614
Anal. Chem. 1984, 56, 614-020
(9) Vysotskii, 2. 2.; Strazhesko, D. N. “Adsorption and Adsorbents”; Strazhesko. D. N., Ed.; Wiley: New York, 1973;No. 1, p 55. (10) Burweli, R. L.. Jr.; Pearson, R. G.; Hailer, G. L.; TJok, P. P.; Chock, S. P. Inorg. Chem. 1955, 4 . 1123-1128. (11) Smith, R. L.; Iskandarani, Z.; Pietrzyk, D. J., submitted to Anal. Chem. (12) Smith, R. L. Ph.D. Thesis, University of Iowa, July 1983. (13) Heifferich, F. “Ion Exchange”; McGraw-HIII: New York, 1962;pp 151, 507.
(14) Bidlingmeyer, B. A.; Del Rios, J. K.; Korpl, J. Anal. Chem. 1982, 54, 442-447. (15) Fritz, J. S.;Gjerde, D. T.; Pohlandt, C. “Ion Chromatography”; Huthig: Heidelberg, 1962.
FbXmVED for review September 19,1983. Accepted December
19, 1983.
Axial Dispersion in Open-Tubular Capillary Liquid Chromatography with Electroosmotic Flow Michel Martin and Georges Guiochon* Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, 91128 Palaiseau, France
Equations are developed In order to express the plate height as a functlon of retentlon when electroosmotic flow Is used to drive the mobile phase through an open tubular chromatographic column. A partly flat, partly parabollc proflle Is used to approxlmate the electroosmotic veioclty profile, uslng a dimensionless parameter, p, whlch Is linked to the electrlcal double layer thickness of the system. I t Is shown that the peak broadening arlsing in the moblle phase is small at low retentlon but increases wlth lncreaslng retention. Numerlcai calculatlons of plate helght are performed and the chromatographlc potentlal of electroosmotic flow Is dlscussed. Attractive chrornatographlcperformances wlil be obtalned only If very large voltages (in the 20-100 kV range or more) can be used. A method Is proposed for the experimental determlnatlon of the electrical double layer thlckness.
There is a strong concerted effort in liquid chromatography (LC) aimed at improving the performance level by achieving more efficient columns for the separation of highly complex mixtures found in natural samples. Conventional packed columns are excellent for the achievement of moderate separation powers, characterized by column efficiency below about 100 000 theoretical plates. Increasing the efficiency without accepting a decrease in the theoretical plate productivity (Le,, at constant number of plates per unit time of analysis) is faced with two major problems: the proper packing and coupling of a large number of column elements and the use of extremely large pressures, in excess of 1 kbar. Barring the advent of a major breakthrough in packed column technology, no good solution to these problems is in sight. An alternative solution has been examined, the use of open tubular columns which are so successful in gas chromatography. Unfortunately, it has been shown by many authors that, in order to achieve a theoretical plate productivity comparable to that of conventional packed columns, the diameter of the open tube should be about twice that of the particles used in packed columns, i.e., it should be smaller than 10 Km (1). Even in this case, the experimental conditions are difficult: a 13 m long open tubular column, 10 Km i.d., could generate 1 X lo6 plates for a compound with k’ = 3, at a reduced velocity of 16 (Le., u = 0.10 cm/s for compounds with a diffusion coefficient of 6 X lo4 cm2/s typical of many compounds in LC conditions). The analysis time would be 15 h (for k’ = 3) and the inlet pressure 42 bars. Narrower columns would be shorter and faster but wodd generate larger 0003-2700/84/0356-0814$01.50/0
pressure drops. Furthermore, very stringent technological requirements would be imposed on the injection and detection devices. For example, the detector cell should have a volume of the order of 1nL (2). Technological difficulties encountered in the development of the suitable instrumentation have led to a systematic investigation of the various possible approaches. The conclusions discussed above regarding the efficiency of open tubular columns and the optimization of their parameters are derived (1,2) from the Golay equation (3) which relates the height equivalent to a theoretical plate to the column diameter, the mobile phase velocity, the diffusion coefficient, and the capacity factor (Le., the retention), assuming the resistance to mass transfer in stationary phase is negligible (4).Then, the major contribution at reduced velocities above 5 is the resistance to mass transfer in the mobile phase, resulting from the parabolic (Poiseuille) velocity profile in an open tube when the flow is induced by a pressure differential between column inlet and outlet. This parabolic profile results from the distribution of shear stress in a viscous fluid in laminar flow. There is another way to move a stream of fluid across a narrow tube, electroosmotic flow. This has been proposed for fractionation applications by Mould and Synge (5, 6) and adapted to column LC by Pretorius et al. (7). The process known as electroosmosis relates to the movement of liquid adjacent to a solid surface under the influence of an electrical field parallel to the solid-liquid interface and is described in standard physical chemistry texts (see, for example, ref 8 and 9). The main advantage of electroosmosis, for chromatographic purposes, besides the fact there is no pressure drop buildup, is that the flow velocity becomes constant at distances, from the wall, of the order of the magnitude of the electrical double layer thickness (10). Accordingly, when this thickness is much smaller than the tube radius, the flow is essentially flat and band broadening is expected to be significantly smaller than for forced flow. Effectively, the plate height values obtained by Pretorius et al. for an unretained solute in a 1mm i.d. tube are about 10 times smaller than for pressure flow. More recently, Tsuda et al. obtained a plate height value of 5.2 Wm in a 132 pm i.d. open column, for an inert compound, which is about 30 times smaller than what should be expected with pressure flow (11). Experiments with electroosmotic flow have also been performed in packed columns (7, 12-14). However, while the above data are quite impressive, the chromatograms presented by Tsuda et al. fail to fully demonstrate a significant advantage of electroosmotic flow. In 0 1964 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
order to evaluate the potential of this flow in open tubular chromatographic columns, we theoretically investigate the peak broadening with such flow in relation to the solute retention and the thickness of the double layer relative to the column radius, in the following.
THEORY OF BAND BROADENING In capillary chromatography, three processes contribute to band broadening of component zones as they migrate through the column, namely, axial molecular diffusion, resistance to mass transfer in the stationary phase, and resistance to mass transfer in the mobile phase. The plate height, H, is, accordingly, given by the following expression:
H = -2Dm
(11)
u=uo
xI1-p;
1 - p Ix I1; u = -[(2p - 1)
Dm where u is the cross sectional average velocity, D, is the solute diffusion coefficient in the mobile phase, a is the column radius, C, is a dimensionless mass transfer term for the mobile phase, and C, is a mass transfer term for the stationary phase. Only the resistance to mass transfer in the mobile phase is affected by the shape of the velocity profiie in the column and, hence, by the type of flow through the coefficient C,. For open tubular columns operated classically with a Poiseuille flow profile, C, and C, are given by the Golay equation (3). In order to evaluate the C, term, one should make use of the Aris generalized dispersion theory (15), already applied to a variety of situations, including turbulent flow (16). According to this theory, C, is given by U
calculation of C,. Therefore, we decided to approximate the profile as follows. The velocity is constant, equal to uo for x < 1- p (0 < p < 1)and decreases quadratically for 1- p < x < 1, with the following conditions: u = uo for x = 1 - p, u = 0 for x = 1 and du/dx = 0 for x = 1 - p . Therefore, this model consists of a core region with constant flow and a parabolic flow in the vicinity of the wall, without any discontinuity of either the velocity or the velocity gradient. It has one adjustable parameter, p, which can be related to the thickness of the double layer relative to the column radius and it can be represented by “0
+ c,- a2u + c,u
615
+ 2(1 - p ) x - x2]
(12)
P2
Equation 10 satisfies the above conditions and is seen to give the usual Poiseuille expression when p = 1, that is, when the wall shear region extends to the whole channel. The average velocity, u, is given by U = 1 - -2p + 1-p2 UO 3 6 When p = 1, eq 13 gives u / u o = 1/2 as required for parabolic flow, while for p = 0, one has u = uo, which corresponds to a plug flow. The evaluation of integrals in eq 7 and 8 using the flow profile given by eq 11and 12 is lengthy, but straightforward. Their fiial and exact expressions are as follows, in combination with eq 13:
with
x = r/a
73 108
(3)
-p3
and
641 + -,,4
743 - -p5
221 + -p6
432
540
360
149 -1260 (14)
P ( x ) = j z02 x ’ Q ( x ’ ) dx’
(4)
and
Here, R is the retention factor, r is the distance to the column axis, and Q ( x ) is the reduced velocity profile Q(x) = u ( x ) / u
(5) where u ( x ) is the velocity at x. The development of eq 2 gives
C, = I1 - 2RI2 + (R2/4)
(6)
Il = & b 2 ( x ) / x dx
(7)
with
Equations 14 and 15 have two interesting limits. Firstly, for = 1, which, as seen above, corresponds to the classical parabolic Poiseuille flow profile, it can be easily verified that I1and I2are equal to 11/24 and 1/3, respectively, which gives the expected following expression for C, (3, 17): p
C, = (11 - 16R
and
+ 6R2)/24
Secondly, when p is very small, a serial development in increasing powers of p gives
1 1 = - 1 + - p4- - p 21- - 3 194 251 p4 4 3 3 945’ - 11340
’[
Alternatively, in terms of the capacity factor k’, which is related to R by R = 1/(1 K’) (9)
+
one has
c,
(11 - 212 =
+ (1/4)) + 2(11 - IZ)k’+ Ilkn (1 + kq2
(10)
Integration of eq 7 and 8 for the parabolic Poiseuille flow profile gives the Golay equation. In order to perform the integrals of eq 7 and 8, one needs an expression for the velocity profile in electroosmotic flow. Exact expressions of this profile can be obtained (see below), but they are rather complex and cannot lead to analytical
(16)
+ ...
] ]
(17)
(18)
Therefore, for a perfectly flat, plug flow, when p = 0, one gets
C, = (1 - R)’/4 (19) as was previously obtained by Giddings (16) and as may be directly obtained from eq 2 by stating Q ( x ) = 1 and P ( x ) = x 2 in eq 5 and 4, respectively. DETERMINATION OF p As mentioned above, the flow profile used for the evaluation of the peak broadening mobile phase parameter C, is an approximation of the exact profile generated in electroosmotic flow. The parameter p in the expression of C, has therefore
616
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
to be related to the physicochemical characteristics of the system. The basic equation of fluid motion in this case is simply
with uo given by (8) uo =
4-E -
477 where E is the applied field and {is the potential at the surface of the capillary (or, more exactly, the potential at the plane of shear, called the zeta potential). The expression of the potential profile, $(r), has been obtained from the PoissonBoltzmann equation with the help of the Debye-Huckel approximation for the capillary geometry (18)
where Io is the zero-order modified Bessel function of the first kind (sometimes called hyperbolic Bessel function) and K is the reciprocal of the Debye length. This length can be viewed as the effective thickness of the diffuse part of the electrical double layer, i.e., the distance from the wall where the potential drops approximately by a factor l i e . It must be noted that the Debye-Huckel approximation applies for relatively small values of the zeta potential for which the dimensionless parameter Ka, called the electrokinetic radius of the system, is relatively large, so that there is only unsignificant overlap of the double layers in the center of the capillary. The average flow velocity for such a profile can be computed from eq 20 and 22, which gives
0.2
1
L
1
1
0.2
OL
I
0.6
I
0.8
r/a
Figure 1. Velocity profiles vs. relative distance from column axis: solid curve, electroosmotic flow; dashed curve, partially parabolic flow. Ka = 10 and p = 0.3083. The dotted curve represents the classical Poiseullle parabolic profile (p = 1).
Table I. Comparison between the Cm Values Obtained for the True Electroosmotic Flow Profile and the Partially Parabolic Flow Giving the Same Flow Rate‘
k’
(true profile)
(partially parabolic profile)
0
0.0071 0.1081 0.1695 0.2449 0.3341
0.1134 0.1761 0.2528 0.3433
Cm
1 2 5 m
a KU =
0.0085
% re1 error
16.5 4.7 3.7 3.1 2.7
10. The corresponding p value is 0.3083,
according to eq 25. where Il is the first-order modified Bessel function of the first kind (18). For relatively large values of Ka, it can be expanded in power series of l / K a , as (19) 1 1 1 -U = I - - +2 Ka
UO
+-+4(Ka)3
(Kay
4(Ka)4
+ ...
(24)
A good estimate of p can be obtained by equating the average velocities given by eq 13 and 23. Accordingly, one has p=2
[ ( 1-
I - - 3- z l ( K a ) ) ” ’ ] Ka Io(Ka)
(25)
As noted previously, p must not exceed 1 , which imposes a condition on Ka
i.e. Ka
2 3.4
This condition will be fulfilled in most practical cases unless very fine capillaries and/or very dilute solutions are used. Besides, it can also be considered as the condition of validity of eq 22. The calculation of p may be more convenient if eq 25 is rewritten as a serial power expansion of 1/Ka r
1
I+-+-+Ka
1
19
+ ...
4 ~ a 4 ( ~ a ) ~6 4 ( ~ a ) ~
-1
(28)
The error made with this approximation limited to the four first terms in less than 2%0for all the values of Ka for which the condition 27 is fulfilled.
In Figure 1 is shown the exact velocity profile u ( x ) / u ovs. x calculated from eq 20 and 22 for the specific case where Ka = 10, as well as the partially parabolic profile given by eq 11 and 12 with p equal to 0.308, according to eq 25 or 28, for the same value of Ka. The classical Poiseuille parabolic profile is also shown on this figure. Obviously, the approximated partially parabolic profile differs slightly from the true one. Accordingly, the C, value resulting from the true flow profile will differ somewhat from the value calculated by using eq 6, 13, 14, and 15. To evaluate this difference, the integrals Il and Iz in eq 7 and 8 were numerically computed from the profile given in eq 20 and 22 for Ka = 10. They equal 0.334 and 0.289, respectively. The values obtained from eq 14 and 15 for the partially parabolic flow with p = 0.308 are 0.343 and 0.292, respectively. It can be calculated that the p value which would give the same Il and I2 values as the true one is, in the above case, equal to 0.274. The corresponding values of C, for both profiles, as well as the error made in the approximation of the true profile by the partially parabolic profile giving the same flow rate, are given in Table I for different k’values. It is seen that the error is the largest for small k’and is less than 5% when k’exceeds 1. In view of the difficulty of obtaining precise values of the Debye length, 1 / ~it, can be considered that the partially parabolic profile, given by eq 11 and 12, describes very adequately the electroosmotic flow profile when p is linked to Ka by eq 25 or 28, i.e., when the average flow velocity is the same for both profiles. Figure 2 shows a plot of p vs. Ka, the electroosmotic radius which is the reciprocal of the ratio of the effective thickness of the double layer to the column radius. The error made when approximating p with 3 / ~ aaccording , to eq 28, is also shown on Figure 2 as a plot of p / ( 3 / ~ avs. ) Ka. It is seen that
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
Lm 0. 4
t
617
1
t
0. 3
0. 2
0. 1
41025 2
Flgure 2. Variations of the partially parabolic flow parameter p (left vertical axis) with the double layer parameter Ka and of p / ( 3 / ~ awith ) ~a (right vertical axis). p and Ka are linked by the fact that the partially parabolic flow profile and the true electroosmotic profile have both the same average velocity.
0. 2
0. 6
0. 4
0. 8
6 Flgure 3. C, vs. p curves: k’values, 0, 1, 2,5, and from the lower to the upper curves; soli curves, exact solution; dashed curves above the correspondlng solld curves, second degree polynomial approximation: dashed curves below the solid ones, fourth degree polynomial approximation.
this error is practically always less than 10%. We note in passing that the thickness of the layer with varying velocity has been indicated to be about 31“ (20).
RESULTS AND DISCUSSION Calculation of the Plate Height Parameter C,. The values of C,, calculated according to eq 6 , 9 , and 13-15, are plotted on Figure 3 as a function of p for the following values of k’, respectively: 0, 1,2,5, and m from the lower to the upper curve. It is seen that the variations are rather smooth. Thus C, is not very sensitive to a change in p which means that an exact determination of p is not critical for a good estimate of C, (cf. also Table I). As expected, C, increases monotonically with p, i.e., with decreasing values of the radius of the constant velocity central core of the partially parabolic flow profile. The dashed curves on Figure 3 represent the approximate values of C, calculated according to eq 17 and 18,respectively. At low p values, they cannot be distinguished from the exact curve. The dashed curve above each solid curve is obtained by using eq 17 and 18 limited to the terms in p2, while the dashed curve below each solid curve corresponds to the same calculations using the terms in p4. Figure 3 shows that the second degree approximation is valid for values smaller than about 0.3, while the fourth degree approximation extends the range of validity up to p equal to about 0.7. More significant than the effect of p is the variation of C, with the capacity factor. For a classical parabolic profile, it
4
k’
6
8
Flgure 4. C, vs. k’curves: p values, 0, 0.05, 0.1, 0.2, 0.5, and 1 from the lower to the upper curve. The dashed horizontal lines near the right w ) axis indicate the asymptotic value of C, at infinite retention.
is well-known that C, increases by a factor of 11 when k’ increases from 0 to m (3,17). On the other hand, for a perfectly flat flow profile (p = 0), C,, which is zero for nonretained solutes, must be finite for retained compounds, because the thermodynamic requirement of concentration equilibration between the mobile and stationary phases obliges the solute molecules to travel radially, to and from the tube wall which is covered by the stationary phase. These exchanges can occur solely by diffusion. Because of the finite rate of solute diffusion, this contributes to band broadening and this contribution increases with increasing retention, in spite of the fact that there are, in this case, no flow inequalities. Therefore, the ratio of the band broadening parameters, C,, for flat and fully parabolic profiles increases with increasing k’from 0 for unretained solutes to 0.33,0.42,0.49, and 0.55 (=6/11) for k’ equal to 1 , 2 , 5 , and 03, respectively. This is illustrated by the plot of C, vs. k’, shown in Figure 4, for different values of p , respectively, 0,0.05,0.1,0.2,0.5, and 1 from the lower to the upper curve. All the curves on Figure 4 show the same trend, whatever the p value. They increase rapidly with k’ at low k’ values and more gradually for high k’values. Whatever the value of p, more than half the C, value at infinite k’is reached when k’equals 2.5. Note that only for p = 0 the curve goes through the origin. Otherwise, there is always a finite, albeit small, contribution of resistance to mass transfer in the mobile phase to the plate height equation for the nonretained peak. Similarly, the ratio, r,, of the C, value for a given p to C, for the fully parabolic flow increases quite rapidly with increasing k’, for k’values lower than about 2, and then increases slowly toward its asymptotic value at k’= m, as can be seen in Figure 5. The two-thirds of the asymptotic values are reached at k’ below 1.25. From Figures 4 and 5, it appears that the electroosmotic flow will prove advantageous compared to the parabolic forced flow principally at k’values lower than 2. Thus, the electroosmotic flow shows some similarities with the turbulent flow, another situation in which the velocity profiles are approximately flat and the C, values, very low for unretained or slightly retained solutes, become comparatively large at high k’ (16). The minimum values of the reduced plate height, h = H/2a, are plotted in Figure 6 as a function of the capacity factor, for different p values, assuming that the term of resistance to mass transfer in the stationary phase, C,, in eq 1 is negligible. In these conditions, the minimum value of h is equal to (2C,)ll2. It can be very small at low k’values when p is lower than 0.2. The shapes of all the curves in Figure 6 are very similar. Moreover, these small h values are obtained at a higher velocity than those for parabolic flow. The ratio, ru, of the optimum velocity of a given p to the one for the par-
618
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
rm
0.8
t
i
t8
5
0. 6
4
0. 4
3
2
1
2
4
k’
6
2
8
Figure 5. Variations with k’of the r mratio of the plate height parameter C, for partially parabolic flow to the corresponding parameter at p = 1 (parabolic flow). p values are as follows: 0, 0.05, 0.1, 0.2, and 0.5 from the lower to the upper curve.
4
k’
6
B
Figure 7. Ratio r , of the optimum velocity (for which the plate height
is minimum) for partlally parabolic flow profile to the corresponding optimum velocity for parabolic profile vs. k’. p values are as follows: 0, 0.05, 0.1, 0.2,and 0.5 from the upper to the lower curve.
eq 16, for fully parabolic flow, applies for a partially parabolic flow by replacing the radius a by the double layer related parameter 6 = pa in eq 1, they obtained a 6 value of 100 pm, which they found not unreasonable (7‘). The above assumption, which implies that C, = p2/24 for an inert compound, is, however, not correct. Combination of eq 6,17, and 18 shows that, for small p and unretained solutes, C, is given by
0.2
Using this equation with the data of Pretorius et al., one finds which gives for the double layer thickness, 1 / K = 21 pm and Ka = 24, so condition 27 is satisfied. A similar comparison can be made for the data of Tsuda et al. ( I I ) , obtained with a 132 pm i.d. Pyrex tube. They found a plate height of 5.2 pm for unretained pyridine, in a 0.025 M Na2HP04-H20 solution at a velocity of 0.09 cm/s. Taking the stated value of 9.2 X lo* cmz/s for D,, one finds p = 0.084, 1 / u = 1.8 pm, and KU = 36. Unfortunately, it is difficult to analyze the data for retained compounds as the retention factor is not indicated. But, it is observed that the plate height values for aromatic hydrocarbons retained on an ODS column with an acetonitrile-water eluent are much higher than the plate height of unretained pyridine in a different eluent. This is not surprising in view of the large increase of C, with retention. In addition, the stationary phase term C, in the plate height equation may not be negligible. More recently, Tsuda et al. presented an electrophoretic separation of pyridinium salts in a 85 pm i.d. X 90 cm long fused silica column with an aqueous solution of 0.05 M NaZHPO4as the eluent (21). Pyridine, which is neutral in this solvent, is eluted by electroosmosis without electrophoretic displacement in 22.7 min and has a plate number estimated from the figure as 150 000 f 20%. Assuming the same value for the diffusion coefficient of pyridine in this solvent as above, one finds p = 0.122-0.186, Ka = 16.4-24.7, and 1 / K = 1.7-2.6 pm. This double layer thickness is similar to the one found above in the case of the Pyrex glass-0.025 M NazHP04 solution interface. Electroosmosis in packed LC columns has been investigated in a recent study by Stevens and Cortes (14).Although no peak broadening data are presented, it is clear that the significant decrease in flow rate observed with decreasing particle size is due to the increasing thickness of the double layer relative to the dimension of the passageway between particles, which indicates that, in this case too, the double layer thickness is on the order of magnitude of 1 ym. It is not possible to compare the 1 / K values determined with data obtained by other means since they are not available to us. However, they do not seem unrealistic as it is known that p = 0.128
2
4
k’
6
8
Flgure 6. Minimum reduced plate helght, H / 2 a I as a function of k‘. values are as follows: 0, 0.05, 0.1, 0.2,0.5, and 1 from the lower
p
to the upper curve.
abolic flow is plotted vs. k’ on Figure 7 , for the p values as on Figure 5, but, in this case, p decreases from the lower curve to the upper one. This ratio is equal to [C,(p = l ) / C m ( p ) ] l / z . When p is small, the optimum velocity is quite large. In the limiting case of unretained solutes with piston flow (p = 0), C , is zero and therefore the optimum velocity is infinite as only axial molecular diffusion contributes to peak broadening. This situation, however, is unrealistic, as, at large velocities, the zone dispersion will then be controlled by extracolumn effects. Moreover, for retained solutes, the contribution of the resistance to mass transfer in the stationary phase will, if practically significant, reduce somewhat the gap between the parabolic flow and the partially parabolic or piston flow curves. Unfortunately, there is not yet a sufficient amount of efficiency data with LC capillary columns to get a clear figure of this contribution. If it can be made negligible, then electroosmotic flow can prove advantageous, especially at small k’values. For example, a solute with a diffusion coefficient of cmz/s and a capacity factor of 0.5 eluting from a 1 m long, 10 pm i.d. column will show a plate number in excess of 420 000 for a retention time slightly less than 15 min if the flow is plug-like. With a parabolic flow, the plate number will be 200000 and the retention time over 30 min. Comparison with Experimental Data and Determination of Electrokinetic Parameters. It is useful to compare the present theoretical results with available data on electroosmotic flow. Pretorius et al., using a quartz tube and methanol as solvent, observed that, for an unretained solute, the plate height increases with increasing flow rates, which they attributed to the shear flow in the double layer thickness. Assuming that
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
the double layer thickness, 1 / ~can , vary very widely from a few nanometers for high ionic strength aqueous solutions to 100 gm for oil (7,22)or even more (20). Moreover, one would have expected the value for methanol to be higher than the one for the Na2HP04solution, in agreement with the above findings. In fact, it appears that the combination of electroosmosis with modern LC capillary chromatographic equipment with on-line detection (11-13) should provide a very efficient method for the determination of electrokinetic parameters. Indeed, in view of the fact that C, is very sensitive to a small variation of p, for an inert solute, as seen from eq 29, the plate height vs. velocity curve can be used for the determination of p, and, hence, the estimation of the electrical double layer thickness, 1 / ~with , the help of the equations developed above. Then, from the measurement of elution time of the inert compound, which gives the average flow velocity, and the p value, the electroosmotic velocity uo can be derived at different values of the electrical field E , which provides the zeta potential l (see eq 21), since e and 11 can be independently determined. An interesting application of this method concerns the variations of the double layer thickness and the zeta potential with the nature and the amount of the ligands attached to the surface of the glass or fused-silica wall of the capillary tube, as well as with the composition of multicomponent eluent mixtures. This may give some useful insights into the mechanism of retention in bonded-phase chromatography. It is important to use an inert solute for such electrokinetic measurements to ensure that there is no stationary phase contribution to peak broadening, although it is for unretained compounds that the extracolumn effects are comparatively the largest. They should be made as small as possible, especially by using an on-column detection technique. This method requires the knowledge of the diffusion coefficient which can be determined independently, for example, by a chromatographic dispersion technique (23). Alternatively, if one uses successively a forced flow and the electroosmotic flow, with the same channel, the a2/D, ratio can be determined from the plate height curve in the former case when C, = 1/24. Accordingly, the ratio of the slopes of the ascending branches of the plate height curves for the two types of flow provides the desired parameter p. One should note that the a radius obtained from forced flow data represents the geometrical radius of the tube, while, in theory at least, for the electroosmotic flow, a refers to the distance from the tube axis to the plane of shear. However, even if the position of the slippage plane is not clearly known, this difference is, in practice, totally insignificant, since the two a values differ only by one or a few molecular diameters. Potential of Electroosmotic Flow in Chromatography. As seen above, the electroosmotic flow leads to lower peak broadening than the pressure flow, especially at small values of k’. As the performances of capillary columns in LC are presently limited by the technological state of injection devices and detectors, due to their significant relative contribution to band broadening, it is useful to compare both types of flow on this basis. Let us assume that a given theoretical plate number is looked for and that the column length is the same in both cases. Then, the plate height must also be the same for both types of flow. If the flow velocity must remain constant in order to give the same analysis time, then the ratio aeo/aparmust be equal to (Cm,par/Cm,eo)1/2 = r,, where the suffiies eo and par refer to electroosmotic and parabolic flows, respectively. Then, the volumetric peak widths, uv, are in the ratio av,Bo/uv,pBI. = C,,par/C,,m = r,2 while the time peak widths are the same. Figure 7 shows what improvement can be expected, as a function of p and k‘. It is significant only at
819
low p values and for small capacity factors. If, now, the columns are operated at their optimum velocity, their radius ratio is, still, aeo/apar= r,, and the uv is ratio still equal to r:, but the time width ratio, which is equal to the analysis time ratio, is then ut,eo/at,par = l / r u instead of 1 previously. Accordingly, if the capacity factor is 1and p is zero, electroosmosis can be performed in a tube with a diameter 1.73 times larger than for pressure flow, which gives a three times larger volumetric peak width than for pressure flow. But the time peak widths, which are in the ratio of the retention times, will then be equal or in the ratio 111.73 = 0.58 depending on whether the columns are operated at the same velocity or at their optimum velocity. This w ill require a similar or smaller time constant for the detector, respectively, but will allow a three times larger detector volume. Therefore, present limitations of capillary columns can be somewhat relaxed by the use of electroosmotic flow. With conventional LC detectors, satisfactory separations, which are impossible to carry out with pressure-driven flow, may be obtained with electroosmotic flow. One should note, however, that, the potential gain in terms of detector cell volumes is in the ratio of a few units, which is significant, but does not reach the order of magnitude which would be necessary to permit the use of present detector cells. One may also try to compare both types of flow on the basis of the peak capacity they can offer since this overall parameter, which is the number of peaks which can be separated with a given (generallyunit) resolution (24),has been, at times, used for comparison of various techniques (25-27). The peak capacity, n, obviously, increases with the analysis time, that is, with increasing k’. The rate of generation of peak capacity is simply (30) where V, is the retention volume, Vo is the column volume, and uv is the peak volumetric standard deviation. Replacing VJV, by (1+ k 3 and av/Vo by (1 + k 3 ( H / L ) 1 / 2and using eq 1, 9, 16, and 19, one obtains, after integration, for the parabolic flow
ti
and, for a plug flow
1 L n = l + - (4- ) 2a
[3 +
(p
96
= 0)
v2
1
(5 9 (([(5 112 In
+
;4k ’ +
.,
+
2)
(3 + [ (; ;);]”’ + ;)
(f + $ ) k f i ] ] ” ’ + ?)/( V
+
$)k!+
+
(32)
where v is the reduced eluent velocity equal to v = 2au/D,
(33)
Equations 31 and 32 assume that the stationary phase contribution to peak broadening is negligible. The variations of
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
620
18 u-
20
u- 50
u-200 V
I
I
I
I
8
4
I
k'
I
I
12
I
I
I
16
Flgure 8. Parabolic flow. k'dependence of the peak capacity related term, n' = 4(n - l ) / ( L / 2 a ) " * , where n is the peak capacity, for
different reduced eluent velocities, v, indicated near the right vertical axis. n 4
3
2
1
W
I
1
I
4
I
I
8
I
k'
I
12
I
I
I
I
16
Piston flow. k' dependence of the peak capacity related term, n' = 4(n - l)/(L12a)'/*, where n is the peak capacity, for different reduced eluent velocities, v, indicated near the right vertical axis. Flgure 9.
n'= 4(n - 1 ) / ( L / 2 ~ ) are ~ / ~plotted a t different v values as a function of k'on Figure 8 for the parabolic flow and Figure 9 for piston flow, assuming that v does not change with k' (constant diffusion coefficient). All the curves show that the increase of n with k'is much larger at low k'and levels off rapidly at large k', but this effect is more pronounced in the case of the piston flow. Besides, the curves for the piston flow are all above the curves for the parabolic flow at the same velocity, except for v = 4 below k' = 1. Accordingly, the potential of the electroosmotic flow lies essentially in the low k'region. With piston flow, changing the k'of the last eluted compound from 1 to 19, at v = 50, will decuple the analysis time but only double the peak capacity. Moreover, fast velocities can be used in the low k'region without detrimental effects. For example, with a 1 m long, 10 pm i.d. column, a peak capacity of 155 (68 for parabolic flow) can be reached up to k' = 2 in 20 min if the average diffusion coefficient is cmz/s. At v = 200, only 149 peaks will be separated at unit resolution in the same time (now up to k ' = 11) on this column; on the other hand, if a 6 m long column is used at this velocity (v = 200), the peak capacity in the same time (20 min, now k' = 1) will then be equal to 233. However, such an analysis might just be impossible, due to the very large voltage required to generate such fast velocity streams in a long channel. Indeed, from the data of ref 11 with a Pyrex tube, one can calculate that the electrical field required to induce a 1 cm/s flow velocity is 71.4 kV/m, 164 kV/m, and 294 kV/m for acetonitrile, distilled water, and methanol, respectively.
In conclusion, the prospect of using electroosmotic flow in capillary columns does not appear as bright as could be expected from the superficial consideration of its piston-like character, because of the fact that the peak broadening increases significantly with retention. Nevertheless, it has been shown that the nonequilibrium mobile-phase band broadening parameter C, is lower, typically by a factor 2 to 5 , and that the peak capacity can be significantly larger for electroosmotic flow than for pressure-driven flow. However, more experimental data are fieeded to evaluate the contribution of the stationary phase to peak broadening, which has been assumed to be negligible in the above calculations. It has been shown that the external volumetric requirements may be somewhat less severe for electroosmosis than for pressure flow but that the detector response time may have to be faster in the former case. Clearly, the technological limitations of parabolic flow capillary columns may be displaced, especially since there is no pressure drop buildup by electroosmosis, while, on the other hand, new technological limitations may be imposed by the establishment of the required electrical field. The full evaluation of the chromatographic potential of electroosmotic flow must wait for more experimental data for the chromatographic behavior of columns with this type of flow and its technological implications, but at the present stage, its prospect seems only moderately encouraging. In addition, it is well-known that the passage of an electrical current can generate thermal effects, which are sometimes detrimental to electrophoretic techniques. These effects can be made small if narrow bore tubes are used (12, 13) and they have been neglected in the above treatment. In cases where they are not negligible, they will make worse the prospect of using electroosmotic flow in capillary columns.
LITERATURE CITED Guiochon, G. I n "Micro HPLC"; Kucera, P., Ed.; Elsevier: Amsterdam, 1983; p 1. Knox, J. H.; Gilbert, M. T. J. Chromafogr. 1979, 186, 405. Golay, M. J. E. "Gas Chromatography"; Desty, D. H., Ed.; Butterworths: London, 1958; p 36. Ishii, D.; Hibi, K.; Asal, K.; Jonokuchi, T. J. Chromatogr. 1978, 151, 147. Mould, D. L.;Synge, R. L. M. Analyst (London) 1952, 7 7 , 964. Mould, D. L.; Synge, R. L. M. Biochem. J. 1954, 58, 571. Pretorius, V.; Hopkins, E. J.; Schieke, J. D. J. Chromatogr. 1974, 99, 23. Aveyard, R.; Haydon, D. A. "An Introduction to the Principles of Surface Chemistry"; Cambridge University Press: Cambridge, 1973; Chapter 2. Hiemenz, P. C. "Princlples of Colloid and Surface Chemlstry"; Marcel Dekker: New York, 1977; Chapter 11. Everaerts, F. M.; Mikkers, F. E. P.; Verheggen, Th. P. E. M.; Vacik, J. I n "Chromatography. Part A"; Heftmann, E., Ed.; Elsevier: Amsterdam, 1983; Journal of Chromatography Library, Vol. 22A, Chapter 9. Tsuda, T.; Nomura, K.; Nakagawa, G. J. Chromatogr. 1982, 248, 241. Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 5 3 , 1298. Jorgenson, J. W.; Lukacs, K. D. J. Chromatogr. 1981, 218, 209. Stevens, T. S.;Cortes, H. J. Anal. Chem. 1983, 55, 1365. Aris, R. Proc. R. SOC.London, PartA 1959, A252, 538. Marlin, M.; Gulochon, G. Anal. Chem. 1982, 5 4 , 1533. Giddings, J. C. J . Chromafogr. 1961, 5 , 46. Rice, C. L.; Whltehead, R. J. Phys. Chem. 1985, 69, 4017. Angot, A. "ComplOments de MathOmatlques", 4th ed.; Editions de la Revue d'optique: Paris, 1961; Chapter 7. Hunter, R. J. "Zeta Potentlal In Colloid Science"; Academic Press: London, 1981; Chapters 2 and 3. Tsuda, T.; Nomura, K.; Nakagawa. G. J. Chromafogr. 1983, 264, 385.
Davies, J. T.; Rideal, E. K. "Interfacial Phenomena"; Academic Press: New York, 1961; Chapter 2. Atwood, J. G.; Goldstein, J. J . Phys. Chem., in press. Giddlngs, J. C. Anal. Chem. 1987, 3 9 , 1027. Giddings, J. C.; Dahigren, K. S e p . Sci. 1971, 6 , 345. Martin, M.; Jaulmes, A. S e p . Sci. Techno/. 1981, 16, 691. Guiochon, G.; Beaver, L. A,; Gonnord, M.-F.; Siouffi, A. M.; Zakaria, M. J. Chromatogr. 1983, 255, 415.
RECEIVED for review August 18, 1983. Accepted December 13, 1983.
