Anal. Chem. 1994,66, 4236-4242
Modeling Flow Profiles and Dispersion in Capillary Electrophoresis with Nonuniform 5 Potential Catherine A. Keely,* Tom A. A. M. van de Goor, and Douglass McManigill Hewlett Packard Laboratories, 3500 Deer Creek Road, P.O. Box 10350, Palo Alto, California 94303-0867
We have reported earlier that in capillary electrophoresis experiments at low pH values, broadening of the sample peak can be unusually high-higher than can be accounted for by diffusion alone. In this paper, theoretical flow profiles inside the capillary have been examined as contributing to band broadening and raising the plate height. The theory is based on the expectation that the 5' potential changes along the length of the capillary, z,due to electrical fields that exist across the wall of the capihy. This paper discusses the particular case in which the radial voltage across the capillary wall is constrained to be a linear function of z. This case applies to the configuration in which the capillary is covered by a conductive shield. The con6guration is of special interest because it can allow control of electroosmotic flow under specific conditions. Gwen that the radial voltage is a linear function of z,it is reasoned that ((2) can also be assumed to be linear. The theory is developed for the resulting velocity profile as a function of z,and the effect on dispersion and plate height is presented. The theory includes addition of pressure and the corresponding influence on plate height. In capillary electrophoresis (CE), high voltage is applied between the ends of the capillary,generating an axial electric field. Due to the electrical environment near the inside wall of the capillary, which is dependent on parameters such as the capillary surface and the buffer, the axial field will generate electroosmotic flow @OF) through the capillary. The 5; potential is the parameter representing the electrical condition at the wall which relates the axial field to the electroosmotic flow. In recent years, a number of papers have been published discussing the existence and effect of fields acting not along the length of the capillary but across the These radially directed electric fields are induced by the high axial electric fields required in CE. Their magnitude has been shown to have an (1) Keely, C. A; Holloway, R R; van de Goor, A A A M.; McManigill, D./. Chromatogr. A 1993, 652, 283-289. (2) Lee, C. S.; Blanchard, W. C.; Wu, C:T. Anal. Chem. 1990,62,1550-1552. (3) Lee, C. S.; McManigill, D.; Patel, B. Anal. Chem. 1991, 63, 1519-1523. (4) Wu, C.-T.; Huang, T.-L.; Lee, C. S. Anal. Chem. 1993, 65, 568-571. (5) Huang. T.-L.; Tsai, P.; Wu, C.-T.; Lee, C. S. Anal. Chem. 1993, 65, 28872893. (6) Ghowski, K; Gale, R J. In Proceedings ofthe International Symposium on Biosensor Technology 1989; Buck, R. P., Ed.; Marcel Dekker: New York, 1990; pp 55-62. (7) Holloway, R R; Keely, C. A; Lux, J. A; Mcmanigill, D.; Young, J. E. 4th Intemational Symposium on High Performance Capillary Electrophoresis, Amsterdam, February 9-13, 1992; Poster PT-27. (8) Hayes, M. A.; Ewing, A G. Anal. Chem. 1992, 64, 512-516. (9) Hayes, M. A.; Kheterpal, I.; Ewing, A G. Anal. Chem. 1993, 65, 20102013.
4236 Analytical Chemistry, Vol. 66,No. 23, December 7 , 7994
effect on the electroosmotic (EO) velocity when using low pH buffers (typically pH 2-5) are used.ls5 Thus, researchers have designed capillary configurations which constrain the radial electric fields to take advantage of this phenomenon. To understand the effect of the radial fields on the EO velocity, several papers have proposed models in which the 1; potential on the inside surface of the capillary is modified by the radial These models treat the capillary wall as a combination of capacitors and resistors which allow the external electrical environment to have an effect on the internal electrical environment. These models provide some insight on the mechanisms by which radial fields affect electroosmotic flow, but they do not address the increased plate height. It is normally expected that the efficiency of the CE system or the number of theoretical plates follows the relation N = pEZ/ 2 D 0 (see Glossary for definition of all terms). This has been verified when the buffer pH is above about 5. We have previously reported that at low pH conditions there is increased dispersion (or decreased efficiency) in the sample peaks in conventional CE, and the above equation does not hold. In capillary configurations where radial fields are constrained, there remains this extra dispersion, resulting in a lower efficiency N or a higher plate height H,where H = Ld,JN.l This paper combines several theoretical models and examines the flow profile in the capillary and the plate height obtained. In this theory, the radial field changes along the length of the capillary, affecting the 5; potential, flow profile, and dispersion which must also change along the length. The effect of a pressure differential on the flow profile is also included in the theory. THEORY In standard CE, the electroosmotic velocity u is given by the Helmhotz-Smoluchowski equation,"
In this equation, the 1; potential is normally assumed to be constant. In the theory developed below, it is shown that this is not always the case, and changing 1; impacts the flow profile and plate height. f Potential. Several models exist describing the 1; potential for a system. For CE, the 1; potential is defined as the electrical potential at the shear plane of the liquid, and it is an experimentally determined quantity. While models may differ in their description (10) Jorgenson, J. W.; Lukacs, K D. Anal. Chem. 1981, 53, 1298-1302. (11) Hunter, R J. Zeta Potential in Colloid Science; Academic Press: London, 1981. 0003-2700/94/0366-4236$04.50/0 0 1994 American Chemical Society
of the potential in the region between the shear plane and the capillary wall, there is typically an exponential drop from the double layer to the center of the capillary,l1 Based on the Debye-Huckel theory, for electrophoresis in bare fused silica, it is reasonable to assume the following relation between 5 and the wall potential Y&11J2
5 = Yo exp[-/cx]
I
5 ,
(2)
The double layer thickness is K-’, where
“ 1d
(3)
-5
-25.0
0.0
-12.5
12.5
25.0
Radial Voltage (kV)
which, as is shown, is dependent on the electrolyte composition and c~ncentration.~~ The value of x represents the distance from the wall to the shear layer and is estimated to be between 3 and 10 nm.13 Since this quantity is not well defined, the value of 5 is correspondingly uncertain. From the Gouy-Chapman model, the wall potential in volts for a 1:l electrolyte at 298 K is dependent on the buffer concentration c and the surface charge density ut and is given by13
Y,, = 0.0514 sinh-’
u 8.513
%
Figure 1. Electroosmotic mobility vs radial voltage as predicted by eqs 1-5 with four different pH values (pKa = 5.8, 10 mM buffer concentration).
cy
E
1
.
(4)
1
Hayes and Ewing@have developed an expression for ut when a radial field is present as the sum of a pH-dependent fused silica surface charge term, osi, and a radial voltage-dependent surface charge term, 0,:
-5
I 2
6
4
a
10
PH Figure 2. Electroosmotic mobility vs pH as predicted by eqs 1-5 with three different settings for the radial voltage (pKa = 5.8, 10 mM buffer concentration).
= c s i ; ( m1 )
The term usi is a simplification of the equilibrium conditions at the inside surface of the capillary, and a more elaborate model may include other variables. The term uv is derived by treating the capillary as a cylindrical capacitor. Combining the above relations for YO,and ut, one obtains the 5 potential (and hence the EO mobility peothrough eq 1) as a function of several variable parameters: pH, radial voltage, inside and outside radius, inside surface material (through K, and y ) , capillary wall material (through E S J , buffer composition and concentration (through K and c), and shear layer location (through x) . ( Potential Dependencies. The relationship between the 5 potential and the radial voltage is of primary interest here, because of the observed effect of radial voltage on EO velocity. Thus, this
c,
(12) Davies, J. T.;Rideal, E. K Interfacial Phenomena; Academic Press: New York, 1963. (13) Hiemenz, P. C. Principles of Colloid and Surfnce Chemistty; Marcel Dekker: New York, 1986.
relation as predicted by eqs 1-5 is now examined by plotting the electroosmotic mobility as a function of a few important variables. Figures 1-4 illustrate the EO mobility dependence. In the equations, the capillary dimensions are assumed to be r, = 0.050 mm and fi = 0.0125 mm. Figure 1 graphs the theoretical curve for the EO mobility vs the radial voltage for four pH values, where the radial voltage is the potential drop measured across the wall from outside to inside. At pH 3, changing the radial voltage causes a large change in EO mobility, but this effect is reduced at high pH values and is nearly gone by pH 6. Figure 2 shows this effect in a different way; the EO mobility is plotted as a function of pH for three values of radial voltage. The mobility range between the lines for VR= -25 kV and VR= 25 kV (voltages chosen strictly for comparison purposes) is a measure of the range over which the mobility may be controlled. Given the fused silica surface in this model, the range is severely reduced above pH 5. In this model, the pH value can only be shifted by changing the silanol surface equilibrium constant, K,. This is shown in Figure 3, where the EO mobility vs pH is given for three equilibrium constants. The center curve uses the value taken from Scales,14where the ~
~
~~~~
~
(14) Scales, P J., Grieser, F , Healy, T W Langmuzr 1992, 8, 965-974
Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
4237
and Idol's secondary flow has radial and axial direction components, is a function of both radius r and distance z, and averages to zero over any capillary cross section. The flow components in the z-direction, zl&) and vm(r,.z),are of interest here and are given in the Appendix along with their reduced forms,
ce
which are justified later by the simplified model. This represents a constant bulk velocity P and a superimposed parabolic profile
" I
-
vzs.
2
a
6
4
10
PH Figure 3. Electroosmotic mobility vs pH as predicted by eqs 1-5 with three different settings for the equilibrium constant ( VR= 25 kV, 10 mM buffer concentration). 10
1
1
5E
If the capillary is subjected to pressure-driven flow in addition to the EOF, pressure flow terms must be added to the profiles. The primary velocity profile is then
and the contributionto the secondary velocity profile is the familiar parabolic laminar flow profile,
\
\ \
(9)
lWmM
I
\
-10
I
-25.0
I
, -12.5
1
I
I
0.0
12.5
so that the complete secondary profile is
'
25.0
Radial Voltage (kV) Figure 4. Electroosmotic mobility vs radial voltage as predicted by eqs 1-5 with three different buffer concentrations (pKa = 5.8, pH = 3.0).
pKa is 5.8. To increase the pH value where the curve rises, a pKa of 6.8 is used, and to shift the curve to the left, a pKa of 4.8 is used. Possible factors influencing this value in an actual capillary may be capillary pretreatment, manufacturing conditions, or contamination. Finally, the effect of buffer concentration is shown in Figure 4,where the mobility is plotted against the radial voltage for three values of concentration c. The graph indicates that a lower concentration buffer would produce a greater change in mobility as the radial voltage is changed. Velocity Profile. The theory development above determines 1; as a function of many parameters, including the radial voltage across the capillary wall. In a CE system, it can be expected that the radial voltage will be different in different parts of the capillary or will change as a function of distance along the capillary length, z. Anderson and IdolI5 developed a theory for the velocity profile in a capillary given that 1; is an arbitrary function of z. The theory produces a stream function, and from the stream function the flow is calculated and expressed as the sum of a primary flow and a secondary flow. The primary flow is constant, is only in the z direction, and is equivalent to the average bulk flow. Anderson (15) Anderson, J. L.; Idol, W. K Chem. Eng. Commun. 1985, 38, 93-106.
4238 Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
Dispersion. In capillary electrophoresis, a sample boundary traveling through the system experiences various forces which cause the boundary to become less sharp and the flow front less pluglike. Thermal dif[usion and velocity profile are two of the main dispersive forces that cause broadened peaks in an electropherogram. Their combined effects are measured by calculating the plate height or efficiency (plate numbers) of the system. For a capillary electrophoresis system, plate height is defined as
Neglecting the injection variance and the variance due to the finite detection window, the sample peak variance is the sum of the dispersion effects over the time,16
where Kd is the dispersion due to diffusion and the velocity profile. The dispersion due to the profile can be calculated according to the method of Datta and Kotamarthi,17 adapting for the zdependent profile, giving a dispersion that is a function of z, (16) Aris, R Proc. R. SOC.London 1956,,4235, 67-77. (17) Datta, R: Kotamarthi, V. R AIChE J. 1990,36, 916-926.
N
which is the diffusion effect added to the effect of concentration changes resulting from the velocity profile perturbations. Since Kd is a function of 2, the time integral in the variance equation above must be changed to reflect the z dependence. The appropriate peak variance expression is then the integral of the dispersion effects over the distance z traveled in the capillary,
- 0.00
B B d
ru h
