalizing Intersegmental Pressure. As discussed above, if each segment were operated individually, it would generate the virtual flow velocity given by eq 11 or 12. However, conservation of volumetric flow rate in the form of eq 10 puts a restriction on the flow velocities. A close analysis of the various variables in the system and of the different possibilities that may occur leads to the conclusion that pressure at the interface of the packed and open segments, Pi, has to be different from that at the inlet and outlet of the CEC, P0. This “flow-equalizing intersegmental pressure”, Pi, provides a mechanism to alter the a a and ueo,open , to satisfy velocities, ueo,packed and ueo,open, to ueo,packed the conservation of volumetric flow rate given by eq 10. Development of similar local pressure gradients inside the packed bed as a result of abrupt velocity changes in the vicinity of packed particles was mentioned in a recent publication.25 Figure 5a illustrates pressure distribution in a typical CEC column when ueo,packedApacked < ueo,openAopen (case I) and the intersegmental pressure, Pi, is smaller than the ambient pressure, P0. In the open segment, this pressure gradient builds up opposite to the direction of the EOF, slowing the flow and making the volumetric flow rate equal to that in the packed segment. There is a continuous dissipation of energy in the system in the form of work done by this induced pressure gradient in equalizing the flow rates in the two segments. The electrical energy being the sole input in the system, part of it is dissipated in this manner. Following earlier analysis of EOF in porous media,20,23 the packed segment is represented by a hypothetical, open tube having the equivalent length of Le. The detailed analysis is presented in the Appendix, and the final expression for the flowequalizing intersegmental pressure, Pi, is given by eq A6, which can be rewritten in terms of the dimensionless packed length, λ, as follows:
P i ) P0 where µeo,packed and µeo,open are the respective mobilities of the EOF marker measured independently in the packed and open segments of the column. Analysis of the Three Possible Cases in CEC. Even though the flow velocities as given by eqs 11 and 12 are “virtual”, they are useful for the estimation of the EOF-generating ability of the packed and the open segments. Thus, they can be used to characterize the two segments and distinguish among the three possible cases in CEC according to the relative magnitude of the volumetric flow rates, ueo,packedApacked and ueo,openAopen. Case I: ueo,packedApacked < ueo,openAopen. This is the case when the zeta potential at the packing surface is such that the volumetric flow rate generated in the packed segment is lower than that in 3072 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
[
µeo,openAopenVopen - µeo,packedApackedVpacked(1 - λ)L/Le [Bo(1 - λ)Aopen/ηλ] + (a2Aopen/8η)
]
(13) Equation 13 shows that the magnitude of the intersegmental pressure depends on the applied voltage and the lengths, free cross-sectional areas, and resistivities of the two column segments. The pressure difference, Pi - P0, as calculated from eq 13 is plotted as a function of the dimensionless packed length, λ, with the conductivity ratio as the parameter in Figure 5b. The pressure (25) Poppe, H. J. Chromatogr. A 1997, 778, 3-21.
intersegmental pressure-driven flow so that, for the most general case, we obtain that
uaeo,packed )
( )(
) ( )[
P0 - Pi BoAopen + Le ηApacked µeo,packed
]
2I1(κae) Vpacked 1(14) Le κaeI0(κae)
and
uaeo,open ) -
( )
( )[
P0 - Pi a2 Vopen 2I1(κa) + µeo,open 1Lopen 8η Lopen κaI0(κa)
]
(15) where I0 and I1 are the zeroth- and first-order Bessel functions. Equations 14 and 15 can further be simplified when the doublelayer thickness is smaller than 2% of the mean channel diameter, i.e., κa and κae > 50.
uaeo,packed )
( )(
)
( )
(16)
( )
(17)
P0 - Pi BoAopen Vpacked + µeo,packed Le ηApacked Le
and
uaeo,open ) Figure 5. (a) Schematic illustration of the pressure gradients across the open and the packed segments of the column when σopen/σpacked ) 3.1. (b) Plot of the pressure difference, Pi - P0, across the packed segment against λ with the conductivity ratio as the parameter. Conditions used in the calculation by eq 13 are Apacked/Aopen ) 0.4; ) 0.4; η ) 10-3 N m-2 s-1, µeo,packed ) µeo,open ) 3.33 × 10-8 m2 V-1 s-1. The rest of the conditions are the same as in Figure 2.
drop across any of the segments, Pi - P0, is maximum at small λ values and becomes zero at the extremity λ ) 1. In the present case, ueo,packedApacked < ueo,openAopen, and so the pressure drop, Pi P0, is always negative. Further, it is seen in Figure 5b that, as σopen/σpacked increases, the pressure drop decreases. This is so because smaller conductivity of the packed segment leads to a higher potential drop across the packed segment and, consequently, a smaller inequality in the flow rates generated in the two segments. On the basis of the above discussion, the role of the flowequalizing intersegmental pressure in cases I-III can be described as follows. Case I. In this case, ueo,packedApacked < ueo,openAopen, and so, as discussed above, Pi < P0 in order to equalize the flow rates in the open and packed segments. Case II. Here, ueo,packedApacked ) ueo,openAopen, and so it follows from eq 13 that Pi ) P0. Case III. This case is opposite of case I in that ueo,packedApacked > ueo,openAopen, and consequently it follows from eq 13 that Pi > P0. Mean Flow Velocities in the Packed and Open Segments. The flow velocities through the packed and open segments in series can be expressed by contributions from the EOF and the
(
)
P0 - P i a2 Vopen + µeo,open Lopen 8η Lopen
When the packed segment is much less permeable than the open segment, the flow velocity in the packed segment is not significantly influenced by the pressure gradient. Then, the actual velocity in the packed segment turns out to be identical to the virtual velocity, as given by eq 10, and we obtain that
( )
uaeo,packed ) ueo,packed ) µeo,packed
Vpacked Le
(18)
Since, in columns most commonly used in CEC at present, the detection window is located right after the packing, the chromatograms will not be palpably affected by the pressure gradient. Therefore, once the potential drop across the packed segment is evaluated, eq 18 gives a satisfactory estimation of the actual interstitial flow velocity through the packed segment of the column. Figure 6 illustrates the dependence of the mean actual mobilephase velocities through the interstices of the packed segment, a a ueo,packed , and in the open segment, ueo,open , when σopen/σpacked ) 3.1. As discussed in the Appendix and expressed in eq A8, even a a though ueo,packed is higher than ueo,open , the velocity of the inert tracer (chromatographic flow velocity) can be lower in the packed segment than in the open segment when the particles of the packing are porous. To illustrate the effect of the intersegmental pressure on the flow velocity in the open segment, the open segment velocity was plotted against λ in Figure 6 by using eqs 12 and 17. The magnitude of this effect is greater at low λ and appreciable also in the range 0.6 < λ < 0.9, which is likely to be of practical Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
3073
As mentioned earlier in discussing eqs 16 and 17, the flow field in the open segment may be affected strongly by the axial pressure gradients, whereas in the packed segment the flow field is expected to remain unaltered. This is illustrated in Figure 7, where the flow velocity in the open segment, ueo,open(r), is plotted against the dimensionless radial coordinate, r/a, with λ as a parameter for the three cases defined earlier. Case I. The conservation of volumetric flow rate yields the following expression for the radial distribution of the flow velocity in the open segment:
uaeo,open(r) ) -
( ) [ ( )] ( )[ P0 - P i a2 r 1Lopen 4η a
µeo,open
Figure 6. Dependence of the flow velocity in the interstices of the packed segment as calculated by eq 18 and the velocities in the open segment as calculated by eqs 12 and 17 on λ for case I and σopen/ σpacked ) 3.1. The inequality of the two velocities in the open segment, ueo,open and uaeo,open, illustrates the effect of the pressure differential, Pi - P0. The schematic of the CEC column at the top illustrates the discontinuity in flow velocity at the interface of the two column segments. The rest of the conditions are as in Figure 5.
2
+
]
Vopen I0(κr) 1(19) Lopen I0(κa)
where r is the radial coordinate. Equation 19 can be further simplified for κa > 50 as
uaeo,open(r) ) -
(
) [ ( )]
P0 - P i a2 r 1Lopen 4η a
2
( )
+ µeo,open
Vopen Lopen
(20)
Equation 20 shows that, in this case, the velocity profile in the open segment is also the sum of two contributions: one from the EOF and other from the intersegmental pressure. As a result, as shown in Figure 7, the flat velocity profile of the electroosmotically driven flow is distorted by the presence of the pressure gradient. The resulting flow profile is reversed parabolic with the lowest velocity in the center line (r/a ) 0). Further, it can be seen in Figure 7 that, with decreasing λ, the flow velocity increases and the flow profile becomes more parabolic. Case II. Since Pi ) P0, eq 20 simplifies to
( )
uaeo,open(r) ) µeo,open Figure 7. Fully developed flow profiles in the open segment of the CEC column for the three cases, as calculated according to eqs 2022, with λ as the parameter. The dimensionless radial coordinate is given by r /a. Conditions are as in Figure 6.
importance. As shown by Figure 6, the velocities decrease about 3-fold in both the packed and the open segments when λ increases from 0 to 1. VELOCITY FIELD AND BAND SPREADING IN THE OPEN SEGMENT Velocity Profiles. So far, we have examined the mean velocities in the two segments averaged over the free crosssectional area of the respective segment. In the following, we discuss the effect of the flow-equalizing intersegmental pressure on the flow profiles in the open segment. The results presented in this section strongly support recent findings26 and explain the flow profiles that were obtained via flow visualization in the CEC column right after the intersegmental frit. (26) Zare, R. N.; Rakestraw, D. J.; Anex, D. S.; Dadoo, R.; Zhao, H.; Nier, D.; Chen, J.-R.; Dulay, M.; Yan, C. Presented at the 49th Pittsburgh Conference, New Orleans, LA, March 1-5, 1998; Lecture No. 1277.
3074 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
Vopen Lopen
(21)
Equation 21 predicts a perfectly flat flow profile, as shown in Figure 7, since the pressure effects are absent. As λ increases, the electric field strength across the open segment decreases, resulting in a reduction of the flow velocity. Case III. For the special case when the inner wall of the open segment is neutral and no EOF is generated there, eq 20 becomes
uaeo,open(r) )
(
) [ ( )]
Pi - P0 a2 r 1Lopen 4η a
2
(22)
which is the equation of parabolic flow profile obtained with pressure-driven laminar flow, as in the open segment only viscous flow prevails. As shown in Figure 7, the pressure gradient increases with decreasing λ, and as a result, the flow velocity increases in the open segment. Effect of Parabolic Flow Profile on Band Spreading. As mentioned above, the flow-equalizing intersegmental pressure has a major effect on the flow field in the open segment, and deviations from the flat flow profile will engender additional band spreading in the open segment. We recall that the flow field in the packed
segment is not affected much by the axial pressure gradient, and, therefore, bandspreading in the packed segment is expected to remain unaltered. In the following, we assume that the column is held straight during operation and that the gravitational effects on the flow profile are negligible. Using the expressions for the velocity profiles given above and following well-established procedures,27-29 to quantify the band spreading by the plate height, H, we obtain the following results. Case I. In the absence of adsorption at the capillary wall, the plate height for the open segment, Hopen, with combined flat and parabolic flow profile, is obtained by eq 20 as
Hopen ) Hopen,LD + Hopen,FP )
2Dm uaeo,open
+
[(
)]
P0 - P i a2 a2 a 24Dmueo,open Lopen 8η
2
(23)
where Hopen,LD is the plate height contribution due to longitudinal diffusion, Hopen,FP is due to the nonflat flow profile, and Dm is the molecular diffusivity of a neutral tracer. Hopen and the two plate height contributions, Hopen,LD and Hopen,FP, are plotted against the dimensionless packed length, λ, in Figure 8. The plate height contribution from the packed segment of the column was assumed to be 4.5 µm. It follows from Figure 8 that band spreading in the open segment due to the parabolic velocity profile is appreciable and needs to be accounted for. Case II. The flow in the open segment is solely electroosmotic; hence, Hopen is expressed by the longitudinal diffusion term, Hopen,LD, as
Hopen ) Hopen,LD )
2Dm uaeo,open
(24)
Case III. For the special case when the inner wall of the open segment is neutral, the flow field is pressure-driven laminar flow and the plate height can be evaluated in a similar manner as in case I. It is the sum of the plate height contributions due to longitudinal diffusion and the parabolic flow profile and is given by
Hopen ) Hopen,LD + Hopen,FP )
2Dm uaeo,open
+
Figure 8. Plots of the plate height contributions due to longitudinal diffusion, Hopen,LD, flow profile effects, Hopen,FP, and their sum, Hopen,total, in the open segment of a CEC column, against λ. The plate height in the packed segment, Hpacked, is assumed to be equal to 4.5 µm. Conditions used in the calculations by eq 23: Dm) 1 × 10-9 m2 s-1. The rest of the conditions are as in Figure 6.
a4(P0 - Pi) (25) 192DmηLopen
CONCLUSIONS The results presented in this work are expected to further the understanding of flow through the CEC column and of the generation and control of the EOF. It has been our goal to provide a set of equations to calculate the various pertinent electrochromatographic parameters including electrical resistivities, actual potential drops, electric field strengths, and flow velocities in the different column segments. In handling and reporting experimental results, it is important that the parameters be evaluated (27) Taylor, G. Proc. R. Soc. London A 1953, 219, 186-203. (28) Aris, R. Proc. R. Soc. London A 1956, 235, 67-77. (29) Golay, M. J. E. in Gas Chromatography; Desty, D. H., Ed.; Academic Press: New York, 1958; p 36.
Figure 9. Sequential evaluation of various electrochromatographic parameters from experimentally measured quantities in the two upper blocks by using the indicated equations of the present work.
in the correct manner. To facilitate this, in Figure 9 we have presented a scheme to carry out such calculations. The flow-equalizing intersegmental pressure may have several important practical implications in some cases besides those discussed above. For instance, when a partly packed column followed by a predetection open segment is used for faster separation of a mixture of neutral and charged components,21 deviations from the flat flow profile in the open segment will engender additional band spreading. Similarly, when the CEC system is connected to another analytical instrument, such as a mass spectrometer (MS), by a long, open capillary, the flow conditions in the connecting tube will determine the efficiency and speed of analysis by the coupled system. The above discussion has focused on the interface of the packed and open segments. Yet, similar discontinuities in the velocity and electric field strength are encountered at any sudden change in the properties of the packing. Silica frits most likely have zeta potentials different from those of the bulk packings, and the discontinuity of zeta potential can lead to development of flow-equalizing intersegmental pressure at such frits with concomitant bubble formation. This can be particularly severe when Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
3075
the zeta potentials of the frit and the packing are drastically different.
(
-
ACKNOWLEDGMENT This work was supported by Grant No. GM 20993 from the National Institutes of Health, U.S. Public Health Service.
( )
)
( )
P0 - Pi a Vopen A + µeo,openAopen (A5) Lopen 8η open Lopen 2
Equation A5 can be solved to obtain an expression for the intersegmental pressure, Pi, as
APPENDIX The conservation of volumetric flow rate for an incompressible fluid under conditions of steady state can be expressed19,24,30,31 as follows:
( )
BoAopen P0 - Pi + η Lpacked
( )[
µeo,packedApacked -
) (
Vpacked BoAopen P0 - Pi + µeo,packedApacked ) η Lpacked Le
( )
]
Vpacked 2I1(κae) 1) Le κaeI0(κae)
( )[
P0 - Pi a2 Vopen 2I1(κa) Aopen + µeo,openAopen 1Lopen 8η Lopen κaI0(κa)
]
(A1)
where κ is the Debye screening parameter, taken as the reciprocal of the double-layer thickness, η is the viscosity of the mobile phase, a is the radius of the capillary, and Bo is the specific Darcy’s law permeability. The Kozeny-Carman equation is often used to evaluate Bo for packed beds and is given by
dp2 3 B ) 180 (1 - )2 o
(A2)
where dp is the particle diameter and is the interstitial porosity of the packed bed. Equation A1 can be simplified by using the property of the zeroth- and first-order Bessel functions that, for κa and κae > 3.5, they are related by
I0(κa) ) I1(κa) and I0(κae) ) I1(κae)
(A3)
Pi ) P0 -
[
µeo,openAopenVopen - µeo,packedApackedVpacked(Lopen/Le) (BoLopenAopen/ηLpacked) + (a2Aopen/8η)
]
(A6)
Analysis of “voltage-assisted chromatography”, or “pressureassisted CEC”, with simultaneous voltage and pressure gradients,8,19,32 can be carried out in a similar fashion. In this case, the pressure differential, P0 - Pi, on the left-hand side of eq A1 is replaced by P1 - Pi, and the one on the right-hand side is replaced by P2 - Pi, where P1 and P2 are the pressures at the column inlet and outlet, respectively. Simplification of eq A1 and estimation of the intersegmental pressure, Pi, can then be carried out in an identical fashion with the result (cf. eq A6) as follows:
Pi ) P1 - {[(P1 - P2)a2Aopen/8η + µeo,openAopenVopen - µeo,packedApackedVpacked(Lopen/Le)]/ [(BoLopenAopen/ηLpacked) + (a2Aopen/8η)]} (A7)
A comparison of eqs A6 and A7 yields that the two equations are identical except for the first term in the numerator of eq A7, due to the applied pressure gradient across the column. Since, in the present case, we are analyzing the system with identical pressures at both the ends, we will be using eq A6 for evaluation of the flow-equalizing intersegmental pressure. As in HPLC, there are three different types of mobile-phase velocities in CEC: superficial velocity, us; interstitial velocity, u; and chromatographic velocity, uc. These three velocities can be related to each other in the following manner:33
uc ) u(/T) ) usT
(A8)
so that eq A1 can be rewritten as
( ) ( )
( )[ ] ( )[ ]
BoAopen P0 - Pi Vpacked 2 + µeo,packedApacked 1) η Lpacked Le κae -
P0 - Pi a Vopen 2 + µeo,openAopen 1(A4) A Lopen 8η open Lopen κa 2
where T is the total porosity of the packing. Evidently, for porous packings, the total porosity being higher than the interstitial porosity, the chromatographic velocity is always lower than the interstitial velocity. LIST OF SYMBOLS AND ABBREVIATIONS a
radius of the capillary tube, m
Equation A4 can be solved for obtaining the intersegmental pressure, Pi, for any double-layer thickness, and when κa and κae > 50, it can be further simplified as
ae
radius of the hypothetical tube, m
Bo
Darcy’s law specific permeability m s2
A
free cross-sectional area, m2
(30) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. (31) Kobayashi, K.; Iwata, M.; Hosoda, Y.; Yukawa, H. J. Chem. Eng. Jpn. 1979, 10, 466-471.
(32) Verheij, E. R.; Tjaden, U. R.; Niessen, W. M. A.; van de Greef, J. A. J. Chromatogr. 1991, 554, 339-349. (33) Horva´th, Cs., Lin, H.-J. J. Chromatogr. 1976, 126, 401-420.
3076 Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
dp
particle diameter, m
Greek Letters m2
s-1
Dm
molecular diffusivity of a neutral tracer,
E
electric field across the segment, V m-1
E*
fictitious electric field across the segment, V m-1
H
HETP of the segment, m
i* open
current generated in an open capillary, A or V Ω-1
i* packed
current generated in a packed CEC column, A or V Ω-1
interstitial porosity of the packed bed
T
total porosity of the packed bed
η
viscosity of the mobile phase, N s m-2
κ
reciprocal of the thickness of double-layer, m-1
σ
conductivity of the segment, Ω-1 m-1
λ
dimensionless packed length, Lpacked/L
µ
mobility of the sample component in the interstitial space, m2 V-1 s-1
µa
actual mobility of the sample component in the interstitial space, m2 V-1 s-1
F
resistivity of the segment, Ω m
I0
Bessel’s function of the zeroth kind
I1
Bessel’s function of the first kind
k′
chromatographic retention factor of a sample component in CEC
L
length of the segment, m
P0
pressure at the two ends of the column, N m-2
Subscripts
P1
pressure at the inlet of the column, N m-2
packed
parameters pertaining to the packed segment
P2
pressure at the outlet of the column, N m-2
open
parameters pertaining to the open segment
Pi
flow-equalizing intersegmental pressure at the interface of the packed and open segments, N m-2
e
parameters pertaining to the hypothetical tube assumed to replace the packed segment
eo
parameters pertaining to electroosmotic flow
r
coordinate in the radial direction, m
Acronyms
R
electrical resistance of the segment, Ω
u
interstitial velocity of the sample component, m s-1
uc
chromatographic velocity of an unsorbed sample component, m s-1
CEC
capillary electrochromatography
CZE
capillary zone electrophoresis
EOF
electroosmotic flow
HPLC
high-performance liquid chromatography
us
superficial velocity of the mobile phase, m s-1
ua
actual interstitial velocity of the sample component, m s-1
Received for review November 18, 1997. Accepted April 17, 1998.
V
potential drop across the segment, V
AC971260A
Analytical Chemistry, Vol. 70, No. 14, July 15, 1998
3077
