Anal. Chem. 1002, 64, 1947-1959
1047
Influence of Analyte Plug Width on Plate Number in Capillary Electrophoresis Scott L. Delinger and Joe M. Davis* Department of Chemistry and Biochemistry, Southern Illinois University at Carbondale, Carbondale, Illinois 62901
A model of peak dlsperslon in capillary electrophoresls Is proposed, in whlch the number N of theoretical plates Is governed by iongltudlnaldlffudon and the lnltlai wldth of the analyte plug Introduced Into the caplilary. To test the model, over 130 experlmental deterrnlnations of N for dansyl-L-lsoleuclne and several determlnations of N for the marker of electroosmotic flow, acetone, were made for a variety of caplilary lengths, voltages, and lnltlal plug wldths. The experlments were dedgned to reduce dlspersion from other sources to acceptable levels. Fromexperhentallydetermlned values of moblllty and current density, the buffer temperature In the caplllary was determlned at different fleld strengths. From thls temperature, the apparent varlatlons of analyte moblilty and dlffuslon coefficient wlth field strength were determined and used In flttlng experimental Ns to the model. Peak shapes were predicted by addltlonal theory to verify that only longltudlnaldlffudon and plug width affected Nunder the experlmental condltlons used here. On the bad8 of this study, lt Is argued that some anomalous values of Nreported In the literature can be explalned by thls model, Instead of Joule heatlng as previously argued.
INTRODUCTION This paper addresses the influence of the initial width of an analyte plug on plate number in capillary electrophoresis (CE). To effect a separation by CE, a small amount of an analyte mixture first is introduced into the capillary by electromigration or hydrodynamic pressure. The components of this analyte-rich plug then are separated by applying a potential difference across the capillary. Theory shows that the initial width of this plug sets an upper limit on the maximum attainable efficiency, as measured by theoretical plate number N.1~2 The theme of this study is that the efficiency realizable by CE is so high that this limitation on N, which we shall call an injection effect, is fairly ubiquitous. This is not the case with many other kinds of separations, e.g., column chromatography, which are incapable of such efficiencies and are more forgiving of such minor perturbations. The motive for initiating this study was our speculative interpretation of some anomalies observed some years ago by Jorgenson and Lukacs. These authors reported plots of N vs voltage V in ref 3 and capillary length L in ref 4 that differed substantially from those expected when longitudinal diffusion is the principal mechanism of band broadening. When longitudinal diffusion dominates, then graphs of N vs V are linear and graphs of N vs L are invariant (see below). In contrast, the data of Jorgenson and Lukacs show a leveling (1) Lauer, H. H.; McManigill, D. TrAC, Trends Anal. Chem. (Pers. Ed.) 1986,5, 11. (2) Hjeren, S. Electrophoresis 1990, 11, 665. (3) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981,53, 1298. (4) Lukacs, K. D.; Jorgenson, J. W. HRC & CC, J . High Resolut. Chromatogr. Chromatogr. Commun. 1985, 8, 407. 0003-2700/92/0364-1947$03.00/0
of N with increasing V and a decrease in N with decreasing L. Furthermore, although the graph of N vs Vis almost linear
for small Vs,the data generate a non-zero intercept. These data are reproduced here as Figure 1; their coordinates were estimated from the published results with a digitizer pad. Jorgenson and Lukacs attributed these behaviors to the nonequilibrium dispersion caused by the dissipation of electrical energy in the capillary as Joule heat, which brings about a radial gradient in temperature and electrophoretic mobility. This explanation is qualitatively reasonable. A more quantitative examination by us (see below), however, suggests that Joule heating alone cannot account for these decreases in N . As we will argue later in the paper, another explanation for these anomalies is that N is controlled by the plug width. This paper reports extensive experimental studies, the interpretation of which leads us to suggest this explanation. In brief, we report a detailed study of injection effects in CE for several capillary lengths, voltages, and plug sizes. In summary, a micromolar solution of dansylisoleucine in a pH = 9 phosphate/borate buffer was injected by electromigration on a single capillary, whose length was varied in four stages from 1.200 to 0.600 m. For each length, the plug size was varied in three stages from 10 to 90 kV-s. For each plug size and capillary length, a potential difference between 1 and 30 kV was applied to sweep the analyte plug past a UVvis detector. Approximately 130experiments were ultimately so generated. The detected peaks were digitally collected, and then plate numbers were calculated with in-house software. The plate numbers so determined were interpreted by a theory, in which plug width and longitudinal diffusion are considered as the only mechanisms of dispersion. The proposed theory is modestly detailed and accounts for the variation of mobility and diffusion coefficient with the temperature of the buffer in the capillary, which in turn varies with the electric field strength. We also predict several peak profiles from additional theory and compare them to experiment as a check on our theory of dispersion. Although our study is both extensive and detailed, we recognize that other studies of injection effects in electrokinetic separations have been reported-kg It furthermore is evident that researchers recognize that these effects influence N.ls2+l7 By and large, however, this recognition appears to (5) Burton, D. E.;Sepaniak, M. J.;Maskarinec, M. P. Chromatographia 1986, 21, 583. (6) Terabe, S.; Otsuka, K.; Ando, T. Anal. Chern. 1989, 61, 251. (7) Otsuka, K.; Terabe, S. J. Chrornatogr. 1989,480,91. (8)Huang, X.; Coleman, W. F.; Zare, R. N. J.Chrornatogr. 1989,480, QA
" I .
(9) Cheng, Y. F.;Wu, S.; Chen, D. Y.; Dovichi, N. J. Anal. Chem. 1990, 62, 496. (10) Mikkers, F. E. P.; Evaraerts, F. M.; Verheggen, Th. P. E. M. J. Chromatogr. 1979, 169, 11. (11) Lauer, H. H.; McManigill, D. Anal. Chem. 1986,58, 166. (12) Sepaniak, M. J.; Cole, R. 0. Anal. Chem. 1987, 59, 472. (13) Foret, F.; Deml, M.; Bocek, P. J. Chromatogr. 1988, 452, 601. (14) Rose, D.J., Jr.; Jorgenson, J. W. Anal. Chem. 1988,60, 642. (15) Grushka, E.; McCormick, R. M. J. Chromatogr. 1990, 471, 421.
0 1992 American Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
When other sources of dispersion are significant, N must be described byzo
500000 r-
400000
300000
1
analyte: hexamine, fluorescene derivafive
300000
N=L
Reprinted wirh permission from. Jorgenson, J.W.; Lukacs, K.D. Anal. Chem. 1981,53,1298
intercept
10
0
v
30
(kVj20
ad2= 2Dt V=15kV
I
v
" n
i 0
0.4
0.8
1.2
1.6
L (m)
Flgure 1. Plots of (a) N vs Vand (b) N vs L determined by Jorgenson and Lukacs. Coordinates of the data were estimated from the published resuits with a digitizer pad. us to be more perfunctory than reflective. Indeed, we believe that the loss of efficiency due to injection effects is not appreciated by many. As supporting evidence, we know of only two references,gJswhich report N values for free-solution CE approaching the theoretical limit predicted by the theory of longitudinal diffusion.'g Yet, this discrepancy between expected behavior and observation does not seem to arouse much interest. While other sources of band broadening (e.g., hydrodynamic flow, adsorption-desorption kinetics, mobility differences between similarly charged analyte and buffer ions, and radial temperature gradients) can be important in specific instances, our study suggests that injection effects are all too common. We hope that our study will emphasize the importance of these effects.
~
u
~
(2)
Number N of Theoretical Plates. As shown by Jorgenson and Lukacs,3 an approximate expression for the number N of plates expected when longitudinal diffusion dominates dispersion is N = pV/2D (la) where p is the apparent mobility, equal to the algebraic sum of the true electrophoretic mobility pep and the electroosmotic flow coefficient peo( p = pep + pea), and D is the analyte diffusion coefficient. If a distinction is made now between the total length Lt of the capillary and the length Ld from the injection end of the capillary to the detection window, a more exact expression is obtained8 N = -p-V Ld 2 0 L,
For a constant ratio plD, plots of N vs V are linear, and plots of N vs Lt (or Ld) are invariant, as long as the ratioLd/Lt is held constant. These trends are not followed by the data in Figure 1. (16) Jones, H.K.;Nguyen, N. T.; Smith, R. D. J . Chromatogr. 1990,
(17)Dose, E. V.; Guiochon, G. A. Anal. Chem. 1991,63,1063. (18) Smith, R. D.; Olivares, J. A.; Nguyen, N. T.; Udseth, H. R. Anal. Chem. 1988,60, 436. (19) Dovichi, N. Personal communication, 1991.
(3)
The time at which the plug migrates past the detection window is designated here as t,, the elution time. This time is expressed by (4)
where v = pE is the average velocity of the analyte plug and E = V/L, is the field strength. The variance due to longitudinal diffusion at this time is obtained simply by equating t in eq 3 to t,
(5) The variance ui2 due to injection effects is related to the width 1 of the analyte plug introduced into the capillary uf = 12/12
(6) where the functional form of 1depends on the means by which the plug is introduced. For injection by electromigration, 1 I 1, equals14 1, = piEiti= piViti/L,
(7a)
where pi, Ei, and Vi are the values of apparent mobility, field strength, and voltage associated with the process of injection. These variables carry the subscript i to distinguish them from their counterparts, p, E, and V, which may (and usually do) differ. The parameter ti is the time over which the injection is made. For injection by hydrodynamic pressure, 1 E lh isI4
THEORY
504, 1.
/
where u,j2 is the variance associated with the jth source of dispersion. In the proposed model, two variances are considered: the variance Ud2 due to longitudinal diffusion and the variance ui2 due to injection effects. The variance ud2 due to longitudinal diffusion at any time t during the plug's migration is
r b)
Reprinted wirh permission from: Lukacs, K.D.; Jorgenson, J.W. HRC 1985,8,407
0
~ I
v n
"
200000
100000
"
c
-
200000
C
1, = pgAhd;ti/32qLt
where p is the buffer density, g is the acceleration due to gravity, Ah is the difference in height between the two ends of the capillary, d, is the capillary diameter, and 9 is the buffer viscosity. In this paper, we shall consider only experiments based on electromigrative injection, although we shall comment later on eq 7b. The number N of plates expected, when longitudinal diffusion and injection effects of electromigrative origin dominate dispersion, is obtained by combining eqs 2,5, and 7a
N=
Ld2 L,2 (piViti)2/1%: + 2DLdLt/pV (piEiti)2/12+ WLd/pE
(8) Equation 8 defines the theoretical model to which we will fit our data. It depends on five controllable parameters-Lt, Ld, V, Vi, and ti-and three variables determined by experimental conditions-p, pi, and D. It is deceptively simple, because p, pi, and D are not constant over a large range of voltages. They are not constant, because the buffer temperature in the capillary increases with increasing field (20) Giddings, J. C. Unified Separation Science; Wiley: New York,
1991.
ANALYTICAL CHEMISTRY, VOL. 84, NO. 17, SEPTEMBER 1, 1992
strength, due to Joule heating. This increase in temperature is accompanied by a decrease in viscosity and an increase in the variables, p, pi, and D, which are inversely proportional toviscosity. Since these variables vary with temperature and temperature varies with E, we properly must consider the variables to be functions of E (or Ei), e.g., p(E). To represent these added complications, we rewrite eq 8 as
N=
Ld2 (~i(Ei)Viti)~/12L; + m(E)LdLt/p(E)V-
where the explicit dependence of mobilities and diffusion coefficient on field strength (temperature) is indicated. Several means by which these variations can be addressed are noted here. First, in the variance Ud2, only the ratio, Dlp, is important, not D and p independently. This ratio varies less with temperature than does either p or D and can be expressed by the Einstein equation as
Dlp = RTk/zeF (10) where R is the universal gas constant, F is the Faraday constant, e is the fundamental charge unit, z is the (signed) number of charges, and T Lis absolute temperature. Although some have used this common equation with confidence,21 others have questioned its accuracyl7 and we shall not use it here. Alternatively, we could approximate both p and D by appropriately weighted constants determined from our data, since the temperature variation of the ratio is small. We shall not do this, either, because we have rather extensive data illustrating the variation of p with temperature. We believe that it would be most unwise to reduce these detailed data to a single, representative value. Rather, we will account explicitly for the variation of p , pi, and D with temperature. In actuality, we will account for their variation with E (or Ei) and simply recognize that this variation exists because the buffer temperature T in the capillary (on which p, pi, and D depend) varies with E. Rather than account for these variations by pure theory, we will allow the experimental data to assist us in their determination. A description of the variation of p with E will be determined easily by fitting experimentally determined values of p to an empirical function of E. This function will also define the injection mobility pi at any Ei. A description of the variation of D with E , however, will require more effort. To determine D , we first must know T. By interpreting the nonlinear variation with E of experimentally determined values of apparent mobility and current density, we will estimate the temperature T in the capillary at different field strengths (i.e., T = T ( E ) ) . From T ( E ) ,we then will determine the variation of D with E in accordance with the relationship22 DqlT = constant from which D = D(E) is determined as
(11)
(12)
and where q,, T,, and D, are the room-temperature values of the buffer viscosity, temperature, and analyte diffusion coefficient. The viscosity q(E) (in kg/(ms)) necessary to the (21)Kenndler, E.;Schwer, C. Anal. Chem. 1991,63, 2499. (22) Sherwood, T.K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975.
1048
determination of D(E) was calculated at temperature T(E) (in "C) from Swindells' equation23
= ?(E) = 1.002 x 10-~108
8=
1.3272(20 - T ( E ) )- 1.053 X 10-3(T(E)- 20)2 (13) T(E)+ 105
which is somewhat superior to the commonly used Andrade equation.24 The actual determination of the functions p(E)and D(E) requires extensive discussion of the experimental data and is deferred to the Results and Discussion. CE is subject to many possible sources of dispersion. Of these, only longitudinal diffusion cannot, in principle, be eliminated. If we are to focus on only injection effects in this study, our experiments must be designed carefully to reduce or eliminate other sources of dispersion. Care was taken to avoid hydrodynamic flow, which would cause a Taylor-like dispersion.25 The electrolyte furthermore was buffered at a high pH to reduce the adsorption of negativelycharged dansylL-isoleucine to the negatively charged capillary wall and, hence, adsorption-desorption kinetics.11 Finally, the ratio of the buffer concentration (phosphate and borate) to the concentration of dansyl-L-isoleucine was kept greater than 400 to reduce the changes in local field strength that cause peak asymmetry.8 Calculation of Peak Profile. To help us determine if other mechanisms of dispersion (e.g., changes in local field strength due to mobility differences between buffer and analyte ions) were active, we calculated the expected shape of many of our peaks. The calculation is based on a onedimensional model; by making it, we are implicitly assuming that any radial concentration gradients that develop are rapidly eliminated by diffusion. In this calculation, dispersion is postulated to arise only from longitudinal diffusion and injection effects. The onedimensional concentration C = C ( x , t )of an analyte expected at time t, when the analyte is deposited at t = 0 as a rectangular plug of width 1 centered about x = 0, is26
where COis the initial concentration of the plug. This result also has been reported elsewhere.8 This profile of C vs x is easily converted into a profile of C vs t by the relationship x = ( t - t,)v = (t - t,)pE
(15)
From eq 14 and values of E , t,, 1, p, and D , the profile CICo can be calculated as a function of time and compared to experimental results.
EXPERIMENTAL PROCEDURES Apparatus. The CE system was constructed in-house from a Spellman UHR 50PN25*BO-kV power supply (Spellman High Voltage Electronics Corp., Plainview, NY), a Linear UVIS 200 UV-vis detector with a CE flow cell (Linear Instruments, Reno, NV), various lengths of a 75-pm-i.d., 375-pm-0.d. fused-silica capillary (PolyMicro Technologies, Phoenix, AZ), and a polycarbonate interlock box for operator safety. The initial experiments were carried out with a 1.200-m capillary. This capillary subsequentlywas deliberately (in one case, accidentally) broken at various stages to generate the other three capillary lengths, (23)Swindells, J. F. In CRC Handbook of Chemistry and Physics, 69th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1988;p F-40. (24)Davis, J. M.J. Chromatogr. 1990,517, 521. (25)Grushka, E.J. Chromatogr. 1991,559, 81. (26) Crank, J. TheMathematics of Diffusion,2nd ed.; Clarendon Press:
Oxford, U.K., 1975.
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
1.046,0.745,and0.600m. By using thesamecapillarythroughout the study, we reduced possible variations of surface chemistries among different capillaries. The detector window was located 0.090 m from the cathodic end of each capillary length and was created by removing the polyimide coating surrounding the capillary with a razor blade. The current was measured between the cathode and ground with a Simpson k200-pA ammeter (Simpson Electric, Elgin, IL). The detector output voltage was sent to both a Kipp and Zonen BD 41 recorder (Kipp and Zonen, Bohemia, NY) and an Epson Equity 1+microcomputer (Seiko Epson, Torrance, CA) outfitted with a DAS-16G112-bitanalogto-digital converter (Keithley-Metrabyte, Taunton, MA). The software for data acquisition was written in-house. The acquired data files were transferred to an Apple Macintosh SE computer for moments analysis. One potential source of dispersion ignored in this study is the effective width of the detector window,6.8 which contributes a variance to N equal to the square of this width divided by 12. The detector specifications state that radiation is focused onto the capillary as a 75-pm band, which corresponds to a variance equal to 4.7 X 10-10m2. The effect of this variance cannot be seen in our peaks, because it is so small (e.g., if this variance alone governed N , one could achieve over 2 X lo9 plates in a 1-m capillary). Reagents. The buffer was prepared from distilled, deionized water and reagent grade Na2B40~10Hz0 and Na2HP04. The buffer composition was 10 mM sodium phosphate and 6 mM hydrated sodium borate; the buffer pH was 8.97. The buffer was passed througha 0.45-pm filter before use and was changed every 2 days (failure to change the buffer with this frequency resulted in the detection of sharp random spikes of unknown origin). A 40.34 pM solution of dansyl-L-isoleucine(Sigma, St. Louis, MO) was prepared from this buffer for the electrophoretic studies; an approximately 0.1 M solution of HPLC-grade acetone (Fisher Scientific, Pittsburgh, PA) was prepared as a marker of electroosmotic flow by dissolving two drops of acetone in 12 mL of buffer. Becauseof the high volatility of acetone,this latter sample was prepared every few days. The samples and buffers were stored in 12-mL amber glass vials capped with Teflon-coated siliconeliners, which were cross-hatchedby a razor blade to permit insertion of the capillary. Protocol. The 1.200-m capillary was conditioned by flushing it with successive solutions of 0.1 M NaOH, 0.1 M HC1, and distilled water for 30 min each. The capillary was then flushed with buffer for 2 h. Flushing was accomplished by placing one capillary end in the solution of interest and connecting the other end to the in-house vacuum. The shorter capillaries generated from this stock were not reconditioned but were flushed with buffer prior to use. Following the installation of a new length of capillary in the flow cell, a vacuum was pulled on the capillary to ensure that it was filled with buffer. To minimize hydrodynamic transport, the anodic and cathodic buffer reservoirs were filled to the same height (f0.5 mm) and the capillary ends were positioned at the same depths (a0.5 mm) in the reservoirs. For similar reasons, the sample vials were filled to the same height as the buffer reservoirs. All vials were set on the flat base of the interlock box, such that gravitational potential energy had no effect on hydrodynamictransport. Unlessotherwisenoted, injections were performed by inserting the anodic end of the capillary through the cross-hatched liner of the sample vial to the same depth as the cathodic end of the capillary and applying 1 kV across the capillary for 10, 50, or 90 s. The capillary was then removed rapidly from the sample vial and rapidly inserted in the buffer vial to the proper depth. These actions ensured that the hydrostatic pressure at both ends of the capillary was the same during both injection and analysis and also minimized hydrodynamic flow during the insertion and removal of the anodic end of the capillary. The electrophoresis was immediately begun to minimize diffusionofthe plug contents into the buffer.l5 Voltages were varied between 1 and 30 kV. All analyses were performed at least twice and up to as many as five times, and a 30-min electroosmotic wash was carried out between analyses (the wash seemed crucial to reproducibility). The detection wavelength, 255 nm, was selected as a compromise to the optimal detection of both dansyl-L-isoleucineand acetone. Data were collected at
either 10 or 40 Hz, depending on the peak width, such that at least 150 points were collected over the duration of a peak. This large number reduced errors in the moments calculation^.^^ We observe that the rate of electrophoresis is not governed by voltage or capillary length but by their ratio, the electric field strength E = VIL,. Rather than design our experiments around this latter variable, however, we chose to work with measurable parameters-voltages, lengths, and time-to developa study that is of pragmatic interest to the user of CE. For example, in eq 1b N depends on the Ld/Lt ratio, but we have chosen to characterize N at different lengths Lt, since one thinks more intuitively in terms of lengths than ratios of lengths. The electrical conductivity of the buffer was measured at various temperatures by heating a buffer-filled,Parafilm-covered beaker containing a thermometer and calibrated conductivity cell in a GCA 253 circulating system with a tolerance of f0.1 "C (theParafilm prevented evaporative lossesat high temperatures). The conductivity cell was connected to a YSI 32 Conductance Meter (Yellow Springs Instrument Co., Yellow Springs, OH). Classification of Data. Our experimental data are partitioned into two sets, which we designate as 'low-temperature" data and 'high-temperature" data. These designations refer to the laboratory temperature, which was approximately 22 or 30 "C in the two respective cases. Our initial experiments (corresponding to L, = 1.20 m, Viti = 10 and 50 kV.s and to Lt = 1.046 m, Viti = 90 kV.s) were carried out at the elevated temperature of 30 OC, because the University's physical plant had not yet turned on the air conditioning (these experiments were carried out in early spring 1991). The remainingexperimentswere carried out with air conditioning, which lowered the laboratory temperature to 22 "C. This issue is relevant, because the transport parameters p, pi, and D vary with temperature. To address this issue, we analyzed these two data sets separately. Momenta Analysis. The digitized peaks were filtered with a nine-point moving boxcar to reduce noise. Each filtered peak then was plotted in KaleidaGraph (Synergy Software, Reading, PA) to determine the beginning and end points of the peak. The 5'12 data points, where S is the sampling frequency (i.e., 10 or 40 Hz), immediately before and after the peak's beginning and end points were fit to a straight line by linear least-squares theory. This line was interpreted as the peak's baseline, and the zeroth, first, and secondmoments of the peak were calculatedwith respect to it. The zeroth moment, or peak area, was used as a measurement of experimental reproducibility (see below). The first moment was identified with the elution time, tarof the peak, and the apparent mobility p was calculated from this time in accordance with eq 4. The number N of plates was calculated from the second moment of the peak, a2,as2O N = t;/a2
The software necessary to these calculations was written inhouse in FORTRAN 77 and executed on a Macintosh SE microcomputer. Two- and three-dimensional graphs were generated on the SE computer with KaleidaGraph and DeltaGraph (Deltapoint, Monterey, CA). Another possible source of apparent dispersion is the time constant T of the detector, which was set to its minimum value, 0.1 s. The minimum variance a2 determined from our dansylL-isoleucine peaks was about 0.1 s2 (corresponding to Lt = 0.745 m, V = 27 kV, and Viti = 10 kV&. Because the observed signal is the convolution of the input signal from the capillary and an exponential response function with T = 0.1, the actual zone . ~ ~ an variance is given by the difference, a2 - T ~ Consequently, error of 10% was incurred in this datum by ignoring the time constant. All other values of Gwere greater than 0.3 s2, resulting in an error of only 3% or less. To minimize spurious results, we report here only those experiments for which both plate numbers and peak areas were fairly reproducible. It was not unusual to have reproducible N's and irreproducible areas, especially for the 10 kV*s runs; occasionally, the opposite pattern was observed. None of these (27) Dyson,N. Chromatographic Integration Methods;Royal Society of Chemistry: Cambridge, U.K., 1990. (28) Bracewell, R. N. The Fourier Transform and its Application, 2nd ed.; McCraw-Hill: New York, 1986.
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, lSQ2
data was used. In some cases, the selection of data seemed almost arbitrary. We justified the integrity of our selections, however, after the completion of our study; this justification is discussed below.
20000
f
a ) low-temperature data
-
J = 0.263E 3.1Se-6E' t
2.34e-IOE'
RESULTS AND DISCUSSION Fidelity to Experimental Conditions of Jorgenson and Lukacs. In our preliminary experiments, we duplicated as closely as possible the experimental conditions by which the data in Figure 1 were determined. Over time, experimental difficulties forced us to modify several of these conditions. For example, we initially adopted a 0.05 M, pH = 7 phosphate buffer similar to that used by Jorgenson and Lukacs. We quickly diluted this buffer by a factor of five (to 0.01 M), however, when we realized how much current was required to maintain a high voltage. This diluted buffer also seemed more consistent with the buffer concentrations commonly used today. With the diluted buffer, however, we experienced an adsorption of analyte to the capillary wall, as characterized by a gradual decrease in N for V values greater than 20 kV or SO.^ Similar decreases in N have been reported by the group of Novotny.30 This decrease is quite different from the leveling of N expected from injection effecta,which indeed is observed in this study. Furthermore, our electroosmotic flow was not very reproducible at this pH. After helpful discussions with others (see Acknowledgment), we ultimately adopted experimental conditions so different from those of Jorgenson and Lukacs that our findings cannot be used to challenge rigorously the interpretation of the data in Figure 1. Rather, we only can make plausible arguments based on the interpretation of the data at our disposal. We now turn to that interpretation. Determination of p ( 4 , ~(4, and D ( 4 . Before we can fit our experimental Ns to eq 9, we first must determine the functions p ( E ) , pi(E), and D(E).These functions will be determined from least-squares fits of various polynomials to the appropriate experimental data. The exact functions determined here depend on our specific experimental conditions, e.g., buffer concentration, composition,pH, and heattransport characteristics of the capillary. In other words, the functions determined here are useful only in this particular study. Of a more general interest are the means by which these functions can be determined, which can be applied to any CE (or micellar electrokinetic) experiment. Determination of Buffer Temperature in the Capillary and p ( E ) . Procedures for the determination of buffer temperature have been suggested by others. Lukacs measured the conductance of the buffer-filled capillary as a function of potential and inferred the buffer temperature from additional measurements of buffer conductivity at different temperatures.31 Terabe et al. measured the surface temperature of the capillary (which is a little less than the buffer temperature) with a therrnoco~ple.3~We outline a different approach,which is based on detailed interpretation of the graphs of current density J and mobility p vs field strength E. Our objective is to determine from them the mean buffer temperature Tasa function of E (the buffer temperature typically varies by less than 0.5 OC or so across the capillary crosssection;33 we can only determine the mean temperature). As an internal check, we will calculate T by two independent means. (29)Delinger, S. L.;Davis, J. M. FACSS XVII Annual Meeting, Cleveland, OH, October 7-12, 1990. (30)Liu, J.; Dolnik, V.; Hsieh, Y.-Z.; Novotny, M. Anal. Chern. 1992, 64,1328-1336. (31)Jorgenson, J. W. In New Directions in Electrophoretic Methods; Jorgenson, J . W., Phillips, M., Ed.; ACS Symposium Series 335;American Chemical Society: Washington, DC, 1987. (32)Terabe, S.;Otauka, K.; Ando, T. Anal. Chern. 1985,57,834. (33)Jones, A. E.;Grushka, E. J. Chrornatogr. 1989,466,219.
1851
0
20000
10000
30000
40000
E (V/m) 15000
b) high-temperature data J = O.315E - 1.99e-6E'
0
10000
+
20000
30000
E (V/m) Flgure 2. Plots of Jvs E determined from the (a) low-temperature and (b) high-temperature data. Solid curves are the unwelghted leastsquares fits of the indicated polynomialsto the data: the circlesrepresent experimental determinations.
The motive for this determination, as outlined in the Theory, is the estimation of D(E). In the course of this estimation, we will also determine p(E) and pi(E). ( a )Low-Temperature Data. Figure 2a is a plot of J = i / A vs E determined from the low-temperature data. The experimental J's were calculated as the ratio of the measured current i (in amperes) to the capillary cross-section A = 4.42 X m2. The mean experimental values are represented by circles, and the vertical error bars represent one standard deviation. These data represent determinations of J from the three shortest capillaries. The solid curve is a unweighted least-squares fit of the function
J = 0 . 2 6 3 3 + aE2 + bE3 (17) to these data, with a = -3.15 X 10-6 and b = 2.34 X 10-10. An unweighted fit was used, because the reproducibility of J among runs was high, as evidenced by the small standard deviations. The coefficient multiplying E in eq 17,0.263, is the approximate value of the buffer's electrical conductivity k (in s2-l m-1) at the approximate room temperature, 22 "C. By postulating a fitting function of this kind, we are assuming implicitly that the relationship J = k E = 0 . 2 6 3 3 is valid at very low field strengths, where the buffer temperature is essentiallyequal to room temperature. In contrast, we assume that the nonlinear variation of J at higher field strengths results from an increase in k with temperature. From the identity, J = k E , we estimate k from eq 17 as k = k ( E ) 0.263 - 3.15 X 104E + 2.34 X 10-'OE2 (18) One observes that, since coefficient a is negative, the predicted k actually decreases with E before it increases ( b is positive). This decrease has no physical origin but rather is attributed to the determining of our coefficientsfrom limited data. As we show later, the predicted decrease is very small. Equation 18 expresses the variation of k with E . Physically, this variation exists, because the temperature T of the buffer in the capillary increases with the amount of dissipated heat, which in turn increases with E. An expression for k alternatively can be determined directly as a function of temperature. Figure 3 is a plot of k vs T (in "C),determined as
1952
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
0.2 r 15
I
35
75
55
95
0
T ("C)
20000
30000
E (Vlm)
Ftgure 3. Plot of k v s T(in "C) determined from the buffer used in this study. The solid curve Isthe unweighted least-squares fit of the Indicated polynomial to the data: the circles represent experimental determi-
nations.
10000
7.0 10'
c
b) mobilities from hightemperature data
3 6.0 IO'
60
T ("C)
2
50
5.0 10' 40
+ 3.42e-17E'
30 an L"
~~
0
~
~
~
10000
20000
30000
E (Vlm) T's determined from high-temperature data
40000 I
T determined from J vs. E 1
0
10000
20000
30000
40000
E (V/m) Flguro 4. Plots of Tvs €determined from the (a)low-temperature and (b) high-temperaturedata. Solid curves represent f s determined from current densky;dashed curves represent Ts determlned from apparent
mobility.
detailed in the Experimental Procedures. The unweighted quadratic that describes the measured values of k and T is
+
k = k ( T ) = 0.126 5.99 X + 1.06 X 10-5p (19) From this equation, the electrical conductivity, 0.263, in eq 17 was calculated. By equating k ( T ) , eq 19, to k(E),eq 18, a unique quadratic expression in T is defined for each numerical value of E. One root is negative and physically meaningless; the other root is positive and is the mean temperature of the buffer in the capillary. The solid curve in Figure 4a is the plot of T vs E so determined. The predicted temperature a t high field strengths, 55-60 O C , is in agreement with other reported values.31-32 In summary, a profile of T was determined by proposing a theoretical function k ( E )for the electrical conductivity and then equating this function to the experimentally determined function, k(7'). Here, k ( E ) was determined from J vs E. It is possible to propose another theoretical estimate of k(E) from completely independent measurements. This estimate is determined from the variation of the apparent mobility 1.1 with E and is based on Walden's rule, which states that the product of the analyte mobility and buffer viscosity is constant over wide ranges of t e m ~ e r a t u r e .Because ~~ conductivity is (34)Morris, C. J. 0.R.;Morris,P. SeparationMethodsinBiochemistry; John Wiley: New York, 1964.
40000
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
Table 1. Ranges of D, Y. and
dansyl-L-isoleucine acetone a
1953
Predicted for This Study
36 200 25 800 33 600
22 30 22
1.020*-2.020fi 1.02Dfi-l.41Dfi (1.25-2.25) X lo4
98
38 80
4.00-7.18 4.79-6.34 5.28-9.74
80 32 84
Percentage change calculated with respect to room-temperature value.
Equation 20 also defines the injection mobility, pi, at any injection field strength Ei = Vi/Lt. Implicit in our use of eq 20 to estimate pi is the assumption that the buffer reaches its steady-state temperature, T, very quickly after the injection voltage Vi is turned on. This assumption is necessary, because the )c expressed by eq 20 represents a steady-state value, in which the variation of T (and hence p ) with time during the initial stages of electrically heating the capillary35 has been washed out. Thus, the equation for p can represent pi only if the transient heating time during the injection is very short. This is a reasonable assumption for the very low Ei's used here (-1667 V/m or less). In fact, we made our injections at these low Ei's to assure the validity of this assumption. Furthermore, injections at low voltages over long times tend to be more reproducible.5 (b) High-Temperature Data. Figure 2b is a plot of J v s E, as determined from the high-temperature data. The solid curve in this figure is an unweighted fit of these data to the function
+
J = 0.315E a'E2 + b'E3 (23) with a' =' -1.99 X lo+ and b' = 2.03 X 10-10. The coefficient multiplying E in eq 23, 0.315, is the electrical conductivity of the buffer at 30 "C, as estimated from eq 19. From eq 23, one infers that k(E) is
k(E) = 0.315 - 1.99 X 1O*E
+ 2.03 X 10-'OE2
(24) When eq 24 is equated to eq 19, the profile of T vs E represented by the solid curve in Figure 4b is obtained. Figure 5b reporta the variation of apparent mobility p with E for the high-temperature data. The estimation of T from these data is somewhat more challenging than for their lowtemperature counterparts. This challenge exists, in part, because of scatter in the data. More important, though, is that the fitting function also must determine the injection mobility pi with accuracy. The data necessary for this accurate estimation, however, are lacking; the room temperature was lowered before p's at low field strengths could be estimated. We ultimately fit the data by weighted least squares to the function
+ +
bE cE2 = p(E) = 4.79 X (25) with b = -2.82 X 10-13 and c = 3.42 X 10-17. In other words, we arbitrarily defined the intercept to equal 4.79 X 10-8 m2/ p
(V-s). This number equals the product of the intercept determined from the low-temperature data, 4.00 X 10-8 m2/ (V-s),and the ratio of the viscosity of the buffer at 22 O C to that a t 30 O C . In other words, we simply corrected the lowtemperature intercept for the variation of viscosity with temperature. While fixing the value of the intercept in this manner is not desirable, the absence of data at low field strengths left us little choice. An indirect check on this intercept does exist, however: the ratio of the estimated electrical conductivity at 30 O C , 0.315, to this mobility is 6.58 X 106, just as it is in eq 21. Since k/r should be independent of temperature, this finding is encouraging. The function k(E) was estimated from eqs 21 and 25 and then equated to eq 19. The dashed curve in Figure 4b is the (35) Coxon, M.; Binder, M.
J. J. Chrornatogr.
1974, 101, 1.
profile of T so estimated. While the agreement of this temperature profile with the profile determined from the current density is not as good as it was for the low-temperature data, it is still reasonable. The quadratic that describes T from the mobility data is
+
T = T(E) = 29.7 - 2.11 X 104E 3.10 X 10-8E2 (26) We shall use this profile to estimate T(E)for the high-temperature data. Its choice over the profile determined from the current density is arbitrary. Equation 25 also defines the injection mobility, pi, for the high-temperature data. Determination of D = D(E). From the functions T(E), we can determine s(E) from eq 13. In accordance with eq 12, these functions determine D(E) in terms of room-temperature diffusion coefficients. ( a ) Acetone. The diffusion coefficient of acetone in water at 20 "C is 1.16 X 10-9m2/s.22From this value and eqs 12 and 22 (only low-temperature data are used here), we calculate that D(E) = 1.25 X lo-' - 1.64 X 10-14E + 1.38 X 10-l8E2 (27) where T* = 21.8 O C in eq 12 and vlt is the viscosity at this temperature. Hence, D is defined for all values of E. (b) Damyl-L-isoleucine. Because the diffusion coefficient of dansyl-L-isoleucine at room temperature is not known to us, we shall consider its room-temperature value, D*, to be a parameter determined by fitting the N's of our electrophoretic peaks to eq 9. For the low-temperature data, eqs 12 and 22 predict D(E) = DJ1.02 - 1.34 X 10-5E+ 1.13 X 10-'E2)
(28)
whereas for the high-temperature data, eqs 12 and 26 predict
+
D(E) = D,,(1.02 - 1.20 X 10-5E 1.05 X 10-'E2) (29)
where D* is the diffusion coefficient of dansyl-L-isoleucineat 21.8 O C (in eq 28) and at 29.7 "C (in eq 29). Here, D is defined for all values of E in terms of Dt. Determination of pw = pW(E). Figure 5a illustrates the variation of the electroosmotic flow coefficient peowith E, as determined by acetone. The squares represent experimental determinations, and the vertical error bars represent one standard deviation. The dashed curve is the weighted fit to these data of the function beo= 5.28 X
+ 6.56 X 10-14E+ 3.77 X lO-I7E2(30)
Note that the coefficient multiplying E is positive. This expression also determines the injection mobility, pw,i, for the acetone. Characterization of Functions for D(E), r ( @ , and pi(I3'). Table I reports the ranges of the variables D, p , and peopredicted by this study. For a given analyte (e.g., dansylL-isoleucine or acetone) and room temperature (e.g., 22 or 30 OC), the minimum values are determined by the appropriate equations with E = 0 V/m; the maximum values are determined by E = E,, the maximum field strength associated with that analyte and temperature. It is apparent that both D and p can vary by roughly a factor of 2 over the voltage ranges considered here. The ratio D / p does not vary
1954
-
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992 L, = 0.600 m
1.2 10' r
N
injection: I k v
01
so s (50 kv.ri
t I
" O lo4 A e c f l o n ,
L , = 0.6 rn
L , = 0 745 In IO kV af IO see (IW PV.sJ
5900
6000
6100
6200
6300
466
489
TIME (s)
0.0 loo y 0
492
495
498
501
504
TIME ( 5 )
I
10000
v
20000
30000
Flgure 6. Plot of N vs V determined from the marker of electroosmotic flow, acetone. Solid curves are predictions of N based on eqs 9, 27,and 30;the circles represent experlmental determinations.
significantly, however; the percentage variations over the ranges considered here are less than 10%. Prediction of N for Acetone as Electroosmotic Flow Marker. As a preliminary evaluation of our fitting functions, the numbers N of plates corresponding to the peaks generated by our marker of electroosmotic flow, acetone, were predicted from eq 9. The predictions are possible, because all variables and parameters on which eq 9 depends are known. Consequently, a good (poor) agreement between measured and predicted values of N will be indicative of good (poor) approximations to the functions T(E), p ( E ) , and D(E) determined from the low-temperature data. Figure 6 is the graph of N vs V so determined. The circles and squares represent values of N determined by experiment, and the vertical error bars on these symbols represent one standard deviation. Each symbol represents the average of two experiments. Two lengths, Lt = 0.745 and 0.600 m, and two plug sizes, 50 and 100 kV.s, are represented in the figure. The solid curves are the predicted values of N determined by combining eqs 27 and 30 with eq 9. The agreement between predictions and data is quite good, especially a t low V's; the largest errors at the high V's are less than 10-15 % This small overestimation of N at high V's for the 0.745-m data is systematic and may be due to the effective value of peo,i being a bit lower than postulated. At the relatively high Ei = 13 400 Vlm used to generate these data, peo,imay not reach its steady-state value on a scale of time very short compared to the injection time, ti = 10 s (we initially did not anticipate predicting these Ks and consequently were not concerned that peo,iwas poorly defined here). The general agreement suggests that we have appropriately modeled the variation of both mobility and diffusion coefficient with field strength. The variation of these data with voltage is easily explained. At low voltages, sufficient time exists for diffusion to substantially broaden the analyte plug beyond its initial width. In this case, diffusion dominates dispersion, and N varies with V in an approximately linear fashion. At large V's, however, the analyte plug is transported so rapidly past the detector that the plug does not have time to broaden much by diffusion. If the initial plug width is large enough, N is predominantly determined by this width and becomes independent of V. In Figure 6, this behavior is exhibited by the 100kV.s plug but not the 50 kV-splug. The data associated with this latter plug, however, are clearly approaching a plateau. Prediction of Peak Profile for Electroosmotic Flow Marker. To affirm further the validity of our modeling of p ( E ) ,D(E), and !(E ' ), we calculated several expected peak profiles from eq 14 and compared the calculations to the peaks generated by acetone. In these calculations, the appropriate values of D,p , and pi were determined from the above empirical fits and calculations, and t , was approximated by the first
.
257
258
259
260
TIME
261 (5)
262
263
I
.
.
.
820
825
830
835
. 840
845
,
324
326
TIME (s)
La = 0.745 rn
496
502
506
TIME
510
(5)
514
316
318
320
322
TIME
(s)
Flgure 7. Graphs of exper!mentally determined and theoretically predlcted peaks of the marker of electroosmotic flow. Solid curves are experimental results: dashed curves are predictions.
moments of the experimental peaks. Figure 7 is a composite of these comparisons for various V's, Lt)s, and plug sizes. The solid curves are the digitally filtered experimental data; the dashed lines are the theoretical predictions. The only adjustable parameter in the predictions, CO,was arbitrarily scaled in each subfigure, such that the maximum height of the experimental peak coincided with the maximum height of the predicted peak. By and large, an extremely good agreement exists between predicted and experimental peak shapes. For large plug sizes and high voltages (e.g., 100 kV-s, 20 0oO V),the peaks exhibit flat tops, because the plug width is so large and the time for diffusion is so short that the center part of the plug never has the opportunity to diffuse along the separation axis. This close agreement between experiment and theory suggests that other mechanisms of band broadening (e.g., adsorption-desorption processes and hydrodynamically induced flow) are negligible under the conditions of our experiments. The agreement is so good that we feel confident in using our expressions for p ( E ) , D(E), and "(E) to model our CE peaks. We now turn to that objective. Least-SquaresFitting of Equation 9 to Ns Calculated from Peaks of Dansyl-L-isoleucine. Figure 8 is a detailed plot of N vs V for the four capillary lengths considered here. The circles, squares, and triangles represent experimental determinations of N corresponding to 10, 50, and 90 kV.s injections (no 90 kV.s data are reported for the longest capillary, which was accidentally broken at the detector window after the acquisition of the 10 and 50 kV.s data). The vertical error bars on these symbols represent one standard deviation. The solid curves are weighted least-square fits of these data to eq 9. Weighted fits were employed because the precisions of the different measurements differ considerably (e.g., consider the imprecisionof the N for the 10kV.s injection at 27 kV in Figure 8c and the precision of the other K s for the 10 kV-e injections in the same subfigure). In general, the
ANALYTICAL CHEMISTRY, VOL. 84, NO. 17, SEPEMBER 1, 1992
1055
m, r) = 0.001 kg/(m.s), pi 4.0 x m2/(V.s),and Vi = lo00 V. Therefore, a 50 kV.s injection in Figure 8 roughly 9.0 105 corresponds to what one would expect for a typical hydro6.0 10' dynamic injection, e.g., an injection for 10 s from a height difference equal to 10 cm. 3.0 10' Several trends are evident in Figure 8. Perhaps the most dominant observation is that N never varies in an exactly 0.0 100 ' 0.0 100 10000 20000 30000s,o loso 10000 20000 30000 linear manner with V, even when ui2 = 0. In the latter case, 8.0 10: this modest curvature is due to the slight increase in the D/p N 6.0 10' ratio with increasing temperature (field strength). Because C 3.0 10' N is inversely proportional to this ratio, the number of plates 4.0 10' decreases slightly with increasing V. As the plug width increases, however, N approaches a plateau with increasing 2.0 10: V and is dominated by injection effects. In one case, cor0.0 10' responding to the 90 kV.s injections on the 0.600-m capillary, 10000 20000 30000 0 5000 10000 15000 20000 25000 0 V V a plateau is actuallyreached (Figure 8d). Similar results were reported by the group of Dovichi and were interpreted Fmn8. Plots of Nvs Vfor peaks of dansyk-isoieuclne,as determined on four different lengthsof capillary. The circles, squares,and triangles correctly as injection effects.9 representexperlmentaiNsdeterminedby 10,50, and 90 kV.s injections. Other trends are also apparent. For the two largest Lt)s, Solid curves are weighted least-squares fittlngs of the data to eq 9, the N's in Figure 8a,b are indistinguishable for the 10 kV.s wlth p(€)and Q€)expressed by eqs 20 and 28 for the low-temperature injections and are barely distinguishable for the 50 kV.s data and by eqs 25 and 29 for the hlgh-temperature data. Dashed injections. This behavior is attributed to the very small curves are graphs of N determlned from eq 9 and the appropriate expressions for p(€), Q€),and D,, wlth ut equated to zero. Values fraction of capillary that is initially occupied by the plugs. In of D,, were determined as least-squares coefficients. For the data in contrast, N clearly decreases as the plug width is increased Figure 8c,d and for the 10 and 50 kV.s data in Figure 8b, T,, = 22 "C from 10 to 50 kV.s, even for the longest length. The decrease and D,, = 5.1 X 10-lom2/s; for the data in Figure 8a and the 90 kV.s is even more marked when the plug size is increased to 90 data In Flgure ab, T,, = 30 "C and D,, = 6.0 X 10-lo m2/s. The kV.s. In Figure 8b, for example, roughly 106 plates can be dlfference L, - Ld = 0.090 m. attained a t high V's for 10 kV-s injections, but only ca. oneimprecision increased with time and was worst for the shortest third that number can be attained for 90 kV.s injections under length. This trend most likely has nothing to do with length identical conditions. In all cases, for the sizes of plugs but with the age and previous use of the capillary. considered here, the influence of injection effects on N is For the fitting of the low-temperature data, p(E)and d E ) measurable for V's greater than 10 kV. In extreme cases were approximated by eq 20 and D(E) was approximated by (e.g., see Figure 8d), the effect is measurable after only 1 or eq 28. For the high-temperature data, p(E) and pi(E) were 2 kV. approximated by eq 25 and D(E) was approximated by eq Another trend is the very minor influence on N of the 10 29. The room-temperature diffusion coefficients, Dt, of dankV-s injections considered here, as compared to the N's for syl-L-isoleucinedetermined by these fits are 5.1 X 10-lom2/s which ai2 = 0. We conclude that plugs of this size have no for the low-temperature data (T, 22 "C) and 6.0 X lO-"J measurable effect on N over the ranges of Vand Lt examined m*/sfor the high-temperature data (Tt= 30 "C). The dashed here. The drawback of injecting plugs of this size, at least curves are graphs of eq 9, as determined by the appropriate by manual means, is poor reproducibility. Our best reproexpressions for D(E), Dt, and p(E),with u? arbitrarily set to ducibilities were attained with 50 and 90 kV-sinjections, which zero. These curves help us gauge the maximum possiblevalues do affect N. of N in the absence of injection effects. A final trend is that the relative influence of injection effects Before discussing these graphs, we observe that the increases as Lt decreases. For example, the differences among estimates of Dfi are quite reasonable. For purposes of the N's for the three plug sizes are more pronounced when comparison, we calculated theoretical estimates of Dt from Lt = 0.600 m (Figure 8d) than when Lt = 1.046 m (Figure 8b). the Reddy-Doraiswamy,Othmar-Thakar, Wilke-Chang,TynThis trend has two origins. First, the influence of any plug Calus, and Nakanishi equations22J6 and then averaged the size becomes more pronounced at shorter lengths, because estimates so determined. The average Dt's were (5.2 f 1.2) the fraction of the capillary that initially is occupied by the X 10-10 m2/s (22 "C) and (6.4 f 1.5) X 10-lO m2/s (30 "C). The plug increases. Secondly, for a constant injection voltage, least-squares estimates of Dfireported above are well within the widths introduced into short capillaries are greater, all one standard deviation of these 9th. other things being equal, than those introduced into long Many users of CE inject analyte mixtures by hydrodynamic capillaries. In other words, a 50 kV-s injection generates a means, instead of electromigrative ones, to minimize b i a ~ . 3 ~ bigger plug on a small capillary than on a large one. This Even though all analytes here were injected by electromiprincipally occurs because the injection field strength, Ei, is gration, we make the following observation to increase the inversely proportional to Lt. In addition, the injection relevance of Figure 8. For our particular system, approximobility pi increases with E;. mately 4.3 times as much analyte would be injected over the Figure 9 is a plot of N vs Lt, as determined by these data same injection time ti by hydrodynamic means than by elecand fits, for V = 10, 15, and 20 kV. These graphs simply tromigrative ones, if a 10-cm difference in height were display the data and fits in Figure 8 relative to a different maintained between the capillary ends. This calculation is axis. The solid curves correspond to the fits determined from based on the ratio of eq 7b to 7a, which is the low-temperature data. Some of the high-temperature data are included to augment the otherwise small pool of lh/le = pgAhd;/32piVir) 4.3 (31) low-temperature data and are indicated by asterisks adjacent with p = lo00 kg/m3, g = 9.8 m/s2, Ah = 0.10 m, d, = 7.5 X to the symbols. For sufficient lengths, N approaches a constant value and becomes independent of plug size. At (36) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of shorter lengths, however, the effect of plug width on N is Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. apparent, resulting in a rapid loss of efficiencywith decreasing (37) Huang, X.; Gordon, M. J.; Zare, R. N. Anal. Chem. 1988,60,375. 1.2 10'
1.2 10'
N
-
-
1956
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992 6.0 lo5 r
I
9 it
v=loooo
700
0.0 100 0
0.4
0.8
1.2
703
706
709
712 398
400
TIME (s)
1.6
402
404
TIME (s)
50 kV-r injection Lc = 0.6 m L8 = 0.6 m
0
0.4
0.8
1.2
1.6 I
L
8.0 10'
2390
V
N
2410
2430
2450 382
384
3%
388
390
TIME (s)
TIME (s)
6.0 10' 4.0 10'
2.0 105
1 / /"
0.0 loo 0
0.4
0.8
1.2
L f = 0.6m
1.6
Lt
Flgurr 0. Plots of Nvs L, for peaksof dansyk-Isoleucine,as determined at different voltages. The circles, squares, and triangles represent experimental Ns determined by 10,50, and 90 kV.s Injections. Solid curves are welghted least-squares fittings of the low-temperature data to eq 9. Data without asterisks correspond to T,, = 22 OC and D,, = 5.1 X m2/s; data with asterisks correspond to T,, = 30 OC and D,, = 6.0 X 10-lom2/s. The difference L, - & = 0.090 m.
length. For sufficiently smallLt's,even the 10 kV-sinjections have a measurable effect on N. To affirm further the integrity of our dispersion model, we compared our dansyl-L-isoleucine peaks to those predicted by eq 14. The predictions were determined as detailed earlier for the electroosmotic peaks. Figure 10 is a composite graph of these comparisons. The solid curves are the digitally filtered experimental data; the dashed lines are the theoretical predictions. By and large, a good agreement exists between the predicted and the experimental peaks. The closest agreements are found at low voltages. The most marked disagreements, although small, are found at large V's and plug widths (e.g., 90 kV-s at 20 kV). Here, the peaks are slightly asymmetrical, with the leading edge somewhat more diffuse than the trailing edge. In such cases, where a substantial portion of the peak center has not been diluted by diffusion, the concentration of dansyl-L-isoleucineis high enough to alter the local field strength over the zone's width (acetone does not behave similarly because it is electrically neutral),*JO even though the buffer-to-analyte concentration ratio exceeds 400. Because the mobility of dansyl-L-isoleucine is less than that of similarly charged buffer ions, the sharpness is introduced at the leading edge of the peak. The sharpness appears at the peak's trailing edge,however,because electroosmosis opposes the electrophoresis and produces a mirror image of the peak.38 With careful examination, this asymmetry also can be seen in the high-voltage 50 kV.s runs. In spite of this asymmetry, the predicted peak shape is fairly good. A P o s t e r i o r i V e r i f i c a t i o n of Data Selection. As observed in the Experimental Procedures, only those data whose (38) Foley, J. Personal communication, 1992.
I
2360
2380
2400
2420
TIME (s)
2440 230
231
232
233
234
235
TIME (s)
Flgure 10. Graphs of experimentally determined and theoretlcaliy predictedpeaks of dansyl-c-isoleucine. Solid curves are experimental results; dashed curves are predictions.
plate numbers and peak areas were both reproducible were used in this study. The greatest irreproducibility in areas was found for the smallest plug sizes (10 kV.s). In some cases, our selection of the data seemed somewhat arbitrary. We verified our selections a posteriori by plotting scaled peak areas against the product of injection voltage and injection time. By modifying a standard formula for the area A of a peak generated by a concentration-sensitive detector,39 one can determine that A (in units of s) is
where eb is the product of the extinction coefficient and the effective path length across the capillary and C ( t ) is the analyte concentration at time t. The areas of the dansyl-Lisoleucine peaks were graphed against a rearranged form of eq 32a, which is ApV = tbC,,piViti= yViti
(32b)
where the slope y of the line equals ebC,pi. A line then was fit by weighted least-squares methods to the appropriate lowtemperature data, with y determined as 4.01 X lo-" m2/ (V-s). The appropriate high-temperature data were similarly fitted, with y determined as 4.92 X lo-" mZ/(V.s). When dansyl-tisoleucine was introduced through the capillary under hydrodynamic pressure and a breakthrough curve was generated, the proportionality constant, tb = 23.8 M-l, was determined. By substituting this constant, the bulk M, and ~i = 4.00 X (or 4.79 concentration Co = 4.03 X x 10-8) mZ/(V.s) into eq 32b, we calculated that y should (39) Karger, B. L.; Snyder, L. R.; Horvath, C. An Introduction t o Separation Science; John Wiley: New York, 1973.
ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
equal 3.84 X lo-" m2/(V.s)for the low-temperature data and 4.59 X 10-11 m2/(V.s) for the high-temperature data. The close agreement between the predicted and measured values of y (thepercentage error is about 5) suggeststhat our selection of data was appropriate. This close agreement also alleviates other concerns. First, it offers additional confirmation that hydrodynamic flow, which could augment or reduce the plug size, is negligible under our injection conditions. Furthermore, it suggeststhat diffusion of dansyl-L-isoleucine,either from the injection vial into the capillary or from the capillary into the buffer vial, was negligible. Finally, it suggests that the rise time of the power supply (about 0.5 s) had a negligible effect on the injection process, even for 10-5 injections. Reinterpretation of Figure 1. The data in Figures 8 and 9 closely resemble the data of Jorgenson and Lukacs shown in Figure 1. In Figures l a and 8, N increases linearly with V for small voltages but approaches a plateau for large V's. In Figures l b and 9, N is independent of length for large Lt's (or generic lengths L's) but decreases rapidly as the length is reduced. Before we argue that the data in Figure 1can be explained by injection effects,we shall evaluate the likelihood that the anomalies in Figure 1 are attributable to Joule heating, as suggested by Jorgenson and Lukacs. First, we acknowledge that we cannot prove that Joule heating effects are absent in these data, because all the relevant experimental parameters were not published. Rather, we can only argue that Joule heating effects are unlikely. To make this argument, we shall examine a worst-case scenario and show that heating effects in this scenario are negligible. Logically, then, they must also be negligible in reality. A calculation based on a worst-case scenario is possible because the dispersion caused by Joule heating increases with increasing temperature. By deliberately overestimating the buffer temperature, we will deliberately overestimate the dispersion. We will focus on Figure lb, because the analyte is dansyl-L-isoleucine. To overestimate the buffer temperature, we will simply assume that it equals its maximum possible value, 100 O C , for all lengths. The actual temperature had to be less; otherwise, the buffer would have boiled and the current would have dropped to zero. Several theories for the nonequilibrium contribution to plate height due to Joule heating have been proposed; some recent ones are by Knox,40 Grushka,41 and Davis.24 For equivalent conditions, these three theories typically agree within a factor of 2 or so. According to Davis, the variance associated with the dispersion is24
where CY is a slightly decreasing function of temperature, B is a heating parameter that increases rapidly with temperature, rc is the capillary radius, and po is the electrophoretic mobility, pep, at the inner capillary wall. Let us suppose that the only sources of dispersion in Figure l b are longitudinal diffusion and Joule heating effects. For lengths greater than 0.9 m or so, N is independent of length, and longitudinal diffusion alone must cause dispersion. Here, N is about 234 OOO. The ratio D / p is determined by this N and eq 1 (which we use, because we are supposing that diffusion is the only source of dispersion) as 3.2 X V. To determine this value, we have assumed that LdLt = 1; the exact ratio of lengths is unknown, but this assumption is reasonable for the longer lengths. This D / p ratio is about twice as large as determined here by our experiments; indeed, (40) Knox, J. H.; Grant, I. H.Chromatographia 1987, 24, 135. (41) Grushka,E.; McCormick, R. M.; Kirkland, J. J. Anal. Chem. 1989, 61, 241.
500000 40;oo
1
1057
Assumptions:
300000
-
200000
-
100000
-
D =6x10'om'ls L , = Ll = 1.0 m
least squares parameters w = 3 . a xlO'drn'lvS Viti I 9 9 kV-r I
0
300000
10000
V
20000
30000
-
N
Assumptions:
200000
-
100000
-
L,=L,
least squares parameters:
0 0
0.4
0.8
1.2
1.6
Flguro 11. Plots of (a) N vs V and (b) N vs L = L, deterrnlned by Jorgenson and Lukacs. SolM curves are unwelghted least-squares flttlng of the data to eq 8. Assurnptlons are reported In both the figure and text. Values of p and V,t, were determlned as least-squares coefflclents. if LdILt were 0.5, then the agreement with our results would be exact. This possibility cannot be ruled out, but it will introduce an error of only a factor of 2 or so. The CY and B terms in eq 33 were estimated using formulas in ref 24. The latter term depends on electrical conductivity. The electrical conductivity of the 0.05 M, pH = 7 phosphate buffer used by Jorgenson and Lukacs was approximated here by multiplying the estimated conductivity of our 0.01 M phosphate buffer at 100 O C by a factor of 5. This approximation, too, is an overestimation, because our buffer also contained borate (which increased its conductivity) and because equivalent conductivity decreases with increasing concentration. Hence, we still are considering a worst-case scenario. We then calculated the number N of plates expected when only ad2 and UJh2are considered as sources of dispersion. We took advantage of the fact that the value of Dlp does not vary much with temperature and used the estimate 3.2 X 10-2 V in our calculations. If we again assume that Ld = Lt, then no measurable dependence of N on UJh2is found over the range of lengths considered here, 0.52 < Lt < 1.54 m. For the worst possible Case, Lt 0.52 m, Ud2 = 1.1 x m2,and nJh2 = 2.5 X 10-9 m2;the variance due to diffusion is 440 times greater than the variance due to Joule heating. Even if Ld is substantially less than Lt, our assumption that Ld = Lt cannot possibly account for a difference of this magnitude. If this worst-case scenario cannot account for the anomalous dispersion in Figure lb, then Joule heating most probably is not its cause. Similar arguments can be made for the data in Figure la. In contrast, the solid curvesin Figure 1la,b are least-squares fits of the data in Figure la,b to eq 8 (not eq 9), with D,p, and pi approximated by constants. Because all relevant parameters were not reported, we had to make some reasonable assumptions to obtain these fita. To obtain the fit in Figure l l a , we assumed that Lt and Ld were both 1.0 m (the capillary lengths varied from 0.8 to 1.0 m in ref 3); to obtain the fit in Figure l l b , we assumed that Ld = Lt. For both fits, the injection mobility pi was approximated by p and D was estimated from the Reddy-Doraiswamyequation as 6 X 10-10
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
m2/s for both analytes. Two least-squares parameters were calculated from these fits: the apparent mobility, p = pi, and the product of the injection voltage and time, Viti. The latter parameter is on the order of 100kV-sfor bothdata sets. These estimates are in good agreement with Jorgenson’s recollection that “in this early work (1981),injection times of 2 to 4 seconds at full voltages (30 kV) were typical“.42 The apparent mobilities are also quite reasonable, especially for N vs V. In this case, one sees that the nonzero intercept described in the Introduction is accounted for rather simply: the data for small V‘s are in fact not linear but only appear to be. In light of the absence of exact values for Ld, Lt, etc., more specific results cannot be determined. There are, therefore, several compelling reasons to believe that the anomalies in Figure 1 are simply due to injection effects. The truth will never be known, however, and given the absence of other facts, it would be foolish to argue further. Our argument must ultimately end on a speculative note. Perhaps we will be forgiven, however, if we echo Orwell: all speculations are equal, but some speculations are more equal than others.
CONCLUSIONS The principal conclusion we draw from our study is that injection effects can have a pronounced impact on efficiency in CE, as measured by theoretical plate number N. The greatest effect is found with large analyte plugs and with short capillaries. This finding is particularly relevant to the analytical community, whose members commonly use short capillaries to reduce analysis times and inject plugs of modest size to increase precision. This study suggests that some commonly accepted rules of thumb in CE may be misleading. For example, one often reads that the volume of the injected plug must be less than 1%of the total column volume, if injection effects are to be reduced to acceptable levels.l3-43 We suggest that the upper limit, 176,is difficult to rationalize, at least in some cases. For example, the width of the 90 kV.s plug introduced onto the shortest capillary here (L, = 0.600 m, Ld = 0.510 m) is pivitil Lt=(4.00X 10-8)(10o0)(90)/(0.600)=0.006m=6mm. While this is indeed a large plug, it is only 0.006/0.600 = 0.01, or 176 , of the full volume of the capillary and only 0.006/0.510 = 0.012,or 1.276,of the volume from the injection end to the detection window. Yet no one would deny the profound impact on N imposed by a plug of this size, as illustrated by Figure 8d. Furthermore, we believe that all too many users of CE invoke Joule heating effects to explain losses of efficiency at high field strengths. In fact, one has to dissipate a good deal of electrical power in the capillary before these effecta become measurable. In the worst case considered here, for example (Lt = 0.745 m, V = 27 kV, and i = 76 PA), the amount of heat dissipated is 2.0 W, uJh2 is only 7.4 X 10-10 m2, ad2 = 5.2 X 10-7 m2, and the expected temperature drop over the capillary cross-section is only 0.35 deg. This dispersion simply is not important under the conditions of our experiments, nor is it important in most CE experiments with similar buffer concentrations. Rather, the losses of efficiency observed at high field strengths in CE are principally due to adsorptiondesorption kinetics, injection effects, or in a poorly designed system, hydrodynamic flow. Finally, the finding that peak asymmetry can be observed, even when the buffer-to-analyte concentration ratio exceeds 400, contradicts the commonly stated rule that such effects (42)Jorgenson, J. Personal communication, 1989. (43)Gordon, M.J.;Huang, X.;Pentoney, S. L., Jr.; Zare, R. N. Science 1988,242,224.
Figure 12. Three-dlmenslonalplots of N vs Vand L, determlned from the low-temperature data for 0, 10, 50, and 90 kV.s Injections. The graphofNftx V;t,=O kV-swasdetermlnedfrcmeq9andtheappropriete expressions for p ( 0 , 40,and D,, with u t equated to zero. The difference L, - = 0.090 m.
are minimal when this ratio exceeds 100 or ~ 0 Others . ~ have reported symmetrical peaks for identical concentration ratios.8 The apparent discrepancy is resolved by noting that the magnitude of this effect is not only controlled by the concentration ratio but by plug width,lO which determines whether the plug center is diluted by diffusion by the time the plug is swept past the detector. Figure 12 is a three-dimensional plot of N vs Lt and V generated by the theoretical fitting of our low-temperature data to eq 9 (the minimum Lt is 0.090 m, because Lt - Ld = 0.090 m). The graph of N for Viti = 0 kV.s was determined from eq 9 and the appropriate expressions for p ( E ) , D(E), and Dd with ui2 equated to zero. By comparing the graphs for the 0 and 10 kV.s data, one sees that even plugs as small as 10kV-shave a pronounced effect on N at very short capillary lengths (Joule heating effects also can be expected at extremely short lengths and high voltages, as shown in ref 13). For greater lengths, however, 10 kV.s injections have virtually no effect on N (similartrends were observed in Figure 8). Nevertheless, Lt must be considerably greater than 1m before N becomes independent of Lt, even for a 0 or 10 kV.s injection. In contrast, the commonly used capillary lengths of today are about 0.5-0.6 m. These observations lead us to conclude that the simpler form of eq 1,N = pV/2D, is rarely valid but that the more specific form of eq 1,N = pVLd/2DLt, can be realized for plug sizes similar to 10 kV.s. This limiting plug size is comparable to those used by Sepaniak and Cole (15 kV.s)’2 and the group of Dovichi (5 kV.s)9 to reduce injection effects to negligible levels. One may ask by what means injection effects can be reduced. Our experience indicates that small-volume injections, e.g., 10 kV.s, are not reproducible by the manual injection procedures used here. A greater reproducibility quite possibly can be achieved by using the automated injectors associated with commercial instrumentation, especially if injections are made by hydrodynamic means. An alternative way to reduce injection effects in free-solution electrophoresis is to concentrate a large-volume injection into a narrow plug by stacking.2~4~Stacking indeed has been employed in CE46.47but does not seem to be a widely (44)Mikkers, F.E. P.; Everaerts, F. M.; Verheggen, Th. P. E. M. J. Chrornatogr. 1979,169, 1. (45)Ornstein, L.Ann. N.Y. Acad. Sci. 1964, 121,321. (46)Chien, R. L.;Burgi, D. S. Anal. Chem. 1992,64, 489A. (47)Chien, R.L.;Burgi, D. S. Anal. Chem. 1992,64, 1046.
~
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 17, SEPTEMBER 1, 1992
appreciated technique. We hope that this study will emphasize its importance. Plugs also can be preconcentrated in gel capillaries by selective sievinge2 In summary, this study shows that, by the appropriate choice of experimental conditions, one can either minimize or eliminate many other potential sources of dispersion in CE, e.g., hydrodynamic flow,adsorption-desorption processes, radial mobility differences due to Joule heating, and differences in plug velocity due to gradients in local field strength (adsorption-desorption processes admittedly are difficult to eliminate for species with large surface areas, e.g., proteins). We demonstrated simple means for estimating the buffer temperature at different field strengths from the nonlinear variations of current density and mobility with field strength. The current density, in particular, is easily determined; one can gather over a wide range of field strengths the data necessary for ita calculation in a matter of minutes. We furthermore showed that the apparent variations of mobility and diffusion coefficientwith field strength can be determined and that the amount of longitudinal diffusion, as measured by the D / p ratio, can be estimated from these variations. This finding is important, in light of a recent study in which others suggested that this ratio was governed by the electrical charge of the analyte.21 As was shown here, however, mobility
1959
and diffusion coefficient can be considered independent of electrical charge. Finally, we demonstrated that a simple one-dimensional model can be used to predict rigorously the peak profile, when diffusion and injection effects are the only mechanisms of dispersion, and can be used as a diagnostic guide to determine if other sources of band broadening are relevant (e.g., as in the last subfigure in Figure 10).
ACKNOWLEDGMENT We thank Ed Yeung and Norman Dovichi for helpful insights into the origins of our initial experimental difficulties. We also thank Stellan Hjertkn and Joe Foley for helpful comments on our peak asymmetry. Finally, we thank Scott Weinberger for addressing some problems with our detector. This work was supported in part by the Office of Research and Development at Southern Illinois University. This work was presented at the 201st National Meeting of the American Chemical Society, Atlanta, GA, April 14-19, 1991, and the 30th Eastern Analytical Symposium, Inc., Somerset, NJ, November 11-15, 1991.
RECEIVED for review January 9, 1992. Accepted May 18, 1992.
