Random-walk theory of nonequilibrium plate height in micellar

Random-walk theory of nonequilibrium plate height in micellar electrokinetic capillary chromatography. Joe M. Davis. Anal. Chem. , 1989, 61 (21), pp 2...
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Anal. Chem. 1989, 61, 2455-2461

flexible solutes (i.e., providing adequate mixing of solute and phase and using stationary phases) results in complete agreement with conventional chromatographic theory. Registry No. Polystyrene, 9003-53-6.

LITERATURE CITED Armstrong, Armstrong, Armstrong,

D.W.; Bui. K. H. Anal. Chem. 1982, 5 4 , 706. D.W.; Boehm, R. E. J. Chromatogr. Sei. 1984, 22, 378. D.W.; Boehm, R. E.;Bui, K. H. J . Chromatogr. 1983, 6 ,

1

Armstrong, D. w.; Boehm, R. E.; Bui, K. H. J . c h ~ o m t o g r .1984, 288, 14. Boehm, R. E.; Martire, D. E. Anal. Chem. 1989, 67, 471. Boehm, R. E.; Martire, D. E.;Armstrong, D. W.; Bui, K. H. Macromole-

cubs 1983. 16,466. Boehm, R. E.; Martire, D. E.; Armstrong,

cubs 1984, 17,400.

D.W.; Bui, K. H. Macromole-

Snyder, L. R.; Stadalius, M. A,; Quarry, M. A. Anal. Chem. 1983, 55,

1412A.

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(9) DeStefano, J. J.; Goldberg, A. P.; Larmann, J. P.; Snyder, L. R.; Stadalius, M. A.; Stout, R. W. J . Chromatogr. 1983, 255, 163. (10) Stadalius, M. A.; Quarry, M. A.; Mourey, T. H.; Snyder, L. R. J . Chromatogr. 1988.358, 17. (11) Glockner, G. Pure Appl. Chem. 1983, 55, 1553. (12) Glockner, G.; van der Berg, J. H. M. J . Chromatogr. 1988, 352, 151. (13) Stadalius. M. A.; Quarry, M. A.; Mourey, T. H.; Snyder, L. R. J . Chrom t o g r . 1986, 358,I. (14) Katz, E. D.;Scott, R. P. W. J . Chfomatogr. 1983, 268, 169. (15) Poulsen, J. R.; Birks, K. S.; Gandelman, M. S.;Birks, J. W. Chromatographic 1966, 22,231. (16) Blrks. J. W.; Frei, R. W. TRAC, Trends Anal. Chem. (Pers. Ed.) 1983, 7 , 361. (17) . , Fiorv. P. J. I n Princides of Polvmer Chemise; Cornel1 University Preis: Ithaca, NY, 1953;p 519: (18) Quarry, M. A.; Grob, R. L.; Snyder, L. R. Anal. Chem. 1986, 58, 907.

RECEIVED for review May 5, 1989. Accepted August 3, 1989. The authors acknowledge the support, in part, of a grant (to c* Lochmuller) from the National Science Foundation CHE-8500658.

CORRESPONDENCE Random-Walk Theory of Nonequilibrium Plate Height in Micellar Electrokinetic Capillary Chromatography Sir: Micellar electrokinetic capillary chromatography (MECC) is a highly efficient separation technique implemented in a narrow-bore capillary tube, along which is applied a large electric field. Separation is based on the partitioning of electrically neutral analytes between a mobile liquid electrolytic phase, whose motion arises from field-induced electroosmosis, and a micellar phase of electrically charged micelles, whose lesser motion arises from the combined fieldinduced effects of electroosmosis and electrophoresis. The differential migration of the two phases spatially separates the components of a mixture, when the components' partition coefficients differ (1-3). A significant distinction between MECC and the parent technique on which it is based, capillary zone electrophoresis (CZE), is that the plate heights of certain analytes resolved by MECC, under typical experimental conditions, appear to be govemed by nonequilibrium effects when the electric field strength exceeds roughly 7-20 kV/m ( 4 , 5 ) . In other words, a contribution to the plate height increases with the electric field strength, apparently in a linear manner, and becomes the dominant source of plate height at field strengths greater than these. In contrast, only small nonequilibrium-likeeffects are observed in CZE at these field strengths (6, 7), unless the analytes readily adsorb to and desorb from the capillary wall, as do proteins (8, 9). Ultimately, plate heights in CZE do increase with field strength, but only when the field exceeds 30 kV/m or so, under typical experimental conditions (10). In general, rapid separations in MECC, as in any electrophoretic method, require large field strengths. Because plate heights ultimately increase with increasing field strength, one can simultaneouslyobtain rapid and efficient separations only if one reduces or minimizes these (apparent) nonequilibrium effects. To minimize them, one must have some physicochemical understanding of their origin. In a recent work, Sepaniak and Cole argued that these effects originate from transchannel mass transfer ( 4 ) . An equation for this mass transfer that agrees fairly closely with experiment, and thus supports this conclusion, is derived below. 0003-2700/89/0361-2455$01.50/0

THEORY In a recent work, Terabe et al. examined several possible sources of nonequilibrium dispersion in MECC, including sorption-desorption kinetics, intermicelle diffusion, the effect of temperature gradients, and electrophoretic dispersion (11). They developed theories for each source of dispersion and concluded that only electrophoretic dispersion could explain their experimental results. Their work is most instructive in demonstrating that many plausible sources of dispersion do not, in fact, govern the nonequilibrium plate height. In this paper, the author will consider again a source of dispersion first addressed by them and also a new source of dispersion, which seems to explain much experimental data. In both cases, the magnitudes of these dispersions will be estimated from the theory of a random walk, and these estimates will then be compared to experiment. In this approximate but heuristic theory, one assumes that any representative analyte molecule is displaced along the flow direction an average distance 1 from the center of a zone formed from a statistically large number of analyte molecules. After n displacements, the breadth of the resultant analyte distribution is described by the variance u2 = 12n. The nonequilibrium plate height H is then calculated as

H

=

." = -12n L

L

(1)

where L is the column length. Several applications of random-walk theory to chromatography are discussed by Giddings (12).

The basic assumptionsare herewith stated. One will assume that any analyte molecule has two characteristic velocities. The first of these, u,, is the velocity of the molecule in the mobile phase; the second, um0 is the velocity of the molecule in a micelle. For dilute zones, the velocities u, and u,, essentially equal the velocities of the analyte-free mobile and micellar phases, respectively. The average velocity v of the analyte molecule, by definition, is a weighted average of u, and u,, 0 1989 American Chemical Society

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ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989

u =

Umtm’

t,‘

+ umctmc’ + t,;

(2)

where t,’ and t;, are respectively the average times the molecule spends in the mobile and micellar phases. The sum t,’ + t,; equals the retention time tr of the molecule (zone) in the capillary. If the molecule occupies both phases a large number k of times, then the average residence times (or lifetimes) t, and t, of the molecule in the mobile and micellar phases, respectively, are t, = t,’/k and t,, = t,;/k. With these substitutions, eq 2 becomes

(9) The identity t, = L / v is substituted in eq 9 to obtain the second equality, and eq 3 and 6 are used to obtain the third. Equation 9 can be more simply expressed as

where the identity t, = Rt,,/(l - R), obtained by dividing eq 4 by eq 5, has been substituted for t, in eq 9. Combining eq 8 and 10 with eq 1, one obtains

where the fractions of time the molecule spends in the mobile and micellar phases, respectively, are represented by

--t m tm

+

-R

(4)

- 1-R

(5)

tmc

and

--t m c

tm + tmc In eq 4 and 5 , quantity R is the analyte retention ratio. The velocities u, and u,, can be expressed as Om = he$; umc = (pea + p m c ) E (6) where pea and bmcare the electroosmotic flow coefficient and the electrophoretic mobility of the micelles, respectively, and E is the magnitude of the electric field strength. Because u, < u,, peoand pmchave opposite signs. For the sake of simplicity, one will assume here that pea is positive and p,, is negative. If the opposite is true, then the plate height equations derived below must be multiplied by -1. Case 1. Micellar Mass Transfer. In their recent work, Terabe and co-workers derived an equation for this plate height contribution by extending the nonequilibrium theory of chromatography to MECC (11). They noted that the same result could be derived from random-walk theory but did not present the derivation. This latter derivation is of interest in its own right, because it differs from its chromatographic analogue, and it represents an approach to this mass-transfer problem that the author had independently undertaken. Furthermore, the resultant equation, while equivalent to that derived in ref 11, is expressed mathematically much more simply. In this case, mass transfer originates from the kinetics of partitioning, Le., the rates of adsorption of analyte molecules to, and their desorption from, the micellar phase. Therefore, 1 is simply the span that develops between the zone center and a typical analyte molecule each time the molecule occupies either phase. The molecule moves in the mobile phase for time t , with velocity u, and in the micellar phase for time t,, with velocity umc. The zone center moves with velocity u a t all times. Thus 1 = (Au)iti = (u, - u)t, = (U - u,,)t,, (7) where ( A u ) ~is the absolute value of the difference between u and the velocity of the ith phase, and ti is the time spent in the ith phase. Combining eq 3-7, one obtains Equation 8 shows that 1 is proportional to -p,J = v, - umc, which is the origin of the differential displacement between the two phases. The number n of steps taken by the analyte molecule equals the number of times the molecule occupies both the mobile and micellar phases during its residence time in the channel. Based on the discussion immediately following eq 2, the step number n = 2k is

where H, is the nonequilibrium plate height due to micellar mass transfer. This result is equivalent to eq 21 in ref 11 but is mathematically much simpler, because reiention is expressed in terms of R, instead of capacity factor k’, and because the mobilities pe0 and pmcare expressed explicitly, instead of as a ratio of elution times. Equation 11 differs significantly from the random-walk equation for chromatography (12), because the analyte residence times in both phases determine the number n of steps, and the span 1 is proportional to the velocity difference u, - u, = -p,J between the phases. In contrast, the number n of steps in a chromatographic random walk is determined by the mobile-phase residence time only, and span I is proportional to the mobile-phase velocity. As u, goes to zero, pm = -pmc (see eq 6), and eq 11 appropriately reduces to the chromatographic random-walk equation

H m = 2R(1- R)pe$tmc

(12)

where u, = peJ3 is the mobile-phase velocity. If H, is now written as H, = C,V, where C, is a nonequilibrium coefficient and V = EL is the voltage applied to the capillary, then coefficient C, is

Equation 13 is actually too small by a factor approximately equal to 3, because micellar mass transfer occurs in the radial as well as the flow direction. In other words, the micellar random walk is three-dimensional, instead of one-dimensional (13). Accordingly, a more correct estimate of C, is

Calculations of this mass-transfer effect will be presented in a later section. Here, one will simply note that the effect is much too small to explain experimental results. Thus, alternative explanations must be considered. Case 2. Transchannel Mass Transfer. One possible origin of nonequilibrium effects in MECC is the reversible adsorption of analyte molecules to the capillary wall. Axial dispersion of this type has its origin in transchannel mobile-phase mass transfer, which has been extensively studied by Giddings (12). One must be somewhat skeptical of this possibility, however, because one can explain analyte retention in MECC without considering adsorption (14). In contrast, nonequilibrium effects in MECC are substantial ( 4 , 5). Consequently, this type of mass transfer is unlikely to be important under typical experimental conditions. Here, one will consider the possibility that some other kind of transchannel mass transfer contributes to nonequilibrium dispersion in MECC. The dependence of plate height on

ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989

capillary radius affirms that this possibility exists (4). If some other kind of transchannel mass transfer exists, it is unlikely to originate from sources of dispersion common to both CZE and MECC (e.g., the variation of electrophoretic mobility with temperature), because plate height minima are commonly observed at lower voltages in MECC than in CZE. One form of mass transfer that is unique to MECC can be attributed to the variation over the capillary cross section of the analyte partition coefficient P. (Quantity P characterizes the thermodynamic distribution of analyte between the mobile and micellar phases.) Such a variation must exist, because partition coefficients decrease rapidly with increasing temperature (151, and the carrier temperature decreases with increasing distance from the capillary center, because of Joule heating (16).

To understand the consequences of this radial gradient in

P, one will first consider the hypothetical case in which analyte molecules are equilibrated between the two phases and do not diffuse radially. One consequence is that the local average velocity of analyte molecules in the capillary center, where P is relatively small, is relatively high, whereas the local average velocity of analyte molecules in the surrounding region, where P i s larger, is lower. These local average velocities differ, simply because analyte molecules spend a greater fraction of time in the mobile phase in the capillary center than in the surrounding region. These velocity differences necessarily disperse analyte molecules along the flow direction, even though (to a first approximation) neither the mobile- nor the micellar-phase velocity varies across the capillary cross section. Rather, what varies is the fraction of analyte in each phase. The transchannel effect proposed here differs substantially from its chromatographic counterpart. One could equally well describe the resultant dispersion as simply arising from a radial temperature gradient, as one does in describing the radial variation of electrophoretic mobility in CZE. Here, one chooses to use the descriptive term, transchannel mass transfer, principally because the analysis below is similar to that developed for transchannel mass transfer in chromatography. In reality, radial diffusion and adsorption-desorption kinetics exist and complicate the phenomenon outlined above. One can, however, draw the following general conclusion. As long as the time td required for analyte molecules to diffuse across the capillary cross section (which typically varies from milliseconds to seconds in micrometer-diameter capillaries) is much greater than the round trip residence time t, + t,, (which typically varies from microseconds to milliseconds), then molecules in immediately neighboring flow streams occupy both phases many times before they substantially diffuse in the radial direction. In this limiting case, the actual analyte velocity in those flow streams approaches the average velocity described above. Because the average analyte velocity varies across the capillary cross section, these molecules consequently diffuse from regions of high (low) average velocity to regions of low (high) average velocity. In the process, axial dispersion is reduced. Furthermore, when td >> t , t,,, then micellar mass transfer is typically much smaller than the transchannel transfer described here, because both dispersions are proportional to the characteristic time required for molecules to change their velocity states. When this inequality holds, one can assume that dispersion due to adsorption-desorption kinetics is negligible in comparison to dispersion due to the radial gradient in P. This assumption greatly simplifies the analysis presented below. The random-walk theory for this transchannel mass transfer is now developed. As stated above, one will assume that the average analyte velocity varies over the radial cross section.

+

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This assumption implies that analyte molecules occupy both phases a large number of times during the time required for radial diffusion, i.e., that td >> t, + t,,. This time is td = ( a r J 2 / 2 D ,where r, is the capillary radius, a is a scaling coefficient less than 1, and D is the effective radial diffusion coefficient of the analyte. In MECC, analyte molecules can diffuse by two means (17). First, as in CZE and other forms of chromatography, they can diffuse through the mobile phase with diffusion coefficient D,. Secondly, in contrast to CZE and most other forms of chromatography, they are also transported by the diffusion of the analyte-micelle adduct, which has diffusion coefficient D., Because analyte moleculbs diffuse with diffusion coefficient D, for the fraction of time equal to R, and with diffusion coefficient D,, for the fraction of time equal to 1- R, the appropriate value of D must be calculated as a time-weighted average of D, and D, (17). The time td consequently equals t d (=a r )2z - 2(RD, 20

+ ( 1 - R)D,,)

(16)

For dilute zones, D,, essentially equals the micellar diffusion coefficient (17). During the time td the molecule diffuses radially, it also is displaced axially the distance I’ = Avtd from the zone center. This distance is different from that in chromatography, in which analyte molecules diffuse from one nonzero velocity extreme to another through the mobile phase. In MECC, molecules diffuse from one nonzero velocity extreme to another through a “soup” containing two uniformly mixed phases. The presence of the micellar phase in this “soup” cannot be neglected in determining Au. An approximation to I’ is easily derived from the following assumptions. This distance depends on retention ratio R, as in chromatography (12). When R = 1, then, as in chromatography, I’ = 0; no nonequilibrium dispersion occurs because all analyte molecules move with mobile-phase velocity u,. When R approaches zero, however, then 1’approaches zero; no nonequilibrium dispersion occurs because all analyte molecules are bound to micelles, which migrate with velocity u,. (In this case, the partition coefficient is so large that its radial variation negligibly affects the velocity of analyte molecules.) Finally, when R = 0.5, then zone velocity v lies halfway between u, and u,, and 1’ = (u, - U,,)td/2. The function of R that describes these three cases is 1‘ = AUtd = -2R(1 - R)p,,tdE (17) One can show that the value of Au in this equation equals the distance 1 (eq 8) divided by the average analyte residence time, (tm + tmc)/2* Equation 17 can be alternatively derived in the following manner, which gives additional insight into the physicochemical processes of dispersion. In chromatography, in which only the mobile phase moves through the channel, Au is the difference between an extreme value of the mobile-phase velocity and the average zone velocity u (12).In MECC, both mobile and micellar phases move through the channel, and an analyte molecule can consequently assume two extreme values of velocity, u, and u., If only the mobile phase were in motion, then Au would equal u, - u; if only the micellar phase were in motion, then Au would equal v - umc. Both phases, in fact, are in motion, and the appropriate value of Au must be calculated as a weighted average of these two velocity differences, just as D was calculated as a weighted average of D , and D,,. In other words l’= AVtd = ((v, - v)R + ( V - umc)(l - R))td (18) which, with eq 3-6, one can show equals eq 17. Finally, one must evaluate the number n of steps, which in this case is simply the number of times an analyte molecule

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ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989

can diffuse the distance arc in the time interval t,(12)

L

n = -t,= - =L td

vtd

(pea

(19)

+ (1 - R)pmc)Etd

where eq 3-6 are used to obtain the final equality. Combining eq 17 and 19, one can show that the transchannel nonequilibrium plate height, Ht= lf2n/L,equals

Equation 20 differs radically from the random-walk equation for the transchannel nonequilibrium plate height in chromatography, which is ((1- R)(cur,))2u,/2RDm(18). If one takes the limit of eq 20, as pea -pmc (i.e., as u, 0) and D,, 0, one obtains an equation closely related to the one just cited. This result should not be particularly surprising, because the physical situation described by these limits is one in which the micellar phase is stationary and MECC closely (but not identically) resembles chromatography. In this limiting case, step length 1’ equals its chromatographic counterpart, which is (1- R)u,td (12),multiplied by the factor 2R. The number n of steps is smaller in MECC than in chromatography by the factor R (12), because time t d is inversely proportional to R. Consequently, in this limiting case, Ht should be ( ! W 2 / R= 4R times the random-walk equation cited above. The limit of eq 20, as u, and D,, approach zero, is indeed 4R times this result -+

-

-+

D,--o

where, as before, u, = p e s is the mobile-phase velocity. A cautious interpretation of eq 21 assists one in assigning an approximate value to coefficient CY and completing the derivation. This limiting equation has the same functional form as the rigorous transchannel plate height equation in chromatography, when the chromatographic mobile-phase velocity is independent of the capillary radius (12). Because eq 21 corresponds to that case in MECC which most closely resembles chromatography, and because the mobile-phase velocity in MECC is also independent of the capillary radius, this equivalence of functional form is unlikely to be coincidental. One can exploit this equivalence by assigning CY the value (2.2lI2)-l, which makes eq 21 equal exactly the rigorous equation. Equation 20 consequently becomes

As before, a coefficient H,= CtV can be defined such that

The above assignment to CY, although somewhat rationalized, is still arbitrary. The basic criterion by which to test this assignment is the agreement between experiment and theory over a wide range of R values. This agreement is found, as discussed below. PROCEDURES

Experimental values of coefficient C were estimated from published data for comparison to the theoretically derived values, C, and C,. Specifically, three C values were estimated from van Deemter-like plots of (total) plate height H,, vs voltage V , which were reported as Figures 2 and 3 in ref 4,

for the 4-ChlOrO-7-nitrObe”n (NBD) derivatized analytes, ethylamine and cyclohexylamine. Both analytes (henceforth described as ethylamine-NBD and cyclohexylamine-NBD) were separated on a 75-pm-diameter, 75-cm-long fused silica capillary column and were eluted with a mobile phase composed of 0.003 M Na2HP04buffer and 0.01 M sodium dodecyl sulfate (SDS) surfactant. Cyclohexylamine-NBD was also retained on a 25-pm-diametercolumn, with all other conditions unchanged. The analytes were monitored by on-column fluorescence. Five additional estimates of C were made from the numbers N of theoretical plates cited in Table I of ref 5, for the selected nucleic acids 2‘-deoxycytidine (dC), 2’-deoxythymidine (dT), 2’-deoxyuridine (dU), 5-methyl-2’-deoxycytidine (MedC),and 8-bromoguanosine(BrG). These analytes were separated on a 60-pm-diameter,68.5-cm-long fused silica capillary column and were eluted with a mobile phase composed of 0.01 M Na2HP04/0.006 M Na2B,0, buffer and 0.075 M SDS surfactant. These analytes were monitored by on-column UV absorbance at 256 nm. The experimental coordinates (HbbV) were evaluated from the figures in ref 4 with a True Grid 1011 Digitizer (Houston Instruments, Austin, TX). The additional coordinates (Hht = L / N , V)were evaluated from the tabulated plate numbers in ref 5. Each set of coordinates was then fit by least-squares methods to the van Deemter-like capillary equation

B H,, = - + CV (24) V to determine coefficients B and C. The minimization was implemented with the IMSL FORTRAN subroutine BCONF on an IBM 3081-GX computer at Southern Illinois University. The retention times t, of the analytes and the void times t o of the mobile phases were estimated from electrokinetic chromatograms in ref 4 and 5. The electroosmotic coefficients pUeowere then calculated as (19) r2

LI

Cleo

=-

(25)

t oV

where V = 15000 V and L = 0.75 m for the NBD derivatives, and V = 10000 V and L = 0.685 m for the nucleic acids. (These voltages correspond to those used to generate the chromatograms.) The mobility sums u,/E = pmc+ pw were then calculated as peo(tO/tmic), where tmicis the estimated elution time of the micelles. The ratio t,/t,i, for the NBD derivatives was equated to 0.4,a value reported in ref 4 and determined by Sepaniak and co-workersfrom previous work under similar conditions. Quantity t o / t d cfor the nucleic acids was equated to 0.27, the ratio of the elution times of the void peak and the final peak. (While the author is not certain that the elution time of the final peak equals tdc, the value of pmc so calculated is consistent with that determined from the work of Sepaniak and co-workers, as shown in Table I.) The mobility of the micelles, b,,was then computed as the difference between p,, + peo and pea. Experimentalvalues of R were calculated from the retention times t , and the expression (1) r

. I

J

The results of these computations are reported in Tables I and 11. RESULTS AND DISCUSSION

Nonideal contributions (e.g., from injection effects) to plate height are usually found in all chromatographic-likeseparation methods. Experimental measurements of H,, may consequently vary somewhat among different laboratories. To

ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989

+

~

Table I. Values of pm, pm p,, and pmcComputed As Described in the Procedures Section

value x lo8. m2/V.s mobility

ref 4 4.6

4.1

( t o = 13.6 min)

(to= 18.7 min)

1.8 -2.8

2.0 -2.1

P,

eo + Wmc U"... ~

~~

ref 5

~

reduce the effects of these variations, the testing and characterization of nonequilibrium theories are often carried out by comparing experimental and theoretical nonequilibrium coefficients, instead of experimental and theoretical nonequilibrium plate heights. One commonly finds that experimental and theoretical C values correlate well in other forms of chromatography, whereas their plate height counterparts differ by roughly constant values (20,21). Accordingly, one will also compare experimental and theoretical C values here, instead of H values. Table I1 reports the seven C coefficients computed from the least-squares fit of eq 24 to the experimental data. The sums of squares computed from the fits range from 1.6 X 10-lo to 8.6 X m2; eq 24 by and large fits these data well. Micellar Mass Transfer. To calculate the theoretical coefficient C,, one must know the average residence times t,, of these somewhat hydrophobic analytes in SDS micelles. These lifetimes are unknown, but are unlikely to exceed the residence times of strongly hydrophobic phosphorescent arenes, which are roughly 4-250 ps (22,23). If one assumes that t,, equals the upper limit, 250 ps, of this range and substitutes the appropriate data from Tables I and I1 into eq 14, one calculates that the C, values range from 1.4 X to 4.8 X 10-l2m/V. These estimates compare most unfavorably with the experimental C values, which range from 1.1 X 10-lo to 9.0 X m/V; the average error between theory and experiment is -99 f 1%. Even if the micellar residence times exceed 250 ps, they most improbably exceed this value by more than 2 orders of magnitude, which is what would be required, on the average, to reconcile theory with experiment. Furthermore, theory indicates that C, is independent of the capillary radius, whereas experimental C values depend on r,. Thus, the observed nonequilibrium effects are most likely not due to micellar mass transfer, in agreement with the conclusion reached by Sepaniak and Cole (4) and Terabe and co-workers (11). Transchannel Mass Transfer. To calculate the theoretical coefficients Ct, one must know the diffusion coefficients D, of these analytes in the mobile phase. These values were estimated from the Reddy-Doraiswamy equation (24) and are reported in Table 11. For the NBD compounds, the micellar diffusion coefficient D, was equated to 1.0 X 10-lom2/s, which approximatelyequals D, in a 0.01 M SDS solution containing

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less than 0.1 M NaCl (17); this number should also be appropriate for a 0.01 M SDS, 0.003 M Na2HP04solution. For the nucleic acids, which were separated with a higher SDS concentration,D, was equated to 8.0 X 10-l' m2/s (17). From these diffusion coefficients and the data reported in Tables I and 11, one can determine the Ct values reported in Table 11, which vary from 0.22 X to 20 X m/V. These values compare favorably, considering the inexactness of the random-walkmodel, with the experimental C values reported above; the percentage errors between theory and experiment are 10,120,16, -95, -38, -69, -32, and 40. The average error is only -5.8%. As observed by Giddings, an order-of-magnitude agreement between experiment and theory (which corresponds to an error between -90 and 900%) is about all one can reasonably expect from a random-walk analysis (12). Seven of the eight computations agree with experiment within, and usually well within, this criterion. Therefore, these calculations are much better than they superficially appear to be and strongly support the hypothesis that the nonequilibrium effects reported in ref 4 and 5 are largely resistances to transchannel mass transfer. All experimental and theoretical nonequilibrium coefficients are reported in Table I1 for purposes of comparison. The substitution of the appropriate data reported above and in Tables I and I1 into eq 4,5, and 16 confirms that the ratio of the diffusive exchange time to the average round trip analyte residence time, t d / ( t , + t,,), is much greater than unity, in all cases but one. As discussed previously, this inequality must hold if analyte molecules are to occupy both phases a large number of times during the radial diffusive step. If one assumes that t,, < 250 ps, the ratio is greater than (and typically much greater than) 41 for all analytes except dC, for which the ratio is greater than 21. One should note that this latter case, in which the fewest number of velocity changes occur during time t d , corresponds to the poorest agreement between theory and experiment. Because the proposed theory has been applied to a limited data set, one must be wary of extensive generalizations and conclusions. To date, however, its predictions are among the only ones that closely agree with experiment, if one uses reasonable physicochemical parameters. Even though the theory is inherently approximate, one must acknowledge that it tracks the reported experimentaldata fairly well over a large range of retention ratios (0.87 < R C 0.21) and therefore most likely contains some elements of truth. In particular, one should note that the two largest experimental C values correlate with the two largest theoretical ones. Such a relatively complex expression, which depends on the independent parameters R,D,, D,, pm, bc, and r,, is unlikely to fortuitously predict answers that are so close to experiment. Yet the author is fully aware that this theory only partially explains nonequilibrium effects in MECC. In particular, any comprehensive theory must explain why such effects are not

Table XI. Data Deduced from Electrokinetic Chromatograms of Ethylamine-NBD and Cyclohexylamine-NBD in Ref 4 and Various Nucleic Acids in Ref 5 WOD,, mz/s

W B ,m.V

W0C, m/V

10loC,,m/V

101oC,~m / V

compound

t,, min

R

(theoret)

(exptl)

(exptl)

(theoret)

(theoret)

ethylamine-NBD cyclohexylamine-NBD

15.3 25.1

0.81 0.23

7.8 7.1

1.5

3.0

3.3

0.048

2.2 3.5 2.3 7.4 3.1 2.4 5.7

9.0 1.9 4.6 1.1 3.6 3.1 5.5

20.0 2.2 0.22 0.68 1.1 2.1 7.7

0.026 0.026 0.015 0.025 0.029 0.036 0.014

75 Pmb 25 pmb

dC dU MedC

dT BrG (It,,

= 250 ws.

19.4 20.1 20.4 21.3 31.8

Capillary diameter.

0.93 0.87 0.84 0.77 0.21

7.7 8.1 7.4 7.6 7.1

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always observed at low voltages. For example, one observes that the plate height minima associated with the analytes examined here typically lie between 5 and 10 kV. (Perhaps one should reference the minima to the electric field strength, instead of the voltage, but in all cases the capillary lengths are comparable.) In ref 5, however, several nucleic acids that differ only subtly in structure and retention ratio R from the ones examined here exhibit no plate height minima at voltages up to 23 kV. Similarly, almost all the analytes separated by Terabe and co-workers do not exhibit minima at comparable voltages (11). The proposed theory is promising but apparently incomplete. Much theoretical and careful experimental work remains to be done. Furthermore, although the author previously argued that the radial variation of partition coefficient with temperature could be the principal origin of nonequilibrium dispersion, he cannot justifiably state that the close agreement between theory and the experimental results cited here constitutes proof of thii hypothesis. Proof must await a study that is more detailed than a random-walk analysis. However, he can, and does, strongly argue that the nonequilibrium dispersion considered here appears to originate from some form of transchannel mass transfer which is consistent with the derivation of eq 16-20 and 22 and that the proposed hypothesis is consistent with this derivation. One should note some fundamental limitations of the present development. First, the theory should work best for intermediate values of R because, when R is near zero or unity, molecules cannot occupy both phases a statistically large number of times during the diffusive exchange time. Hence, one should not be surprised that the theory cannot account well for the observed nonequilibrium dispersion recently reported by Terabe and co-workers (11),which corresponds to R = 0. Secondly, the theory predicts that H varies linearly with electric field strength E and voltage V. Experimentally, this dependence is observed only when the electrolyte concentration is low ( 4 ) . Nonlinear variations of H with V are found with higher electrolyte concentrations and must be explained by a more detailed theory. In part, such variations might be attributable to Joule heating. Role of Longitudinal Diffusion. As an aside, if one assumes that the term B / V in eq 24 represents the longitudinal diffusion plate height, H I = u t / L = 2Dt,/L, one can show that

(The expression H,= 2 D t J L differs from the longitudinal diffusion plate height proposed by Sepaniak and Cole, in which micellar diffusion is neglected (4)J By substituting the appropriate data cited above or reported in Tables I and I1 into eq 27, one calculates that B ranges from 0.026 to 0.012 m.V. These estimations agree fairly well with the B values reported in Table 11. Approximations in the Proposed Theories. 1. Radial Variation of Electroosmotic Flow. In the above analyses, one assumed that the electroosmotic flow velocity equals the constant u, at all radial positions in the capillary. In fact, the flow varies over about five or so thicknesses of the electrical double layer at the capillary-solution interface, from zero at the capillary wall to u, in the solution bulk (25). Because typical double-layer thicknesses (e.g., 0.0014 pm for an ionic strength of 0.05 M (25))are much less than typical capillary diameters (e.g., 50 wm), the fraction of the capillary diameter over which flow varies (about 0.03% in the above case) is negligibly small. This variation cannot be neglected, however, when the ratio of the capillary radius to the double-layer thickness is on the order of unity (26).

One might intuitively expect the flow to vary with distance from the capillary center, because the radial temperature gradient brought about by Joule heating produces a radial viscosity gradient, and u, depends on viscosity. As shown by HjertBn, however, this viscosity gradient affects the rate of flow only near the capillary-solution interface, where the electrical charge density is nonzero (27). This variation can be ignored, for the reason given above. 2. Variation of p,,, D,, and D,, with Temperature. In the analyses presented above, the transport parameters hc, D,, and D,, were treated as constants. Because of radial temperature and viscosity gradients, these parameters actually vary across the capillary cross section. These variations may be important, especially at high field strengths, and one shortcoming of the present theory is their neglect. This preliminary study suggests that fortuitously these variations have small effects, at least under experimental conditions similar to those reported in ref 4 and 5. Similar variations of these parameters are ignored in simple theoretical studies of CZE, apparently without adverse effects (8, 19). 3. Variation o f t , and t, with Temperature. The analyte residence times t, and t, vary continuously over the capillary cross section, because of the radial temperature gradient. In the above analyses, they are treated as constants. Therefore, eq 3 does not properly describe the average zone velocity v, and consequently both Av and R are functions of temperature. While these preliminary results indicate that the resultant error is probably small, one cannot easily estimate its magnitude. 4 . Micellar Polydispersity. Because micelles are polydisperse structures, a distribution of micelles with different aggregation numbers exists, and micelles with different aggregation numbers exhibit different mobilities. This distribution of mobilities results in an additional nonequilibrium dispersion beyond that addressed by the above equations. This additional dispersion is expected to equal zero, when R equals 1, and to reach a maximum as R approaches zero. Terabe and co-workers recently proposed a random-walk theory for the polydispersity contribution to plate height in MECC (11). The theory exhibits the expected dependence on retention ratio R described above. However, to obtain agreement between theory and the one experimental nonequilibrium effect reported in their work, the authors were forced to assume that the analyte Sudan I11 remains dissolved in SDS micelles for the full lifetimes of the micelles, Le., 10 ms. In light of the short residence times t,, (e.g., 4-250 p s ) exhibited by other hydrophobic analytes, this physicochemical parameter may not have been realistically estimated. An alternative analysis of the polydispersity contribution to plate height will not be considered here. Instead, the relative importance of polydispersitywill be gauged indirectly by a propagation of errors, which will relate the expected statistical fluctuation uH in H to the statistical fluctuation ub, due to polydispersity, in hc. The fluctuation UH so calculated cannot and should not be interpreted as a measure of nonequilibrium dispersion due to polydispersity; rather, a proper expression of this dispersion would be an additive term to the above H equations. The fluctuation UH, however, is in some manner proportional to this term, because a large ub results in a large uH, as determined by propagation of errors. Both eq 11 and 22 can be written in the general form

where a, b, and c are constants and for which

(29)

ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989

In other words, the relative variation of the plate height roughly equals the relative variation of the mobility. If one assumes that the micelles are spherical and that the friction coefficient is proportional to the cubic root of the number average aggregation number m (In,then

(30) where a,,, is the standard deviation of the aggregation number. Both m and a,,, are complex functions of surfactant and salt concentration,which have not been reported for SDS solutions with the compositionsdetailed above. Therefore, the relative error in H must be estimated from eq 30 with other data. In general, both aggregation numbers and micellar polydispersity increase with increasing surfactant and salt concentration (28). Measurements of polydispersity based on quasielastic light scattering yield results proportional to the z-average aggregation number (29-31), whereas those based on time-resolved fluorescence quenching yield results proportional to the number average aggregation number (32), which is the more useful quantity here. Measurements based on the former method tend to be larger, and on the latter method to be smaller, than theoretical estimates of a,,,, which equal about 10-14 for low surfactant and salt concentrations (28,33). If one assumes that a,,, equals 14 (and m = 561, then the final equality in eq 30 predicts that the relative error in H is, at most, about 8%, Although this calculation does not yield negligible results, it does suggest that the polydispersity of the micelles has only a second-order influence on nonequilibrium dispersion.

CONCLUSIONS The analysis of the limited data set presented here suggests that transchannel mass transfer is a likely origin, at least in some instances, of the low-voltage nonequilibrium dispersion observed in MECC. Clearly, the proposed theory is preliminary at this stage and must be tested against a more substantial body of experimental data to gauge its universality. Of equal importance is to discover why all analytes are not subject to this effect. This work will be undertaken shortly in the author’s laboratory. One conclusion that can be drawn from this work is that nonequilibrium effects should in general be reduced by decreasing the capillary radius. If the author’s hypothesis is correct, a reduction of capillary radius will decrease the radial temperature gradient and consequently the radial variation of P that generates local differences in average analyte velocity. Furthermore, the diffusive exchange time of analyte molecules will also be reduced. An alternative conclusion drawn from this work is that nonequilibrium effects can be reduced by decreasing the micellar mobility pmc. One should not commonly decrease the plate height in this manner, however,

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because the peak capacity of the chromatogram will also be reduced.

LITERATURE CITED Terabe, S.; Otsuka, K.; Ichlkawa, K.; Tsuchiya, A.; Ando, T. Anal. Chem. 1984. 5 6 . 111. Armstrong, D. W: Sep. Purif. Methods 1985, 14, 213. Sepaniak, M. J.; Burton, D. E.; Maskarlnec, M. P. I n OrderedMedle in Chemical Separations; Hlnze. W. L., Armstrong, D. W., Ed.; ACS Symposium Series 342; American Chemical Society: Washington, DC, 1987; p 142. Sepanlak, M. J.; Cole, R. 0. Anal. Chem. 1987, 59, 472. Row, K. H.; Grlest, W. H.; Maskarinec, M. P. J. Cbromatogr. 1987, 409, 193. Jones, H. K.; Nguyen, N. T.; Yonker, C. R.; Smlth, R. D. 39th PMsburgh Conference and Exposition on Analytical Chemistry and Applied Spectroscopy, New Orleans, LA, February, 1988 No. 843. Liu, J.; Dolnik, V.; Novotny, M. 1st International Symposium on H!gh Performance Capillary Electrophoresis, Boston, MA, April, 1989; No. MP-103. Jorgenson. J. W.; Lukacs, K. D. Science 1983, 222, 266. L a w , H. H.; McManlglll, D. Anal. Chem. 1986, 5 8 , 186. Fwet, F.; Deml, M.; Bocek, P. J. Chromafogr. 1988, 452. 601. Terabe, S.; Otsuka, K.; Ando, T. Anal. Chem. 1989, 61, 251. Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965. Giddlngs, J. C., personal communication, 1989. Terabe, S.;Otsuka. K.; Ando, T. Anal. Chem. 1985, 5 7 , 834. Balchunas, A. T.; Sepanlak, M. J. Anal. Chem. 1988, 60, 617. Bird, R. 6.; Stewart, W. E.; Lightfwt, E. N. Transport phenomena; Why: New York, 1960. Stigter, D.; Williams, R. J.; Mysels, K. J. J. Phys. Chem. 1955, 5 9 , 330. Giddings, J. C. I n Treatise of Analyrical Chemistry: Pari I , 2nd ed.; Koithoff, I.M., Elvlng, P. J., Eds.; John Wlley: New York, 1982; Vd. 5. Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 5 3 , 1298. Tsuda, T.; Novotny, M. Anal. Chem. 1978, 5 0 , 632. Giddings, J. C.; Chang, J. P.; Myers, M. N.; Davis, J. M.; Caldwell. K. D. J. Cbromafogr. 1983, 255, 359. Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. SOC. 1979, 101, 279. Cline Love, L. J.; Habarta, J. G.; Dorsey, J. G. Anal. Chem. 1984, 5 6 , 1132A. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGrawHIII: New York, 1975. Hunter, R. J. Zeta Potent&/ in C o I M Science: Principles and Applications; Academic: London, 1981. Rice, C. L.; Whitehead, R. J. J. Phys. Chem. 1965, 6 9 , 4017. Hjerth, S . Chromatogr. Rev. 1967, 9 , 122. MukerJee, P. I n Micellizafion , Solubilization, and Microemulsbns ; MItal, K. L., Ed.; Plenum Press: New York, 1977; p 171. Mazer, N. A.; Benedek, G. 6.; Carey, M. C. J. Phys. Chem. 1976. BO, 1075. Corti, M.; Degiorgio, V. Chem. Phys. Len. 1978. 5 3 , 237. Missel, P. J.; Mazer, N. A.; Benedek, G. 6.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 8 4 , 1044. Warr, G. G.; Greiser, F. J. Chem. SOC.,Faraday Trans. 1 1986, 8 2 , 1813. Tanford, C. The Hydrophobic Effect; Wlley-Interscience: New York, 1973.

Joe M.Davis Department of Chemistry and Biochemistry Southern Illinois University Carbondale, Illinois 62901

RECEIVED for review October 4, 1988. Accepted August 3, 1989. The author thanks Southern Illinois University for the start-up funds with which this work was conducted.