Anal. Chem. 198Q,61, 251-260
also thank 0. Hindsgad for assignment of the NMFt spectrum of DHA.
LITERATURE CITED Janata, J.; H u h , R. J. Ion SelectiveElectrodes In Analytkxl ChemisQ: Plenum Press: New York, 1980 Voi. 2, Chapter 3. Janata, J. In SdM State Chem/cal Sensors; Janata, J., Huber, R. J., Eds.: Academic Press: London, 1985. Blackburn. 0.;Janata, J. J . Ekb.ochem. Soc. 1982. 129. 2580. Slbbold, A.; Whaiiey, P. D.;Covington, A. K. Anal. Chim. Acta 1984, 4 7 , 159. Satchwlli, T.; Harrison, D. J. J . Electroanal. Chem. 1988, 202, 75. Harrison, D. J.; Li. X.: Permann. D. Electrochemical Sensors for Blomedlcai Applications; Li, C. K. N.; Ed.; Roc.-€lectrochem. Soc. 1986, 8 6 - 1 4 , 74. Harrlson, D. J.; Cunningham, L. L.; Li, X.; Teclemariam, A.: Permann, D. J . Electrochem. Soc. 1988, 135, 2473. Oesch, U.; Simon, W. Anal. Chem. 1980, 52, 692. Oesch, U.; Slmon, W. LMgest of Technlcal Papers. 4th Int. Conf. on SolM State Sensors and Actuators, Tokyo 1987, 755.
25 1
(10) Bezegh. K.; Bezegh, A.; Janata, J.; Oesch, U.; Xu, A.; Simon, W. Anal. Chem. 1987, 57, 2846. (11) Bond, D. M.; Kratochvii. J.; Treasure, T. Roc. Physbl. Soc. 1978, 265, 5P. (12) Li, X.; Verpoorte, E. M. J.; Harrison, D. J. Anal. Chem. 1988, 60, 493. (13) Moody, 0. J.; Thomas, J. D. R. Selective Ion Sensitive Electrodes; Merrow: London, 1971. (14) Scholer, R. P.; Simon, W. Ch/m/a 1970, 2 4 , 372. (15) Llndner, E.; &&f, E.; Niegreisz, 2.; Toth. K.; Pungor, E.; Buck, R. P. Anal. Chem. 1988. 60, 295. (16) Harrison, D. J.; Cunningham, L. L., unpublished results, University of Alberta, Sept 1986. (17) Senturia, S. D. Dlgest of Technical Papers. 4th Int. Conf. on SolM State Sensors and Actuators, T w o 1987, 11.
RECEIVEDfor review July 25,1988. Accepted October 28,1988. We thank the Natural Sciences and Engineering Research Council of Canada for support of this research.
Band Broadening in Electrokinetic Chromatography with Micellar Solutions and Open-Tubular Capillaries Shigeru Terabe,* Koji Otsuka, and Teiichi Ando
Department of Industrial Chemistry, Faculty of Engineering, Kyoto University, Sakyo-ku, Kyoto 606, Japan
The bamCbroadenlng phenomena that occur In the separation column during electroklnetlc chromatography were studled. Flrsi In order io let column band broadening domlnaie, extracolumn effects were studled on the bask of theoretlcai conslderatlons and experimental data. The llmit of InJection, whlch is the length of the tube occupied by a sample sdutlon, was 0.8 mm and that of deiectlon bandwidth along the tube axls was also 0.8 mm. Observed minimum plate height with a 50 Nm 1.d. open-tubular column was 2 pm under the above condiilons. The causes of thls band broadenlng were ascribed to five mechanlsms: longitudlnal dlffuslon, sorptiondesorpilon klnetlcs, intermicelle mass transfer, radlal temperature gradient effect on electrophoretlc veloclty, and electrophoretlc dlsperslon of the micelles (heterogeneity In mlcelle mobllltles). The relatlve slgnlflcance of each contrlbuilon was estlmated by comparing the theoretlcal conslderatlon wiih the observed dependence of the plate height on the veloclty. The longitudinal diffuskn was a dominant factor when the veloclty was slow; sorptlon-desorptlon klnetlcs and heterogeneity became slgnlflcant factors as higher voltages were applbd. The lntermkelle mass transfer mechanlsm and temperature gradient effect owing to Joule heating were not Important.
Electrokinetic chromatography with micellar solutions (I, 2) is a newly developed chromatographic method that utilizes the technique of free zone electrophoresis in open-tubular capillaries ( 3 , 4 ) . The micelle of an ionic surfactant participates in the distribution of a solute as a moving "stationary" phase and an aqueous solution as the other phase. The micelle and the aqueous phase migrate at different velocities due to the electrokinetic forces, i.e., electrophoretic and electroosmotic effects, which permit chromatographicelution in a time window determined by the velocities of the aqueous phase and the micelle (2). That is, the two phases between which the solute is distributed constitute a homogeneous solution, but 0003-2700/89/0361-025 1$01.50/0
they move differentially. Although this method obviously belongs to a chromatographic technique, it can be said that this approach has widened the electrophoretic separation technique to include the separation of neutral molecules by an indirect process. The high efficiencies attainable by capillary zone electrophoresis have been discussed by Mikkers et al. (3, 5 ) , Jorgenson and Lukacs ( 4 ) ,and Lauer and McManigill (6). The migrational dispersion has been ascribed to differences in electrophoretic mobilities between carrier constituents and sample constituents (5), and it has been shown that the asymmetric concentration distribution of the band frequently observed in free zone electrophoresis can be suppressed only by the application of very small amounts of samples ( 3 ) . Jorgenson and Lukacs (4) have discussed the possibility that since the electroosmoticflow can approximate to a plug flow, longitudinal molecular diffusion should contribute much to the band broadening in capillary zone electrophorsis, although that contribution is generally negligible in conventionalliquid chromatography. We have reported in a previous paper (I)that plate heights of 1.9-3.6 pm were obtained for alkylphenolsin electrokinetic chromatography. Similar efficiencies have been observed for (pheny1thiohydantoin)amino acids (7), chlorinated phenols (8),aromatic sulfides (9),and nucleosides and oligonucleotides (IO). In order to attain the maximum performance, we have systematically studied band broadening in this chromatographic method. Sepaniak and Cole (11) have recently reported on column efficiency in electrokinetic chromatography and we will compare our results with theirs when it is appropriate. Instrumental conditions that are necessary for obtaining the chromatogram without any loss of performance characteristics of electrokinetic chromatography are presented first; that is, extracolumn effects such as injection volumes, the cell volume of the detection part of the tube, the time constant of the detector, and the response of the recorder pen are discussed. The observed plate heights under the minimal extracolumn conditions are considered as the results of band 0 1989 Amerlcan Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
I
I
q . 4
Table I. Asymmetry Factors4 in Electrokinetic Chromatogramsb current/pA voltagec/kV vwd/mm s-I
40 21.3 2.15
30 18.0 1.74
25 15.7 1.33
20 12.9 1.20
1.00 1.03 1.02 0.98 0.92 0.89 1.06
0.90
1.12 1.06 1.06 1.05 0.93 0.89 1.03
0.68 1.02 1.21 0.93
15
10.0 0.90
10 6.4 0.48
asymmetry factor resorcinol phenol
0
2
I v,,/irm
s-’
Flgure 1. Dependence of plate height on electroosmotic velocity. Solutes used were as follows: (1) resorcinol, (2) phenol, (3)p-nitroanillne, (4) nitrobenzene, (5) toluene, (6) 2-naphtho1, (7) Sudan 111.
broadening only in the column. The in-column band broadening is discussed in terms of the total contribution of five causes: longitudinal diffusion, the kinetic process of the solute distribution between the micelle and the aqueous phase, mass transfer in the intermicelle aqueous phase, radial temperature gradient effect on electrophoretic velocity, and electrophoretic dispersion of the micelle. These mechanisms are assumed as independent phenomena, and the relative significance of each contribution is evaluated by comparing the calculated results with the experimental data and followed by a discussion on the possible coupling of these mechanisms. It should be noted that the discussion below is limited to the separation of uncharged solutes by electrokinetic chromatography with micellar solutions. EXPERIMENTAL SECTION Apparatus, Reagents, and Procedure. An apparatus similar to that employed previously was used (I,2, 12). A Shimadzu Chromatopac C-R3A data processor (Kyoto, Japan) was used to measure retention time tR,peak height h, and peak area A, while plate number N was calculated by the following equation (13):
F) 2
N =
27r(
The sampling frequency of the data processor was 10 Hz. The time constantsof the detectors employed in this study were 0.05 s for a Jasco UVIDEC-100-V(Tokyo,Japan) and about 0.1 s for a Jasco UVIDEC-100-11. The latter detector’s electronic circuit was slightly modified to reduce the original time constant. Injected sample volume was estimated as follows: The capillary tube was filled with a phenol solution of a known concentration (ca. 1mg mL-l) by a 100-pL gas-tight microsyringe (Unimetrics, Shorewood, IL), and the absorbance of the solution at 220 nm was measured against a blank solution in order to determine detector response. After substitution of a sodium dodecyl sulfate (SDS) solution in the borate-phosphate buffer (I)for the phenol solution inside the tube, a phenol solution of a known concentration was injected into the positive end of the tube by holding it up for a given period of time at a level higher than that of the SDS solution at the negative electrode, as described previously (I),and a chromatogram recorded. The total amount of the injected phenol was calculated by assuming Gaussian peak shape, making it easy to determine the injection volume or the corresponding length of the tube. Other procedures and reagents are the same as given in previous papers (I,2). RESULTS AND DISCUSSION
Observed Plate Heights. The dependence of plate height
H on electroosmotic velocity u, is shown in Figure 1, which was obtained with a 0.05 M SDS solution in 0.1 M boratephosphate buffer (pH 7.0) (1)and a 50 pm i.d. X 650 mm fused silica tube at ambient temperature (ca. 25 “C). The solutes that migrated through the effective tube length of 500 mm (from the injection end to the detedor cell) were detected with a spectrophotometer (Jasco UVIDEC-100-V) at 210 nm. The
p-nitroaniline nitrobenzene toluene 2-naphthol Sudan I11
1.13 1.07 1.09 0.79
1.11 0.81
0.90 1.03 0.59
0.91 0.95 1.25 1.05 0.49 1.04 0.49
1.05 1.13 1.12 1.06
“Calculated from peak widths at 10% of peak heights (11). bConditionswere the same as those given in Figure 1. “pplied voltage for the 65-cm tube. Electroosmotic velocity. solutes were the same as those previously described (2) as was the example of the chromatogram (2). Since peaks were very narrow and the trace of the recorder pen was not always of a constant thickness, it was difficult to obtain reproducible values for plate heights if they were calculated from retention times and peak widths at half heights measured on recorded chromatograms. However, the data processor gave satisfactorily reproducible values for every peak. The reproducibilities of retention times and relative peak area have been reported elsewhere (14) for some chlorinated phenols. The coefficient of variation of retention times was less than 1.2% (14).Plots shown in Figure 1 show averages of three to five runs, and coefficients of variation of plate heights were on the average 5.7% for resorcinol, phenol, p-nitroaniline, and nitrobenzene, and 17% for toluene, 2-naphthol, and Sudan 111. It is important to note that eq 1is valid only for symmetrical peaks (13). Examples of peak symmetry factors that were calculated from peak widths at 10% of peak heights (13)are given in Table I. Although symmetry values were not very reproducible,most values were in the 0.80 and 1.20 range and were independent of current or electroosmotic velocity. Table I indicates that the plate numbers or plate heights found in this study can reasonably represent column efficiency. As shown in Figure 1,the minimum plate heights observed were 2 pm for most of the solutes, 2.5 pm for 2-naphthol, and 3 gm for Sudan 111. In addition, the electroosmotic velocity which gave the minimum plate height was different among the three kinds of solutes mentioned above. These plate height values correspond to plate numbers of 250 000,200 000, and 167000 for a 500-mm tube. It should be noted that the velocity of every solute, including electroosmotic and electrophoretic velocity, was linearly dependent on electric current (21, although electroosmotic velocity is adopted in Figure 1. Results shown in Figure 1are substantially different from those reported by Sepaniak and Cole (11): Minimum plate heights are 4.5 pm for 7-nitrObenZOfUrazan(NBD)-ethylamine and about 10 pm for NBD-cyclohexylamine, and these values are obtained when less than 10 kV is applied to the 75 pm i.d. X 75 cm tube. Rapid increases in plate height are observed with increases in applied voltages above about 10 kV. These differences will be discussed later. Instrumental Considerations. In order to rule out the possibility of the significant contribution of extracolumn effects t~ the results shown in Figure 1,instrumental conditions are discussed first. In capillary zone electrophoresis employed in this study, an additional tube is not required to connect the injection port to the separation tube (column) and the column to the detector cell, because a sample solution is introduced at the inlet end of the tube and the solute band is detected while the solute is migrating in the tube. That is, on-column injection and detection are employed as described previously (1).
ANALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
0
2
1
',",
3
L
5
'mm
Flgure 2. Dependence of plate height on the Injection length of the sample solution. Solid line is calculated according to H = 0.167112 2.1.
+
Because of the additivity of variances, the observed total peak variance ubt can be expressed as
+ a i 2+ bde:
= ?,a
(2)
where urn? is the variance generated while the solute stays in the column and uinj2 and ode: are extracolumn variances originating in the injection and detection systems, respectively. It is generally accepted that column efficiency and resolution are not seriously impaired by the 5% increase in peak width (15). Thus ~b:
I( 1 . 0 5 ~ ~ 1 ) ~
(3)
From the data shown in Figure 1,we can take the minimum value of the plate height to be 2 pm, in which case the distance variance?,u becomes equal to 1mm2 for a 500-mm column (2 pm X 500 mm). If we assume the condition described in eq 3, which requires that ub,t" be less than 1.103u,?, the sum of extracolumn variances should not exceed 10.3% of the column variance, Le., 0.103 mm2 in this work. The following discussion mainly describes the variance and the standard deviation in the distance unit, which is easily converted to volume units by multiplying it by a cross section, e.g., 1.96 X mm2 for 50 pm i.d. Injection Volume. It is reasonable to assume that a certain length of the inside of the tube is completely displaced by a plug of the sample solution because the sample solution is very slowly siphoned into the injection end of the tube, e.g., 0.03 mm s-l, in this technique. Under these circumstances, the distance variance ad2is written as that of the rectangular plug uinj2 =
(1/12)linj2
(4)
where 1, is the length of the sample plug inside the tube. If it is assumed that the contribution of uinj2is equal to half the total extracolumn variance, the sample length 1, should be leas than 0.79 mrn ([(12X 0.103 mm2)/2]1/2},which corresponds to 1.55 nL in volume. Although this volume may seem too small to be injected, the procedure employed gave fairly reproducible results (14). The dependence of observed plate height H on injection length 1, is shown in Figure 2 where the applied voltage is 20 kV and current is 37 pA. Experimental conditions were the same as those given in Figure 1 except for the detector employed (Jasco UVIDEC-100-11) and temperature (35 "C). Different volumes of the phenol solution of a known concentration (12 mM in 0.1 M boratephosphate buffer, pH 7.0) were introduced into the tube by changing the siphoning time. Since the total variance ub: is expressed as the product of observed plate height H and column length lcol, the combination of eq 2 and 4 gives 1 1 H=lh? + -(uco? + udet? (5) 12401 401 The second term of the right-hand side of eq 5 can be con-
253
sidered as remaining constant under the conditions shown in Figure 2. The solid line in Figure 2 is calculated with eq 5 where lcolis 500 mm and the value of the second term is 2.1 pm. The good agreement between the calculated and found plate heights supports the above view. The dependence of H on the electromigration injection procedures has been reported by Burton et al. (16). Their results also can be approximately described by eq 5, if the injection lengths are assumed to be much longer than their values. Detection Volume. The above discussion on injection volume can be equally applicable to the role of the detector cell, which consists of a portion of the tube, because of the additivity of variances. Therefore, cell length along the tube axis should be less than 0.79 mm, hence the cell volume of 1.55 nL, although the actual length employed was 0.75 mm. It should be mentioned that light path length is not 0.79 mm, but ranges from 50 pm to zero, averaging 44 pm, because UV absorption is measured through the tube at right angles to the tube axis. The Jasco UVIDEC-100-11found the detection limit of phenol to be about 20 pg in the injected amount at 220 nm. Extracolumn band broadening in the on-column detection technique has been discussed by Yang (17),who gave the percentage loss of resolution (% AR) as
1 ( z;:::)'21
% A R = 1 - 1+-
X 100%
(6)
where Hdetand HmIare the plate heights for the flow cell and the column, respectively, and and 1,, are the length of the flow cell along the tube axis and the column length, respectively. It should be noted that eq 6 is not valid for the evaluation of band broadening attributable to the detector cell. In the derivation of eq 6, the following relationship is used: ude?
= Hdetldet
(7)
The variance given by eq 7 expresses the band broadening observed when an impulse band migrates from the entrance to the exit of the cell. That is, eq 7 is the same as that which describes column variance and does not apply to the variance necessary to evaluate band broadening ascribed to slit length or cell volume. In other words, two peaks separated by the length less than ldet cannot be discriminated with the detector, because the two peaks stay simultaneously in the detector cell. It is apparent that eq 6 is not valid if the following exaggerated example is assumed for on-column detection: When two bands separated by 40 mm (R, = 1)in a 2-m column (N = 4 X lo4) pass through a 50 mm long cell, coalescence of the two bands must be observed. On the other hand, eq 6 predicts a resolution loss of 1.2% if Hdet = Hcol. Extracolumn effect generated in the on-column detector cell can be estimated with a different approach from the variance method as follows: By assuming a Gaussian profile of the peak, the concentration profile observed through the whole cell window can be approximated as (see Figure 3)
where C ( t ) is the concentration measured at the time t , Co is the maximum concentration, ut is the standard deviation of an original Gaussian profile in time units, T is a dummy time variable, and T i s the time the solute stays in the cell, Le., ldet divided by the linear velocity of the solute. Peak profiles calculated according to eq 8 are shown in Figure 4 for T = ut, 2ut, and 3ut. Peak widths at half height are also shown in Figure 4. The peak width for T = 0.70~is calculated to be
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
Band Broadening i n Column. Extracolumn effects are kept minimal in this study as described above, and thus the observed band broadening shown in Figure 1 can be ascribed to in-column effects. We will break down the causes of band broadening into five mechanisms
t-k
(9) + Ht + Hep where Hwl is overall column plate height, and H,, H,,,Haq, H,,and Hepare plate heights generated by longitudinal difHcol
+ -
Flguro 3. Illustration of relationship between the concentration observed through the slit and original Gaussian profile In the on-column detection.
0 855
0 724
= H I + Hmc + Haq
fusion, sorption-desorption kinetics in micellar solubilization, intermicelle mass transfer in the aqueous phase, radial temperature gradient effect on electrophoretic velocity, and electrophoretic dispersion of the micelles, respectively. The relative significance of each contribution is discussed below. Equation 9 assumes that the terms in the right-hand side are independent of one another, but this assumption is not strictly correct. The possible coupling among these mechanisms will be discussed later. Although extracolumn effects are evaluated in terms of variance, it is convenient to discuss band broadening in the column in terms of plate height, because plate height is independent of column length. Plate height Hwl is related to distance variance u,,? by UCO?
ClCO
= Hm1401
(10)
Thus, eq 10 indicates that variance and plate height are interchangeable if column length remains constant. Longitudinal Diffusion. Although longitudinal diffusion is negligible in conventionalliquid chromatography while the solute is sorbed in the stationary phase, it should be mentioned that the solute is subject to the diffusional effect even when it is incorporated into the micelle, as well as when it stays in the aqueous phase, because the micelle itself is also subject to molecular diffusion. According to the Einstein equation, the distance variance caused by longitudinal diffusion u t is given as
0 500 0 LEO 0 427 0 362
U?
ti q
Flgure 4. Peak profiles calculated according to eq 8, where t is taken as zero when the center of the band coincides with the center of the cell: original Gausslan profile (a) and profiles when Tis equal to ut (b), 20, (c), or 3u,(d). Peak widths at half-heights are also shown in t / u , unit.
broader than the original width by 2.4%,or the time variance for the calculated peak is larger than the original ut2 by 4.9%. Thus, ldet should be less than the approximate 0.7 mm obtained from the relationship in which T/ut is equal to lbtluwl, because ldet = Tu,and u,,l = upB,where v, is the velocity of the solute. This result agrees well with the value obtained from the calculation based on the variance only. Time Constant. In order to evaluate time-dependent terms, we will take a condition in which plate height is 2 km when solute velocity is equal to 2 mm s-l. Thus, the total time variance becomes 0.25 s2 for a tube length of 500 mm. In this case, peak width at half height is 1.18 s. The effects of the time constant of the detector amplifier and the balancing time of the recorder on band broadening have been described in detail by Scott (15);therefore, the required values of these constants are discussed only briefly. Since eq 3 is also applicable to time variances, the s u m of time variances of the detector and the recorder must be kept below 10.3% of the time variance of the column. The time constant of the detector employed to obtain data shown in Figure 1 was 0.05 8,Le., ut&: = 0.0025 s2. The balancing time of the recorder employed in this study was about 0.5 s for the full scale deflection; therefore, the time variance of the recorder was equal to (0.5 ~ ) ~ / li.e., 8 , 0.0139 s2 (15). The sum of time variances produced by the detector and the recorder was 0.0164 s2, which was less than 10.3% of 0.25 s2 (0.026 s2).
=2Dt~
(11)
where D is the overall molecular diffusion coefficient of the solute in a surfactant solution and t R is retention time. The variance in eq 11 can be divided into two components 2Daq (ul,aq) 2 = -
+
krt~
(12)
where D, and D,, are the diffusion coefficient of the solute in the aqueous phase and the apparent diffusion coefficient of the solute in the micelle, respectively; k’ is the capacity factor defined by the ratio of total moles of the solute in the micelle to those in the aqueous phase ( I ) ,and and ( u d 2 are the distance variances produced by longitudinal diffusion when the solute is in the aqueous phase and in the micelle, respectively. Here, D,, can be regarded as the diffusion coefficient of the micelle itself in the aqueous phase. The retention time is expressed by the capacity factor as follows (1):
1+R’
tR
= 1
+ (to/t,,)L’tO
where to and t,, are the retention times of an insolubilized solute and the micelle. The former is measured experimentally with the methanol peak and the latter with a tracer of the micelle such as Sudan I11 (2). Plate height HIis described by combining eq 10 with eq 11-14 and by making the substitution, u,, = lcol/tO as (15)
ANALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
255
and k d are the sorption and desorption rate constants, respectively. According to the nonequilibrium theory described by Giddings (21),the set of two equations is derived to find the nonequilibrium departure terms eaq and em, in the two phases
+
Xaq*taq Xmc*€mc =0 v ~ / s’ mrn-’
Flguro 5. Dependence of plate height on the reciprocal of electroosmotic veloclty. Solutes are represented by the same numbers as given in Flgure 1. Dashed lines are calculated (see text).
Equation 15 indicates that plate height HIis directly proportional to the reciprocal of electroosmotic velocity. The slope of the plot of HIagainst u-’, depends on the capacity fador as well as the diffusion coefficient of the solute, provided D,, and to/t,, are constant: If the capacity factor is large, the slope becomes gentle, because the diffusion coefficient of the micelle, D,,, is roughly 10 times as small as that of the solute, D, (18,19). The data shown in Figure 1 are replotted as shown in Figure 5, where the abscissa indicates the reciprocal of electroosmotic velocity, in order to examine the contribution of the term H I on the overall plate height H. Only three compounds are selected from Figure 1 to avoid overcrowding with the data points. Approximately linear relationships are obtained between Hand u-,’ as shown in Figure 5, except for one data point on 2-naphthol. Sudan I11 does not give the linear plots. The straight lines drawn for the three compounds in Figure 5 do not pass through the point of H = 0, which is expected from eq 15 for every solute to pass through when uW-’ = 0. Dashed lines given in Figure 5 are calculated according to eq 15, where to/t,, and D,, are assumed to be 0.22 (2) and 5.0 X 10” mm2 s-’ (18,19), respectively, and each D, is determined so as to give the identical slope with the corresponding experimental one: D, = 8.7 X lo4, 8.5 X lo“‘, and 3.6 X mm2 s-’ for phenol, nitrobenzene, and 2-naphthol, respectively. Judging from the literature values found for some weak and nonelectrolytes (20), the values of the diffusion coefficients found above seem reasonable. The results shown in Figure 5 strongly suggest that (1) longitudinal diffusion is operative as a main cause of band broadening throughout the experimental range of electroosmotic velocities, except for Sudan 111, (2) a constant plate height, e.@;., 1.3 pm for phenol and nitrobenzene, should be added to each Hl irrespective of electroosmoticvelocity in order to explain the observed plate heights, and (3) other electroosmotic velocity-dependent mechanisms should be considered at least for 2-naphthol and Sudan 111. The equation derived by Sepaniak and Cole (11)for longitudin-al diffusion is similar to eq 15 but does not include the term k ’D,, because the diffusion of the solute incorporated into the micelle was not taken into account. Although the diffusion coefficients of micelles are roughly estimated to be one-tenth of those of the solutes (18,19),it is evident from eq 15 that the contribution of the term k’D,, to H, cannot be neglected for solutes having large b’ values. Sorption-Desorption Kinetics. The contribution of sorption-desorption kinetics to the plate height in conventional chromatography has been discussed by Giddings (21). Sorption and desorption in this system may be represented by the kinetic analogue (21) k
A,,(aqueous phase)
A,,(micelle)
(16)
kd
where A, and A,, denote a solute molecule in the aqueous phase and that incorporated into the micelle, respectively, k,
(18)
where C is the sum of all concentrations of a solute and z is the distance along column axis. X,,* and X,,* are the fractions of the total number of solute molecules in the aqueous phase and in the micelle undef equilibrium conditions, respectively; thus, Xaq*= 1 / ( 1 + k’) and X,,* = b’/(l k ’ ) in this system. The velocity of the solute zone us is expressed as
+
u, =
u,,
+ R’u,, 1+R’
where u, is the velocity of the micelle. Equation 17 is derived on the basis of mass conservation in mass transfer and eq 18 means a balance of nonequilibrium. The plate height contribution is given as (21)
H=-
2(Xaq*eaqUeo + Xmc*erncUmc) u,(d In C / d z )
(20)
By substituting the solutions eaq and ern, to the set of eq 17 and 18, and eq 19 for eq 20, plate height ascribed to the nonequilibrium process is given as
The identical equation can be derived from the random walk theory (21),provided that the random walk step length 1 takes the mean of the displacements of the sorbed and desorbed solutes relative to the zone center. The complex capacity factor term in the right-hand side of eq 21 compared with that in the corresponding equation for conventional chromatography arises from the situation in which both aqueous and micellar phases move at different velocities in electrokinetic chromatography. Equation 21 predicts that the plate height caused by the kinetic process increases with an increase in velocity u, and with a decrease in the rate constant kd. The capacity factor term in the rght-hand side of eq 21 has the maximum value (0.2_6when k’_= 0.75 for to/t,, = 0.22) and becomes zero when k’ = 0 or k’ = a. The desorption rate constants of some aromatic hydrocarbons from the SDS micelle have been reported by Almgren et al. (22)and the values of rate constants range from 4.1 X lo2s-l for perylene to 4.4 X lo6 s-l for benzene. In addition, Almgren et al. (22)have found that exit rates approximately parallel solubility in water, i.e., the greater the solubility in water, the higher the exit rate from the particular micelle. This means that the solute with the greater capacity factor has the most reduced desorption rate. This relationship makes it difficult to evaluate H,, contribution with eq 21, because the large capacity factor leads to a small value for the capacity factor term but it simultaneously results in a large value for l / k d . Although Sepaniak and Cole (11) have qualitatively predicted that the term H,, (denoted by H , in their paper) increases with an increase in the retention ratio, R, eq 21 does not always predict such a tendency. The data shown in Figure 1 indicate that the mass transfer dispersion is not the most significant a t least for the solutes employed, because the observed plate heights do not increase with an increase in the electroosmotic velocity. The term Hmc*
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NO.3, FEBRUARY 1,
1989
must be zero for Sudan 111, because the capacity factor of Sudan I11 is assumed to be infinity. If we suppose that the desorption rate constant of 4.1 X lo2s-l reported for perylene (22) belongs to one of the minimum values, H,, is found to be 1.27 pm from eq 21, even under unre-alistically unfavorable conditions in which u,, = 2.0 mm s-l, k’ = 0.75 (the smaller k d should provide much larger k’), and to/tmC = 0.22. This trial calculation strongly supports the fact that the kinetic process does not mainly contribute to band broadening if the desorption rate is not extremely slow. However, very slow desorption kinetics may cause significant band broadening in some solute-micelle systems, in which the solute incorporates with the micelle by electrostatic force or ionic interaction (23, 24). Intermicelle Diffusion. The expression for the plate height contribution of intermicelle diffusion in the aqueous phase, Haq,can be easily derived from the random walk theory (21). If we assume a critical micelle concentration of 8 X M and an aggregation number of 62, the average intermicelle distance is estimated to be ca. 10 nm for a 0.05 M SDS solution. Although intermicelle distance fluctuates owing to the Brownian motion of the micelle, we can suppose that relative positions of micelles remain approximately fiied during a short period of time when solutes diffuse between micelles, because the diffusion coefficient of the solute is much larger than that of the micelle as described above. The average diffusion time t D for the solute to diffuse half of the intermicelle distance d is given as tD
= (d/2I2/2Daq
(22)
The number of random walk steps, n,, is obtained from the total time that the solute stays in the aqueous phase, taq,and t D as
The random walk step length 1 is given as the product of t D and the difference in the velocities between the aqueous phase u,, and the solute zone u,
Since the variance u2 is equal to n,12 and u2 = HL, we have
Temperature Gradient Effect. The velocity profile of the electroosmotic flow in open-tubular capillary has theoretically been studied by Rice and Whitehead (Z), who described the velocity profile as being practically flat when the reciprocal of the Debye-Huckel parameter 1 / is~much smaller than the radius of a capillary. Under the experimental conditions employed in this study, the thickness of the electrical double layer ( 1 / ~is) estimated to be 0.8 nm (2),and this is negligible compared with the radius of the tube. Therefore, the velocity profile of electroosmotic flow must be almost completely flat. Joule heating in capillary zone electrophoresis will produce a radial temperature gradient where the temperature is highest along the axis of the tube. The electrophoretic velocity profile caused by the temperature gradient has been investigated by H j e r t h (26) and the parabolic velocity distribution shown. The electroosmotic velocity u,, and electrophoretic velocity ueP are given as (27‘)
2d2 ueP = - f ( ~ a ) E 31 where E is electrical field strength, and q are the permittivity and viscosity of the solution, and 5; and l2 are the zeta potential of the inside walls of the tube and the micelle, respectively. For a sphere the function ~ ( K uwhere ), a is the radius of the micelle, approaches 1for small KU and 3/2 for large KU. It should be noted that t, 7, and E are common to both eq 18 and eq 19, provided both 6 and 7 retain their normal bulk values throughout the electric double layer. However, 7 in eq 26 is the viscosity in the double layer at the tube wall but in eq 27 is the viscosity of the bulk of the solution; therefore, the temperature gradient will affect I) only in eq 27. Permittivity and zeta potentials vary only slightly with temperature (26), and the product t{ is virtually constant regardless of temperature (2, 26). Accordingly, only the temperature dependence of the viscosity essentially determines the velocity profiies of uep, hence the velocity of the solute ( I ) . Then, we will derive the expression for the plate height contribution of the temperature gradient, H,,on the basis of HjertBn’s treatment (26). By assuming the temperature dependence of viscosity as q = A exp(B/T)
(28)
where A and B are constants, HjertBn (26) has derived Although the more exact expression may be derived from the nonequilibrium theory @ I ) , the derivation of the corresponding equation of eq 25 is difficult. However, we suppose that eq 24 is useful for evaluating the contribution of the Hag term. I t is noticed that the capacity factor term in eq 25 is different from chat in eq ,21and the value of this term ranges between zero (k‘= 0 or k ‘ = m) and 0.21 ( k ’ = 3.6 for to/t,, = 0.22). The average intermicellar distance is estimated to be ca. 10 nm for a 0.05 M SDS solution as described above. This small value of d evidently causes Hbqto be negligibly pm), because Daqis about mm2s-l and small (1.0 X u,, is 2 mm s-l. Sepaniak and Cole (1I ) have mentioned the significant contribution of the intermicelle diffusion judging from the improvement of efficiency with increasing micelle concentrations, but this speculation seems unreasonable as discussed above. We suppose that the effect of increasing micelle concentrations on efficiency is due to decreasing the solute to micelle concentration ratio or reducing the overloading effect, because the micelle concentrations are very low under their experimental conditions (11).
where u, and u, are electrophoretic velocities a t the wall and at the distance r from the axis of the tube, rc is the radius of the column, and G indicates the extent of the differences in velocities between the axis and the inside wall of the tube as is easily shown by setting r equal to zero; G is given as
(30) where K,, and h are electrical conductivity and thermal conductivity of the solution, respectively, I is current, and Tois the temperature of the solution. Average electrophoretic velocity of the micelle, ueP, is calculated from eq 29 as ueP = uw(l + G/2)
(31)
It is noted that the parabolic flow profile expressed by eq 29 is different from that of the laminar flow, where the velocity at the wall is zero.
PrNALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
The plate height in open-tubular capillary chromatography with laminar flow was theoretically studied in detail more than 30 years ago (21),and open-tubular liquid chromatography with electroosmoticflow has also been discussed recently (28, 29). Here, it should be remembered that there is no immobilized stationary phase on the inside wall of the tube in this method; consequently electrokinetic chromatography corresponds to a special case where the capacity factor is equal to zero in open-tubular chromatography. The plate height contribution of the parabolic velocity profile given in eq 29 can be obtained by using a slight modification of the Golay equation (21). Equation 29 means that the velocity u, is the sum of the plug flow of the velocity u, and the parabolic flow expressed by u,G(l- r2/r:). Since only the parabolic component of the total flow contributes to the plate height, we first substitute the velocity difference, ueP - u,, rather than uep for the mobil? phase mass transfer term of the Golay equation, where k’ = 0, yielding ”2
H = - -‘ c
240 2
n
cr
+ GUep
where D is the total diffusion coefficient of thtsolute in the micellar solution and is represented as (Daq+ k’D,,) (see eq 15). The plate height thus obtained is generated while the solute migrates through the whole length of the column at the velocity uep - u,. However, since the solute actually migrates at the average velocity uep as described above, H in eq 32 must be multiplied-by (uBp u,)/uw, and by taking into account that the fraction k’/(l k’) of the solute stays in the micelle, the temperature gradient contribution to the plate height is given as
+
257
the micelle. Therefore, electrophoretic dispersion of the micelle containing the solute should be taken into consideration in this method. According to Wieme (31), the following factors are responsible for band broadening in zone electrophoresis: linear thermal diffusion, microheterogeneity,electrodiffusion, eddy diffusion, and electrosorptive spreading. The last two factors are not involved in this method, and the linear thermal diffusion and electrodiffusion have already been discussed above in terms of longitudinal diffusion and kinetic process, respectively; consequently only the problem of microheterogeneity is discussed below. Microheterogeneityis related to randomness in size, shape and/or charge of the migrating species (31) and the variance resulting from microheterogeneity is derived by the random walk theory. The step length 1 is given as
1 = a(ueP)*Atkc
(35)
where u(uep) is the standard deviation p f the electrophoretic micelle velocity distribution and At kcis the mean time the solute stays in the micelle at a time. The time At’mc is equal to the desorption time of the solute from the micelle or the reciprocal of the desorption rate constant kd, because the liietime of the micelle from ita formation to breakup is usually longer than the residence time of the solute (32). Then, eq 35 is rewritten as
1=
(36)
a(uep)/kd
The step number n, is obtained by dividing the total residznce tiEe of the solute in the micelle, tLC, which is equal to tRk’/(l + k’), by AtLC
Ht = where v, - u, is substituted for ueP When G is less than 2%, the second term of the right-hand side of eq 33 can be regarded as G2/4, and Ht is approximated as
Ht =
(1- to/tmc)R’ 24(D,q + R D m c ) ~
B ~4 I ~ K ~
Since u2 = n,12 from the random walk theory and H = $/lco1
(34)
~
should be remembered that eq 21 has the identical variables, F It ~ X ~ T ~ ~ ~ ~ ~ ueo/kd, with eq 38. On the right-hand side of eq 38, the value
Equation 34 means that Ht is proportional to I and inversely proportional to r:. Since current increases in proportion to the square of the radius, the second term of the right-hand side in eq 34 is independent of current and the tube radius provided the voltage is kept constant. Thus, Ht is proportional to u, hence proportional to the electrical field strength (eq 26) regardless of the tube radius. If we take the values B = 2400 K, To= 325 K, X = 5.73 X lo4 W mm-’ K-l, K~ = 5.71 X lo4 S mm-l (found for the 0.05 M SDS solution), rc = 2.5 X mm, I = 4.0 X A, u, = 2.17 mm s-l, to/t,, = 0.22, D,, = lom3 mm2 s-l, and D,, = 5, X mm2 s-l,the G value becomes 0.45% and Ht 0 for k’ = 0,8.9 X lo-’ mm for k’ = 5, and 4.5 X lo4 mm for k’ = co. These considerations suggest that the effect of a temperature gradient on plate height is negligible, although the solution temperature may rise considerably high (2). Electrophoretic Dispersion. Electrophoreticdispersion has been discussed in detail by Boyak and Giddings (30) and Wieme (31),and the concentration distribution in free zone electrophoresis has been evaluated from the viewpoint of conductivity changes by Mikkers et al. (5). It should be remembered that the micelle, which can be transported by electrophoresis, fills the whole tube in this method, i.e., the concentration of the micelle is always kept constant throughout the tube, even when voltage is applied. The solute itself is not directly subject to the electrophoretic force but is indirectly transported by the electrophoretic movement of
of the first term containing E’ranges from zero to (1 - to/ tmc)2/(t0/tmc), e.g., from zero to 2.8 for to/tmC = 0.22, while the second term should be constant. Thus, Hepincreases linearly with increases of u, and kd-’. The polydispersity of the micelle size seems dominant as the origin of microheterogeneity,and although polydispersity is controversial (33), the aggregation number n and the standard deviation of the distribution width u, for the SDS micelle have been reported to be 64 and 13 at 25 “C,respectively (32). Consequently, the relationship between the distribution of electrophoretic velocity and that of the aggregation number or the micelle size is needed to evaluate the effect of the polydispersity. The function f(m) in eq 27 is very complex, so a rough approximation for a sphere is employed
~
T
+
(9.32 X in ( K C I ) 1.109 (39) Equation 39 gives a correlation coefficient of 0.987 for the reported values of the Henry correction factor f ( ~ ain) the 10 I KU I 1 range (27). The value KU is estimated to be in this range for the SDS micelle. In order to express the micelle diameter, a, in terms of n, it is assumed that a is directly proportional to the cube root of n ~ ( K C I )=
a / n m = 0.5n1I3
(40)
where the coefficient of 0.5 is arbitrarily chosen to adjust a to 2.0 nm when n is 64. The standard deviation of a function
258
ANALYTICAL CHEMISTRY, VOL. 61,
NO.3, FEBRUARY
1, 1989
f ( x ) can generally be calculated from the standard deviation of x , ux, through
Then, by inserting eq 39 and 40 into eq 27, the following expression is obtained through eq 41:
Equation 42 is divided by eq27 and eq 39 is combined, yielding the relative standard deviation of ueP
-u(uep) - uep
3.11 X on (9.32 X In ( ~ a ) 1.109 n
+
(43)
I 0 81
1
v e o / mms’
2
,
The zeta potential lZ of the micelle having different aggregation numbers is considered to be the same. By insertion of the values 1 / = ~ 0.8 nm and a = 2.0 nm into eq 39, the value of f(Ka)becomes 1.19; by use of n = 64 and u, = 13, the value of 2.6% is obtained for u(uep)/uepfrom eq 43. Thus, eq 38 can be rewritten for the SDS micelle solution as
By use of to/tmC = 0.22, u,, = 2.0 mm s-l, apd kd = lo3,the plate hejght Hepbecomes 2.6 X_lO-z pm for k’ = 1,7.5 x pm for k’= 5, and 0.14 pm for k’= m. The validity of these estimated values largely depends on the accuracy of the kd value, as is that of H,, described above. The effect of concentration distribution on the peak width discussed by Mikkers et al. (5) is not considered to be significant in this method, because most peaks have practically symmetrical shapes (Table I) irrespective of the amount of solute injected. This observation for neutral solutes implies that the electrophoretic mobility of the micelle incorporating the solute is almost the same as that of the micelle free from the solute, although micelle size should increase with solubilization. This view is consistent with the evaluation of the effect of micelle size on the electrophoretic velocity mentioned above. Relative Significance of Each Contribution. Among five band broadening mechanisms described above, only the longitudinal diffusion contribution becomes increasingly significant with a decrease of the electroosmotic velocity u,, (eq 15). On the other hand, the contributions of the other causes (eq 21,25,34, and 44) to the plate height increase with an increase in v,. Therefore, the linear increasing part of the observed plate height with an increase in urn-’ shown in Figure 5 can be ascribed to the contribution of longitudinal diffusion, HI. The dependence of H on shown in Figure 5 seems to be explained in terms of the s u m of longitudinal diffusion and one more mechanism which generates a constant plate height (e.g., 1.3 pm for phenol) regardless of v, as described in the section LongitudinalDiffusion. However, since no mechanism discussed above is independent of u, the apparent constant part must be merely an accidental result from a few contributions. In order to evaluate the relative significance of each contribution discussed above, some representative plots of plate height versus electroosmotic velocity are constructed according to the five in-column plate height equations (eq 15, 21, 25, 35, and 44): Parts A, B, and C of Figure 6 show examples, where capacity factors are equal to 0 and 1, 5, and infinity, respectively. In Figure 6 the following parameters are employed throughout the calculations: D, = 1.0 X mm2&, D,, = 5.0 X mm2 s-*, to/t,, = 0.22, kd = 1.0 X lo3 d for
0
1 v e o / mm
0
1
v,,
