Anal. Chem. 1985, 57, 559-561
559
CORRESPONDENCE Peak Broadening in Open-Tubular Liquid Chromatography with Electroosmotic Flow Sir: Some experiments recently carried out on liquid chromatographic separations in open tubes (1-3) as well as in packed columns (4-6) with electroosmotic flow have raised a great deal of interest among chromatographers, as they showed improved performances over pressure-driven, laminar flow systems. The nearly flat velocity profile of the electroosmotic flow is expected to lead to a very small contribution to peak broadening from the resistance-to-mass transfer in the mobile phase. A quantitative determination of this contribution as a function of retention in open-tubular columns has been performed in a recent study (7). It was shown that, although very low for unretained compounds, the contribution increases significantly with increasing retention. At each value of the retention, it remains lower than in conventional chromatography. Because the exact analytical expression of the velocity profile in electroosmotic flow is too complex, an approximate expression has been used for the computation of the contribution to peak broadening of the resistance to mass transfer in the mobile phase. In spite of the simple form of the flow profile used (a partially flat, partially parabolic velocity profile), the resulting contribution to the plate height equation is relatively complex and must itself be approximated for practical purposes (7). The aim of the present work is to provide another approximate expression of the true electroosmotic flow profile which leads to both a much simpler and more accurate peak broadening term and which can easily be used for system characterization as well as for optimization and comparison purposes. RESULTS AND DISCUSSION The expression of the velocity profile of the electroosmotic flow in capillary tubes has been derived by Rice and Whitehead as (8)
velocity is then approximated as (7) -
[
4a7
1--
211(Ka)
KaIo(Ka)
]
Ka
1 (Ka)2
1 +-+4(Ka)3
(3)
(4)
uo is the maximum velocity along the axis, and n is an approximating exponent. The cross-sectional average velocity is easily determined as u/uo = n/(n
+ 2)
(5)
This profile can be used for the determination of the dimensionless parameter C, in the plate height term H ,
H, =
c, a2u urn
using the framework of the Aris generalized dispersion theory (9) which gives
(A1 - 2A2 =
+ (1/4)) + 2(A, - A J k ' + (1 + k ' ) 2
A1hr2 (7)
where k'is the solute capacity factor and Al and A2 are given by 1
P 2 ( x ) / xdx
A, =
(2)
0003-2700/85/0357-0559$0 1.50/0
+ *..
u / u o = 1- ( r / a ) "
and
In eq 1 and 2, lo(%) and Zl(x) are, respectively, the zero-order and first-order modified Bessel functions of the first kind. It must be noted that eq 1 and 2 apply only for relatively small values of the zeta potential for which the dimensionless parameter K a , called the electrokinetic radius of the system, is relatively large, so that there is insignificant overlap of the double layers in the center of the capillary. In this case, the velocity along the tube axis, uo, is nearly equal to e { E % / 4 ~ 7 . Indeed, the relative difference between these two values is less than 5 X when Ka exceeds 10. The relative average flow
1 4(Ka)4
Due to the nature of Bessel functions, the mathematics required to solve for the contribution of the resistance to mass transfer in the mobile phase are formidable. Thus it is desirable to approximate u ( r ) with a simple polynomial, in the derivation of an expression for the corresponding plate height term, H,. In the previous study the flow profile was approximated by a composite profile, part flat, near the axis, and part parabolic, near the wall. In spite of the simplicity of this quadratic expression which depends on one parameter, reflecting the extent of the constant velocity region, the resulting H , term is relatively complex (7). It is proposed that the following approximation of the flow velocity profile for electroosmosis in a capillary be used:
c, where e, {, E , 7,K , r , and a represent the eluent dielectric constant, the zeta potential, the applied electrical field, the eluent viscosity, the reciprocal of the electrical double layer thickness at the tube-eluent interface, the distance from the tube axis, and the column radius, respectively. The crosssectional average flow velocity, u, is then equal to
= I - - +2-
"0
with x = r/a
and
1985 American Chemical Society
560
ANALYTICAL CHEMISTRY, VOL. 57, NO. 2, FEBRUARY 1985
4
0 V
2
0 L
2 m
-
0
0,
> o
-m
E
s 0 -2
r la
Flgure 1. Velocity profiles vs. relative distance from column axis: (soli curve) exact calculation for electroosmotic flow wRh ~a = 10; (dotted curve) approximative flow profile given by eq 4 with n = 8.5413; (dashed curve) partially flat, partially parabolic flow profile with p = 0.3083 (ref 7). The three profiles give the same value of the crosssectional average velocity relative to the velocity along the axis.
Table I. Comparison of the C , Values for the Exact Electroosmotic Flow Profile and the C , Values Obtained for Two Approximate Flow Profiles Giving Identical Relative Average Velocities, Ka = 10
true electroosmotic flow profile C,
k'
0.0162 0.0605 0.1081 0.1695 0.2449 0.2837 0.3341
0.1 0.5 1
2 5 10 m
partially flat, partially parabolic flow profileb 70 re1 C, error
flow profile given by eq 4" % re1 C, error 0.0169 0.0619 0.1099 0.1718 0.2476 0.2867 0.3373
" n = 8.5413 according to eq 19. of ref 7.
4.3 2.3 1.7 1.4 1.1 1.1 1.0 *p
0.0183 0.0645 0.1134 0.1761 0.2528 0.2922 0.3433
13.0 6.6 4.9 3.9 3.2 3.0 2.8
= 0.3083 according to eq 28
+ 2)2
4(n + 2)
4n2
n2(n + 4 )
(n
A 2 = -n- -+ 2 4n
+
2 n 2 ( n+ 2 )
(15)
2
n(n + 4 )
The plate height parameter C, is then obtained by combining eq 7 , 15, and 16
c,
=
4
+ ( 4 n + 16)k'+
ria
Figure 2. Relative errors in the approximate flow profile as a function of radial position, Ka = 10: (curve a) flow profile given by eq 4 with n = 8.5413; (curve b) partially flat, partially parabolic flow profile with p = 0.3083 (ref 7).
Table 11. Comparison of the C , Values for the Exact Electroosmotic Flow Profile and the Values Obtained for Two Approximative Flow Profiles Giving Identical Relative Average Velocities, Ka = 50
true electroosmotic flow profile
cin
k'
0.004093 0.034 41 0.072 30 0.124 1 0.189 7 0.224 2 0.269 3
0.1 0.5 1
2 5 10 m
Using the combination of eq 4 and 5 into eq 11 to perform the integrations 11, 9, and 8, one gets
AI=--------
-4
(n2+ 10n
+ 20)k'2
4(n + 2)(n + 4)(1 + kq2
(17)
partially flat, partially parabolic flow profileb 70 re1 C, error
flow profile given by eq 4" 70 re1 C, error 0.004175 0.034 50 0.072 40 0.124 2 0.189 9 0.224 3 0.269 4
= 48.51 according to eq 19. of ref 7. O n
2.0 0.3 0.1
0.09 0.06 0.05 0.05
*p
0.004219 0.034 63 0.072 59 0.124 4 0.190 2 0.224 6 0.269 8
3.1 0.6 0.4 0.3 0.2 0.2 0.2
= 0.06031 according to eq 28
It is worthwhile to point out that if n = 2 in eq 4, it becomes the equation for the velocity profile for laminar flow in a cylindrical capillary. Substituting n = 2 into eq 17 should give the Golay equation (IO),which is in fact the case. Equation 17 is much simpler than the equation obtained for a partially flat, partially parabolic flow profile. In order to apply it, it is necessary to obtain a good estimate of the parameter n. There is an infinite number of ways to estimate it. Indeed, as the flow profiles given by eq 1 and 4 are different, n can be chosen so that the relative velocities, u / u o , are equal for any one given radial position rla. In spite of the arbitrary character of the mode of selection of n, it is thought that a logical one is to assume that the relative average velocities, u / u o ,are the same for both profiles which gives, by combination of eq 3 and 5
- -n n + 2
- I - - +2 Ka
1 (KU)'
1 +-+4(Ka)3
1 (18) 4(Ka)4"'
Anal. Chem. 1985, 57, 561-563
and consequently 3
3
2
8Ka
n=Ka--+-
(
1 1 + Ka -+
...)
(19)
In the limit of large KU values, n becomes equal to KU - 3/2. The relative velocity profiles given by eq 1 and 4 are plotted on Figure 1 for the case where KU = 10 and n = 8.5413, according to eq 19. The partially flat, partially parabolic flow profile, giving the same u / u o value is also plotted on Figure 1. It corresponds to a p value of 0.3083, where p represents the fractional distance from the wall where a radial velocity gradient exists. On Figure 2, the relative errors in these approximate flow profiles are plotted as a function of radial position. It is obvious from Figures 1 and 2 that the flow profile given by eq 4 better approximates the true electroosmotic flow than the partially flat, partially parabolic velocity profile. I t should be noted that the differences between the true and approximative flow profiles decrease with increasing Ka values. It is therefore expected that the C, value given by eq 17 will be in closer agreement with the true C, value than the one obtained with the previously studied velocity profile (7). The C, values for the true electroosmotic profile have been numerically computed according to the h i s dispersion theory for the two cases KU = 10 and KU = 50. They are compared with C, values given by eq 17 and by the previous study (eq 10, 14, and 15 of ref 7) in Tables I and 11, respectively, for various k'values. It is seen that the C, values given by eq 17 are all better estimates of the exact values for the electroosmotic flow profile than the values previously obtained with a partially flat, partially parabolic velocity profile. In addition, the relative errors are decreasing with increasing retention and, for a given k ' value, they are, as expected, smaller a t larger K U . In all cases studied here, they are less than 570,which is quite satisfying, as precise values of K a are difficult to obtain. The conclusions which can be derived from these results are the same as those previously discussed (7). Electroosmotic flow provides a lower contribution to peak broadening than
561
pressure-driven flow, at any given value of k'. However, while this contribution is very small for unretained solutes, it increases significantly with increasing k '. consequently, the potential of open tubular liquid chromatography with electroosmotic flow lies essentially in the low k'region. It should lead to better separations of weakly retained compounds than laminar flow, while it should offer a small advantage in separations of more strongly retained compounds. The eq 4, for the flow profile, and 17, for the mobile phase nonequilibrium contribution to plate height, are simple and accurate descriptions of the electroosmotic flow processes which can be used for fully estimating the potential of electroosmotic flow, optimization of separations and comparison with other techniques or types of flow.
LITERATURE CITED (1) Tsuda, T.; Nomura, K.; Nakagawa, G. J. Chromatogr. 1982, 248, 241. (2) Tsuda, T.; Nomura, K.; Nakagawa. G. J . Chromatogr. 1983, 264, 385. (3) Jorgenson, J. W.; Lukacs. K. D. Anal. Chem. 1981, 53, 1298. V.; Hopkins, B. J.; Schieke, J. D. J. Chromatogr. 1974, 99, (4) Pretorius. nn LJ.
Jorgenson, J. W.; Lukacs, K. D. J. Chromatogr. 1981, 278, 209. Stevens. T. S.;Cortes, H. J. Anal. Chem. 1983, 55, 1365. Martin, M.;Guiochon, G. Anal. Chem. 1984, 56, 614. , Rice, C. L.; Whitehead, R. J. fhys. Chem 1965, 69,4017. (9) Aris. R. R o c . R . SOC.London, Ser. A 1959, A252. 538. (IO) Golay, M. J. E. "Gas Chromatography"; Desty. D. H., Ed.; Butterworths: London, 1958; p 36. ~
Michel Martin* Georges Guiochon Ecole Polytechnique Laboratoire de Chimie Analytique Physique 91128 Palaiseau, France Yvonne Walbroehl James W. Jorgenson Department of Chemistry University of North Carolina Chapel Hill, North Carolina 27514
RECEIVED for review September 4,1984. Accepted October 30, 1984.
Determination of Uranium by Reversed-Phase High-Performance Liquid Chromatography Sir: In spite of the large number of methods already reported ( I ) , the determination of the uranyl ion is still a relevant analytical problem because the usual techniques of atomic absorption and emission spectroscopy are not suitable for the trace analysis of U02'-+,owing to their high detection limits. Also recently spectrophotometric methods have been proposed (2-4); in some cases laser sources (5, 6) or flow injection systems (7, 8) were used. In this paper a new chromatographic procedure is proposed; this procedure is based on the separation and determination of U02'+ as a neutral complex, using reversed-phase chromatography and spectrophotometric detection. Examples of chromatographic separation of UOZ2+as a complex have been already reported, in most cases using TLC and paper chromatography. Besides the widely used organophosphorus ligands, other complexing agents were used in the mobile phase such as TTA (9),formic acid (IO),p(diethy1amino)aniline of phenylgliossal (11, 12) and in the stationary phase such as EDTA (13),NTA ( 1 4 ) , and anilines (15).
In this work the ligand 2,6-diacetylpyridine bis(benzoy1hydrazone) was used for the solvent extraction and chromatographic determination of uranium. The compounds of this
H,DIB
ciass of bis(aroy1hydrazone) derivatives are potential pentadentate chelating agents which can behave as neutral or
0003-2700/85/0357-0561$01.50/0 1985 American Chemical Society
