Anal. Chem. 1998, 70, 3397-3406
Predicting Peak Symmetry in Capillary Zone Electrophoresis. Background Electrolytes with Two Co-ions: Schizophrenic Zone Broadening and the Role of System Peaks Petr Gebauer, Pavla Borecka´, and Petr Bocˇek*
Institute of Analytical Chemistry, Academy of Sciences of the Czech Republic, Veverˇ´ı 97, CZ-611 42 Brno, Czech Republic
This paper investigates fundamental phenomena in the migration behavior of analytes in the very frequent case in current capillary zone electrophoresis practice of using background electrolytes with two co-ions. It is shown that in such background electrolyte systems there always exists a regular system phenomenon which brings enormous electromigration dispersion to certain analytes. The analytes which yield to this phenomenon may be predicted by calculating the so-called velocity slope SX, which is a measure of zone symmetry and broadening. On the example of a simplified model system, it is shown explicitly that these analytes coincide by their mobilities with the mobility of the system peak. The explanation for these effects is given on the basis of schizophrenic behavior where one and the other coions force the analyte zone to be fronting and tailing simultaneously. Detailed theoretical description of the behavior of analyte zones in binary co-ion background electrolytes is presented using peak shape diagrams that provide a comprehensive view of their electromigration dispersion characteristics. Experimental verification of the theory is given, together with a detailed discussion of the effect of the background electrolyte composition on the magnitude and extent of appearance of schizophrenic zone broadening. The more capillary zone electrophoresis (CZE) becomes an important tool for routine analyses in many fields, the more its theoretical background becomes important in practice to be able to use the method in an efficient and correct way. Although there has already been a large piece of work done in the field of CZE theory, phenomena can still be found that are known but insufficiently described or even still unrecognized at all. One group of such phenomena can be addressed under the collective term “system peaks” or “system zones”. In its presently accepted meaning,1 this designates such types of zones (i.e., migrating concentration discontinuities) that do not contain any analyte from the sample but where some background electrolyte (BGE) component is completely or partially absent. A number of papers have addressed this topic in the past few years. It has been shown that the formation and/or presence of system zones is an inherent feature of CZE in BGEs comprising more than one ionogenic species of the same sign of effective charge.1-3 A simple (1) Beckers, J. L.; Everaerts, F. M. J. Chromatogr. A 1997, 787, 235-242. S0003-2700(98)00301-1 CCC: $15.00 Published on Web 07/07/1998
© 1998 American Chemical Society
theoretical approach was presented,3 defining the system zone as a migrating vacancy in the concentration of a BGE component. This has led to the concepts of vacancy3,4 and differential4 CZE, where the roles of the sample and BGE are reversed, i.e., the “analysis” of pure BGE in a sample-filled capillary provides a series of vacancy peaks that can be assigned to individual sample components. Experimental studies of BGEs with two co-ions (binary co-ion BGEs)5,6 have revealed some properties of the system zones, especially as far as the shape and sign of the related peak on the record are concerned. An important conclusion was made6 concerning the transition between a single co-ion and binary co-ion BGE, stating that an addition of about 5% of the second co-ion causes the resulting BGE to behave macroscopically as a regular binary co-ion one. This was demonstrated on the practically important case of bicarbonate coming from atmospheric CO2. Another theoretically accessible problem important in practice comprises the zone width and zone shape resulting from the given electromigration-dispersion (EMD) characteristics of a system. Because EMD is system-dependent, it cannot be affected by a change of the operational variables as effectively as other types of dispersion. Hence, its knowledge and the possibility to predict EMD in a fast and simple way are of major practical interest. Recently,7 the velocity slope (SX) was introduced as a useful quantity characterizing the EMD of a given analyte in a given BGE. SX was defined as the change in the local analyte migration velocity with the analyte molar fraction at infinite analyte dilution; its sign determines whether the analyte zone is of a fronting or a tailing shape, and its numerical value relates to the EMD zone width. Using a pK vs ionic mobility coordinate system, the SX values calculated for a given BGE can be plotted in the form of iso-SX contours; the result, called a peak shape diagram (PSD), allows fast and easy prediction of the appearance of CZE records. A similar approach was presented recently by Xu et al.,8 who introduced the EMD constant as a parameter making it possible (2) Poppe, H. Anal. Chem. 1992, 64, 1908-1919. (3) Gebauer, P.; Bocˇek, P. J. Chromatogr. A 1997, 772, 73-79. (4) Mikkers, F. E. P. Anal. Chem. 1997, 69, 333-337. (5) Desiderio, C.; Fanali, S.; Gebauer, P.; Bocˇek, P. J. Chromatogr. A 1997, 772, 81-89. (6) Macka, M.; Haddad, P. R.; Gebauer, P.; Bocˇek, P. Electrophoresis 1997, 18, 1998-2007. (7) Gebauer, P.; Bocˇek, P. Anal. Chem. 1997, 69, 1557-1563.
Analytical Chemistry, Vol. 70, No. 16, August 15, 1998 3397
to express separately the influence of conductivity and pH effects on the direction and degree of EMD of a zone. Only limited attention has been paid so far to the EMD properties of systems with a binary co-ion, although their knowledge has practical importance because many buffers used in CZE practice (e.g., those used with indirect detection or in the analysis of DNA fragments) contain at least two anionic co-ions. It is known that binary co-ion BGEs may potentially cause problems by an extraordinary broadening of some sample zones. This effect was described by Bullock et al.9 and investigated more in detail by experiments and computer simulations by Williams et al.10,11 The latter authors revealed the nature of that type of peak distortion to be an additional EMD dispersion and concluded that it occurs when the analyte zone comigrates with the system zone of the binary co-ion BGE. This agrees with the conclusion made by Beckers12 that the interaction of analyte and system zones leads to their enlargement. Theoretical calculations by the same author13 revealed that the analyte zone in the vicinity of the system zone causes a disturbance in both the BGE composition and the local mass balances, this resulting in heavy broadening of the sample zone. The concept of SysChart14 was proposed recently to be used for a complete description of the properties of a BGE: the SysChart consists of a system of eight plots showing the dependence of pH and concentrations in the analyte zone and of the ratio of the analyte velocity in the BGE to that in the analyte zone on the analyte mobility. From the plots, the properties of an analyte zone can be estimated and the apparent mobility of the system peak in a binary co-ion BGE can be predicted. This paper is aimed at presenting a comprehensive theoretical study of EMD effects in binary co-ion BGEs. The concept of velocity slope (SX) is used as the quantity making it possible to express the EMD properties (shape, width) of an analyte zone in a binary co-ion BGE in a quantitative manner. For a simplified case of strong electrolytes, an analytical solution is derived showing that in each binary co-ion BGE such an analyte mobility can be found for which SX ) (∞ and that this mobility exactly equals the apparent mobility of the system zone. For a general case, the peak shape diagram is utilized that makes it possible to show (in a single plot) for a binary co-ion BGE the SX values of all possible analytes and to recognize the region of those exhibiting extraordinary zone broadening in particular. The effect of the composition of the binary co-ion BGE on the EMD characteristics of the system is revealed, and the conclusions are illustrated by model experiments. THEORETICAL SECTION Characteristics of the Model. In this paper, we investigate the behavior of an analyte ion X migrating zone-electrophoretically in a binary co-ion BGE. The analyte is generally assumed to be a weak acid, HX. The binary co-ion BGE comprises two weak acids, HA1 HA2, and a weak base, B, as the two co-ionic and one (8) Xu, X.; Kok, W. T.; Poppe, H. J. Chromatogr. A 1996, 742, 211-227. (9) Bullock, J.; Strasters, J.; Snider, J. Anal. Chem. 1995, 67, 3246-3252. (10) Williams, R. L.; Childs, B.; Dose, E. V.; Guiochon, G.; Vigh, G. Anal. Chem. 1997, 69, 1347-1354. (11) Williams, R. L.; Childs, B.; Dose, E. V.; Guiochon, G.; Vigh, G. J. Chromatogr. A 1997, 781, 107-112. (12) Beckers, J. L. J. Chromatogr. A 1994, 679, 153-165. (13) Beckers, J. L. J. Chromatogr. A 1996, 741, 265-277. (14) Beckers, J. L. J. Chromatogr. A 1997, 764, 111-126.
3398 Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
Figure 1. Scheme of the model system.
counterionic species, respectively. The setup of the model is shown in Figure 1. The treatment of the problem is based on the same algorithm as used previously7 for the description of zone properties of single-co-ion systems. The analyte zone is investigated under conditions modeling infinite analyte dilution, and the zone composition is accessed via a simple moving-boundary model. This model calculates the composition of the analyte zone (zone AX, see Figure 1) from the known composition of the BGE (zone A) assuming that the Kohlrausch regulating function ω is kept constant with time at any point of the separation path. The mathematical description of the model is given in section 1 of the Appendix. As shown previously,7 the most useful output of the calculations using such a model are the values of the velocity slope, a quantity that has proved useful for the description and prediction of zone properties such as zone symmetry and zone width. The velocity slope SX is defined7 as the change in the local analyte migration velocity with the analyte molar fraction at infinite analyte dilution:
SX )
( ) dνX,AX daX
(1) aXf0
where aX is the molar fraction of X in its zone defined by eq A10 of the Appendix and
νX,AX )
iu j X,AX κAX
(2)
is the local analyte migration velocity; i is the electric current density, uj X,AX is the local effective mobility of the analyte in its zone, and κAX is the local specific conductivity in the analyte zone. The numerical value of SX for analyte X in a given binary co-ion BGE is accessible by the mathematical procedure described in section 1 of the Appendix. In this way, it is possible to investigate how the velocity slope depends on the mobilities of the involved ionic species and on the composition of the BGE (namely, on the concentration ratio of both co-ions A1 and A2). To get, however, a telling insight into the nature of the resulting dependencies, we investigate first a simplified case that keeps the description on the level of simple explicit equations. Analytical Solution for the Simplified Case. The simplifications assume that all species involved are fully ionized and that the effects of H+/OH- are neglected. In such a case, most equations simplify considerably, such that the velocity slope can be expressed by one explicit equation (for a detailed description of how this equation was derived, see section 2 of the Appendix). For a BGE with A1 as the single co-ion, this equation has the form
differs substantially from those obtained for single-co-ion systems. The curve passes through zero at both points of uX ) uBGE, but in the interval between uHIBA and uGA the function passes through a singularity point, near which it shows high positive or negative values of SX. This finding has significant practical importance because it predicts extraordinary zone broadening for analytes whose mobilities are close to the singularity point. It is very important for practical purposes to examine eq 4 in detail and to find the mobility uX that provides an infinite SX value. It follows directly from this equation that
SX ) ∞ S k 7 ) 0
(5)
By expressing k7 from eqs A30, A23-A25, A27, and A21 of the Appendix, we obtain after rearrangement
uX ) )
Figure 2. Calculated dependence of the velocity slope SX (in 10-3 m s-1) of analyte X on its ionic mobility uX (in 10-9 m2 V-1 s-1) for three different BGEs: (a) HIBA (20 mM HIBA + 20 mM Na+), GA (20 mM GA + 20 mM Na+); (b) HIBA + GA (10 mM HIBA + 10 mM GA + 20 mM Na+).
SX )
i (uX + uBH)(uA1 - uX) FcA1,A (u + u )2 A1
(3)
BH
where F is the Faraday constant, uX, uA1, and uBH are the mobilities (unsigned quantities) of the analyte, BGE co-ion, and counterion, respectively, and cA1,A is the concentration of A1 in the BGE. Note that the expression (uX + uBH)(uA1 - uX)/(uA1 + uBH)2 in eq 3 is equal to the active part of the EMD constant as derived under similar simplifications by Xu et al.8 It follows directly from eq 3 that in a single-co-ion BGE the function SX ) SX (uX) becomes zero at uX ) uA1. Only when uA1 > uBH, the function also reaches a local maximum at uX ) (uA1 - uBH)/2. Figure 2a shows the course of this function for a model system with either R-hydroxyisobutyric acid (HIBA) or glycolic acid (GA) as the BGE co-ion. Both curves show a monotonic course and pass through zero at uX ) uBGE. For a binary co-ion BGE, the explicit equation for SX has the form
SX )
iuX uX + uBH 1 FcA1,A uBH k7
(4)
where the quantity k7 is defined by eq A29 of the Appendix. The value of k7 is a function of only the ionic mobilities uA1, uA2, uBH, and uX and of a1, the molar fraction of A1 in the binary co-ion BGE (see eq A21 of the Appendix). Figure 2b shows the course of the function SX ) SX(uX) for a binary co-ion BGE containing a 1:1 mixture of HIBA and GA. It is seen that the obtained curve
(1 - a1)uA1uBH + a1uA2uBH + uA1uA2 a1uA1 + (1 - a1)uA2 + uBH (cA1,A/uA1 + cA2,A/uA2 + cBH,A/uBH)uA1uA2uBH cA1,AuA1 + cA2,AuA2 + cBH,AuBH
(6)
It can be shown that eq 6 implies that uA1 > uX > uA2. Using eq A4 and the simplified eq A11, we then obtain
uX )
uA1uA2uBHωAF κA
(7)
When defining the mobility usys of the system peak in a binary co-ion BGE by analogy with the classical mobility definition, we get
κA usys ) νsys i
(8)
When expressing further the velocity of this system peak according to ref 3 as
νsys )
uA1uA2uBHωAFi κA2
(9)
we get again eq 7. This demonstrates that the found value uX of analyte X with infinite velocity slope is identical to the mobility of the system peak of a binary co-ion BGE, i.e., the discontinuity region of SX is identical with the migration region of the system peak. Peak Shape Diagram of the General Case. The mathematical model described in section 1 of the Appendix opens the way to investigate also systems involving weak monohydric acids and bases and without neglect of the effects of the H+/OH- ions. Since our theoretical approach is based on the velocity slope, SX, we can make use of the peak shape diagram (PSD) that we introduced recently7 for the description of single-co-ion systems. In comparison with the PSD of a single-co-ion BGE, that of a BGE with two co-ions has a more complex shape, and its features can be characterized as follows (see Figure 3): Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
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Figure 3. Schematic view of a PSD for a binary co-ion BGE. For explanation, see text.
(i) There are two “zero” contours corresponding to SX ) 0 (the two thick lines in Figure 3). (ii) The dislocation of the SX values is in principal accordance with the course of the dependence shown in Figure 2b, i.e., for both zero contours it holds that when going left the velocity slope increases (SX > 0) and when going right the velocity slope decreases (SX < 0). (iii) Between both zero SX contours the singularity region is found. The dashed line shows which substances should theoretically have SX ) (∞ in this BGE, i.e., exhibit infinite EMD. The above-mentioned features of the PSD of a binary co-ion BGE can be demonstrated on a model example using again the HIBA-GA binary co-ion but now with Tris as the counterion at pHBGE near 4.5. Figure 4a shows the PSD of a BGE consisting of a 1:1 mixture of the HIBA and GA co-ions. The two zero SX lines split the PSD into three regions, two of which may be characterized by the predominant effect of one of the two co-ions. The left-hand-side region of the PSD resembles that of a PSD of a single-co-ion BGE with GA only (data not shown) and can thus be assigned to the predominant effect of the GA co-ion. This region comprises low-mobility samples with tailing zones. Similarly, the right-hand-side region of the PSD can be assigned to the predominant effect of the HIBA co-ion and comprises highmobility samples that form fronting zones. In the intermediate region of the PSD, the mixed effects of both BGE co-ions on the analyte zone apply. Samples lying in this region exhibit either zone fronting or zone tailing. Analytes the points of which lie near the dashed line must be expected to show extraordinary zone broadening. Figure 4b shows the appearance of the PSD for a binary coion system when one of the two co-ions is present in a minor amount (90% GA and 10% HIBA). In comparison to the 1:1 mixture (Figure 4a), a pronounced shift of the zero SX line corresponding to HIBA to lower uX values is seen; in addition, this line gets very close to the dashed line for SX ) (∞. The appearance of this PSD can be interpreted as follows: the minor amount of HIBA in the GA BGE demonstrates itself on the PSD as the insertion of a narrow disturbance region affecting the shape and width of the zones of analytes whose points it comprises. This is even more true in the case when one of the two co-ions is present at an 3400 Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
Figure 4. Calculated peak shape diagrams for BGEs composed of HIBA and GA (all containing 16 mM Tris): (a) 10 mM GA + 10 mM HIBA; (b) 18 mM GA + 2 mM HIBA. The mobilities uX are in 10-9 m2 V-1 s-1, and the indicated values of the iso-SX curves are in 10-3 m s-1.
impurity level only (e.g., at a concentration ratio of 99:1). The shape of the PSD (data not shown) is almost identical with that for a single-co-ion BGE, except for the presence of the second zero SX line surrounded by a very narrow region where the shape and width of zones of analytes concerned are disturbed/modified with respect to a pure BGE, i.e., the effect of extraordinary zone broadening is theoretically predicted to remain active even at such low concentrations of the second co-ion. EXPERIMENTAL SECTION The theoretical calculations were made using a simple program written in QBasic. The velocity slope (at current density 1000 A m2-) and other zone parameters were calculated from eq A15 using aX ) 10-8 (except where stated otherwise). The mobilities and pK values used for the calculations are shown in Table 1. An automated capillary electrophoresis instrument P/ACE 2100 system, equipped with a UV detector set to 214 nm, was used for experiments (Beckman Instruments, Fullerton, CA). Electrophoretic separations were performed in an uncoated capillary of 27 cm total length, 20 cm effective length, and 0.075 mm i.d.
Table 1. Constants of the Substances Used for the Calculations (Taken from Ref 15) substance H+ OHNa+ glycolic acid R-hydroxyisobutyric acid picrate salicylate tris(hydroxymethyl)aminomethane a
abbreviation
uXa
pKHX
GA HIBA Pic Sal Tris
362.5 202.5 51.9 42.3 33.5 31.7 35.3 29.5
3.85 3.97 0.71 2.94 8.08
In 10-9 m2 V-1 s-1.
Figure 6. Experimental CZE records of Sal and Pic in various BGEs composed of HIBA and GA (all containing 16 mM Tris). The numbers above the records indicate the concentration of HIBA (that of GA was the complement to 20 mM). The composition of the sample was 0.10 mM Pic + 0.10 mM Sal + 0.17 mM Tris. Figure 5. Calculated dependence of the velocity slope SX (in 10-3 m s-1) of Pic and Sal on the HIBA concentration in binary co-ion BGEs with cHIBA + cGA ) 20 mM and cTris ) 16 mM.
(Polymicro Technologies, Phoenix, AZ). The capillary inserted into a Beckman capillary cartridge was thermostated at 25 °C. Samples were pressure-injected by 0.5 psi for 10 s, corresponding to a 1.0-cm-long injection zone. A constant voltage of 20 kV was applied for anodic flow. Between runs, the capillary was purged with the appropriate BGE. All chemicals were of analytical reagent grade. Salicylic acid (Sal) and picric acid (Pic) were from Lachema Chemapol (Brno, Czech Republic). Tris(hydroxymethyl)aminomethan (Tris) and glycolic acid were from Sigma, and HIBA was from Fluka (Buchs, Switzerland). All BGEs used contained 0.005% hydroxypropylcellulose. The solutions were prepared in deionized water produced by trapping ions in a mixed-bed ion exchanger using an Aqua Purificator G 7749 (Miele, Gu¨tersloh, Germany). pH was measured by a 240 model pH meter from Corning (Sudbury, U.K.). RESULTS AND DISCUSSION For the experimental verification of the theory, we have selected electrolyte systems which were already discussed as examples in the Theoretical Section. Picrate (Pic) and salicylate (Sal) were selected as the model analytes, both fully ionized at pHBGE around 4.5. Figure 5 presents the calculated dependence of SX of Pic and Sal on the composition of the binary co-ion BGE. The broadening effect is predicted to apply at approximately 13 mM HIBA + 7 mM GA and at approximately 18 mM HIBA + 2 mM GA for Pic and Sal, respectively. Figure 5 also shows that
the range of BGE composition where the broadening effect applies is relatively wide, being of the order of 1 mM; note that for Sal it is narrower than for Pic because for Sal the binary co-ion BGE composition is already quite asymmetric (9:1). In Figure 6 are shown the results of the related experiments, showing the records of analyses of Pic and Sal. At 0 mM (singleco-ion BGE with GA only) and 6 mM HIBA, the records show, in accordance with theory (see Figure 5), slightly tailing zones of both analytes. At 10.5 mM HIBA, the tailing of the zone of Pic is more pronounced. The record at 13 mM HIBA confirms the predicted broadening of the Pic zone. At 15 mM HIBA, the Pic zone is still broadened but now clearly fronting, as predicted. When the HIBA concentration is further increased to 18 mM, the point of maximum broadening is reached for the Sal zone, whereas the Pic zone is already sharp and symmetrical. At 18.5 mM HIBA, the Sal zone is narrower again but still broadened with pronounced fronting. The last pattern for 20 mM HIBA (single-co-ion BGE with HIBA only) shows for both Pic and Sal sharp zones, the latter being fronting, as predicted by Figure 5. The experiments shown in Figure 6 have confirmed the existence and rules of appearance of the discussed type of broadening. Its maximum magnitude was found to be high enough to cause considerable zone broadening, but far from the infinity predicted by the theory. The explanation for this fact can be found in the differences between the ideal conditions assumed by the theory and the real experimental conditions, especially as far as the analyte concentration in the sample zone is concerned. The theoretical SX values were obtained for infinite analyte dilution (aX f 0 in eq 1, approximated in our calculations by aX ) 10-8). In actual systems we have to count with much higher aX values, lying usually in the 10-3-10-1 range, and eq A15 of the Appendix Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
3401
Figure 7. Calculated dependence of the binary molar fraction of GA in the analyte zone, aGA,AX, on that in the binary co-ion BGE, aGA,A, for various values of the molar fraction of the analyte X aX in its zone: (1) aX ) 10-8, (2) aX ) 0.01, (3) aX ) 0.1.
should be used to obtain an estimate of the actual SX. Obviously, the analyte velocities in both BGE and sample zone (νX,A and νX,AX, respectively) are mutually different finite numbers, and their difference must be finite, too. An interesting result has come from the trial to generate the plot from Figure 5 using aX values between 10-4 and 10-1 (data not shown). The calculations provided almost same SX values as shown in Figure 5, except around the singularity region, where the model did not provide any physical solution. The range of BGE composition without solution was larger with higher aX values used. These findings can be explained in terms of natural properties of electrophoretic systems. The sample zone is, in fact, a ternary co-ion electrophoretic system of anions A1--A2--X-, the local composition of which is governed by the equations given in section 1 of the Appendix. Among the input conditions, the composition of the binary co-ion BGE (a1, expressed as the molar fraction of one of the two co-ions, see eq A21 of the Appendix) and the molar fraction of the analyte in the sample zone (aX, eq A10 of the Appendix) are the most important. Obviously, we cannot find a sample zone composition that fits with the required ωA value (see eqs A11 and A12 of the Appendix) for any a1 and aX. This is a consequence of the fact that the binary co-ion system is extended by the presence of a third (analyte) anion that modifies not only the concentrations of both binary co-ions but also their ratio. This is documented by Figure 7, which shows the calculated compositions of the Pic zone in the HIBA-GA binary co-ion BGE for different aX values in the form of the dependence of the molar fraction of GA in the Pic zone on that in the binary co-ion BGE. Because both mentioned molar fractions, aGA,AX and aGA,A, are expressed as binary ones (aGA ) 1 + cGA/cHIBA), their values are between 0 and 1. As expected, for very small aX the concentrations of both GA and HIBA are almost the same in the BGE and in the analyte zone (solid line 1 obtained for aX ) 10-8). The results for aX ) 0.01 (dotted curve 2) and aX ) 0.1 (dotted curve 3) show that around the singularity region (13 mM HIBA for Pic from Figure 5 corresponds to aGA,A ) 0.35) the fraction of GA in the sample zone increases with respect to that in the BGE when approaching the singularity value from the left and leaves a considerable range of aGA,A values unassigned. On the right of 3402 Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
the singularity value, the GA fraction in the analyte zone drops to (almost) zero. The presented findings agree with the results presented recently by Beckers,13,14 whose SysCharts show unassigned regions for the case of a binary co-ion. We may conclude from Figure 7 that the existence of concentrated analyte zones is forbidden at the composition near their singularity point. The reason for this can be seen in that the extraordinary broadening sample zones coincide by their mobilities with the system zone of the binary co-ion BGE (see Theoretical Section). A (more or less pronounced) system zone is at the same time formed always when a sample zone is introduced into a binary co-ion BGE because the concentration ratio of both co-ions in the sample differs from that of the BGE and/or the presence of the analyte in the sample usually represents a disturbance to this ratio. On the basis of the above discussion, it may be expected that, when sampling higher concentrations of an analyte at conditions where the extraordinary broadening of its zone should proceed, obscure results may be obtained. This is really true, as was shown recently in ref 16, where the behavior of the analyte zone under the above-mentioned conditions was called “schizophrenic”. This seems to be the right term to describe the behavior of an analyte zone that is forced by the system to be simultaneously fronting and tailing and to migrate under nearly “forbidden” conditions. We therefore assign the type of extraordinary zone broadening investigated in this paper the term “schizophrenic zone broadening” or “schizophrenic electromigration dispersion” (SEMD). It may be expected that, under conditions where a strong system zone is formed (that can be originated by selecting a suitable composition of the sample, especially as far as the contents of both BGE co-ions is concerned), its comigration with the zone (of the same mobility) of the analyte exhibiting SEMD will distort the patterns even more. Figure 8 shows the results for a series of experiments similar to those in Figure 6 with the only difference that the sample contained also a considerable amount of GA, which induced the formation of a strong system zone. As is seen from the records in BGEs with 13 and 18 mM HIBA that correspond to maximum SEMD of Pic and Sal, respectively, these broadened peaks show some additional distortion induced by interference with the comigrating system zone. This is a proof that the “mobility” of the system zone equals the mobility of the analyte zone exhibiting SEMD. The record of the BGE with 15 mM HIBA shows that the system zone migrates between the Sal and Pic zones, slightly broadening the former and distorting strongly the latter. This is in accordance with the theory3 predicting that a sample with an excess of GA should induce the formation of a fronting system zone with sharp rear boundary. Finally, we show experiments related to the situation where one of the BGE co-ions is present at an impurity level only. To keep the ratio of both co-ions in the BGE constant (cHIBA/cGA ) 99, 19.8 mM HIBA + 0.2 mM GA), the pHBGE (controlled by the concentration of Tris) was selected as the variable to find the point of maximum SEMD of Sal. The result of calculation in Figure 9 shows that this point is reached at 15.33 mM Tris. It can be also seen that the range of theoretically predicted SEMD of Sal is very (15) Pospı´chal, J.; Gebauer, P.; Bocˇek, P. Chem. Rev. 1989, 89, 419-430. (16) Gebauer, P.; Desiderio, C.; Fanali, S.; Bocˇek, P. Electrophoresis 1998, 19, 701-706.
Figure 8. Experimental CZE records of Sal and Pic in various BGEs composed of HIBA and GA (all containing 16 mM Tris). The numbers above the records indicate the concentration of HIBA (that of GA was the complement to 20 mM). The composition of the sample was 0.10 mM Pic + 0.10 mM Sal + 0.17 mM Tris + 2 mM GA.
10a), the record of CZE of Sal and Pic in the 99:1 BGE (Figure 10b) does not show any difference as far as the peak width is concerned. The same results were obtained in BGEs with the Tris contents being slightly higher or lower (by one or several tenths of millimolar). This indicates that, at concentrations of one of the co-ions of a binary co-ion BGE comparable with that of the sample, the SEMD effect may become negligible. This conclusion has, however, to be applied with care as is demonstrated by Figure 10c. Here a sample comprising besides Pic and Sal also an excess of GA was analyzed in the 99:1 BGE. The formed system zone (owing to the low GA concentration in the BGE resembling already more a normal zone of GA) strongly interfered with the major sample component, this resulting in an extreme broadening of the Sal zone as well as in sharpening of the zone of Pic. Figure 9. Calculated dependence of the velocity slope SX (in 10-3 m s-1) of Sal on the concentration of Tris in the binary co-ion BGE containing 19.8 mM HIBA + 0.2 mM GA.
Figure 10. Experimental CZE records of Sal and Pic in various BGEs composed of HIBA and GA (all containing 15.33 mM Tris): (a) 20 mM HIBA; (b,c) 19.8 mM HIBA + 0.2 mM GA. The composition of the sample was (a,b) 0.10 mM Pic + 0.10 mM Sal + 0.17 mM Tris or (c) 0.10 mM Pic + 0.10 mM Sal + 0.17 mM Tris + 2 mM GA.
narrow. Figure 10 presents the related experiments. It can be seen that, with respect to the BGE with 20 mM HIBA only (Figure
CONCLUSIONS It is an inherent property of BGEs with two co-ions (binary co-ion BGEs) in CZE that (i) they form a system zone and (ii) analyte zones migrating close to this system zone show extraordinary broadening. Both these effects, which are part of the EMD characteristics of any binary co-ion BGE, can be explained and predicted qualitatively and quantitatively using a simple theoretical approach. This approach is based on application of a movingboundary model on a low-concentration analyte zone and utilizes the concept of velocity slope (SX, eq 1) to get a complete and telling description of the EMD characteristics of a binary co-ion BGE in the form of the PSD. For a simplified model comprising strong electrolytes, only SX of an analyte X can be expressed explicitly (eq 4). It can be shown that the dependence SX ) SX(aX) involves a singularity point (SX ) (∞) and that the uX value of this point is the same (eqs 7-9) as the apparent mobility of the system peak of the given binary co-ion BGE. The PSD of a binary co-ion BGE in the general case (Figures 3 and 4) shows three regions delimited by two zero SX contours. Both outer regions comprise analyte zones whose EMD properties are governed predominantly by the BGE co-ion of closer mobility. This is not the case of the central region, where the effect of both co-ions strongly superimposes and where the curve characterizing Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
3403
the course of the SX singularity also lies. As in single-co-ion BGEs, the PSD makes it possible to read out the SX value of any analyte of known uX and pK and thus to predict an estimate of its peak shape and width. For binary co-ion BGEs, moreover, the analytes lying near the singularity curve can be recognized. Such analytes are predicted to show strong fronting and at the same time strong tailing, this resulting in extraordinary broad zones. Due to the character of this broadening, it can be called schizophrenic EMD. The investigation of binary co-ion BGEs of various ratios of both co-ions has shown that even if one co-ion is present in a minor amount, the system keeps its EMD characteristics, including SEMD of some analyte zones. The smaller the contents of the second co-ion, the more the BGE and its PSD resemble a single-co-ion system. If the second co-ion is present at trace levels (1% and less) only, the central region (including that of the singularity) of the PSD becomes very narrow and can be described in terms of disturbance in the PSD of a single-co-ion BGE. Model experiments confirm the coincidence of the theoretical predictions of SEMD with the conditions of its actual appearance. The extent of peak deterioration depends also on the magnitude of the comigrating system peak. If the second co-ion is present at a trace level only, the SEMD effect may become too slight to be experimentally detected. ACKNOWLEDGMENT This work was supported by the Grant Agency of the Czech Republic (Grant No. 203/96/0124), the Grant Agency of the Academy of Sciences of the Czech Republic (Grant No. A4031703), and the Ministry of Education of the Czech Republic (MSˇ MT, Grant No. VS 96021). APPENDIX 1. Procedure of Calculation of the Parameters of a Sample Zone of a Given Concentration Level in a BGE Comprising Two Co-Ions. For the model system shown in Figure 1, the following set of moving-boundary equations involving components A1, A2, B, and X can be written:
j X,AX jcA1,Au j A1,A jcA1,AXu j A1,AX u ) (cj - jcA1,AX) κA κAX κAX A1,A
(A1)
jcA2,Au j X,AX j A2,A jcA2,AXu j A2,AX u ) (cj - jcA2,AX) κA κAX κAX A2,A
(A2)
jcB,Au j X,AX j B,A jcB,AXu j B,AX u ) (cj - jcB,A) κA κAX κAX B,AX
(A3)
where jci,j is the total (analytical) concentration of substance i in zone j, ci,j is the concentration of ion i in zone j, ui is the mobility of ion i, uj i,j is the effective mobility of substance i in zone j, and κj is the specific conductivity of zone j. Removal of κA and κAX results in
jcA1,Au j A1,A u j B,AX + u j X,AX jcA1,AX ) jcB,AX jcB,Au j B,A u j A1,AX - u j X,AX jcA1,A
jcA2,Au j A2,A u j B,AX + u j X,AX jcA2,AX ) jcB,AX jcB,Au j B,A u j A2,AX - u j X,AX jcA2,A
)
Kw u cH,A OH
[
jcB,AX
)
3404 Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
Kw ) 0 (A8) cH,AX
(
)
j X,AX aX jcA1,Au j A1,A u j B,AX + u j X,AX u j A1,AX u + + jcB,Au j B,A u j A1,AX - u j X,AX uA1 uX 1 - aX
(
)
]
j X,AX aX u j B,AX j A2,A u j B,AX + u j X,AX u j A2,AX u jcA2,Au + ) jcB,Au j B,A u j A2,AX - u j X,AX uA2 uX 1 - aX uBH
( (
) )
u j X,AX u j A1,A + u j B,A u j A1,AX u j X,AX aX jcA1,A + + u j B,A u j A1,AX - u j X,AX uA1 uX 1 - aX j X,AX aX u j X,AX u j A2,A + u j B,A u j A2,AX u jcA2,A + + u j B,A u j A2,AX - u j X,AX uA2 uX 1 - aX cH,AX -
Kw (A9) cH,AX
where the molar fraction of X in the mixed zone AX is expressed as
jcX,AX jcX,AX + jcA1,AX + jcA2,AX
(A10)
The regulating function for zones A and AX can be expressed as
ωA )
κAX ) F cA1,AXuA1 + cA2,AXuA2 + cX,AXuX + cBH,AXuBH + Kw u (A5) cH,AXuH + cH,AX OH
)
where Kw is the water constant. Combination of the above equations provides
(A4)
(
(
u j X,AX u j A2,A + 1 (A7) u j A2,AX - u j X,AX u j B,A
cA1,AX + cA2,AX + cX,AX - cBH,AX - cH,AX +
with
(
)
The electroneutrality condition for zone AX can be expressed as
aX )
κA ) F cA1,AuA1 + cA2,AuA2 + cBH,AuBH + cH,AuH +
(
u j X,AX u j A1,A + 1 (A6) u j A1,AX - u j X,AX u j B,A
ωAX )
jcA1,A jcA2,A jcB,A jcH,A + + + uA1 uA2 uBH uH
(A11)
jcA1,AX jcA2,AX jcX,AX jcB,AX jcH,AX + + + + + b ) ωA uA1 uA2 uX uBH uH (A12)
uA2 uX + uBH cA2,AX ) cBH,AX(1 - a1) uBH uA2 - uX
where
uA1 - u j A1,AX uA2 - u j A2,AX jcH,AX ) jcA1,AX + jcA2,AX + uA1 uA2 uX - u j X,AX u j B,AX + jcB,AX + cH,AX (A13) jcX,AX uX uBH
and b is a correction term expressing the contribution of BH+ to the transport of H+:
b)
(
κAX jcBH,AXu j BH,AX - jcBH,Au j BH,A u j X,AXuH κA 1
)
(A14)
(1 - a1) uX uA2 + uBH (A19) cA1,A a1 uBH uA2 - uX
[
cBH,AX ) cA1,A
[
]
uA1 uX + uBH uA2 uX + uBH a1 + (1 - a1) - 1 + aX (A20) uBH uA1 - uX uBH uA2 - uX
where
a1 ) The combination of eqs 1-11 eliminates all unknown variables except cH,AX; its correct value is found by solving the above set for cH,AX by looking for such a value of ωAX that equals the ωA value of the original BGE. As soon as cH,AX is known, all other quantities (in particular uj X,AX, κAX, and the velocity νX,AX) can be obtained from the above equations. The velocity slope SX can be then calculated by approximating the derivative (see eq 1) by a finite differences ratio,
SX ≈
νX,AX - νX,A aX
(A15)
νX,AX )
iuX κAX
νX,AX )
(
)
aX κAX/F ) cA1,AX uA1 + uX + 1 - aX
(
(A21)
iuX k2 + (1 - k2)aX - aX2 FcA1,A k + k a + (k - k )a 2 7
8 X
5
6
(A22)
X
with
uX uA1 + uBH 1 - a1 uX uA2 + uBH + (A23) uBH uA1 - uX a1 uBH uA2 - uX
uA2 uX + uBH uA1 uX + uBH k2 ) a1 + (1 - a1) - 1 (A24) uBH uA1 - uX uBH uA2 - uX uA22 uX + uBH uA12 uX + uBH k3 ) a1 + (1 - a1) + uBH uBH uA1 - uX uBH uA2 - uX (A25) uA1uX uX + uBH uA2uX uX + uBH k4 ) a1 + (1 - a1) ) uBH uA1 - uX uBH uA2 - uX uX(k2 + 1) (A26) k5 )
(A16) k6 )
(i is the current density) must be expressed as a function of known variables and aX only. For this, eqs 5, 6, 7 and 9 are simplified for the present system and expressed as follows:
cA1,A cA1,A + cA2,A
Combination of eqs 16-20 provides
k1 ) where νX,AX relates to a sufficiently small aX. 2. Explicit Expression of the Velocity Slope for a BGE Comprising Two Co-Ions, Simplified to the Case of BGE Formed by Strong Electrolytes Only. This simplified case assumes the two BGE co-ions A1- and A2- and the counterion BH+ to belong to strong electrolytes, so that all expressions given in eqs 1-10 hold in simplified forms where the total concentrations and effective mobilities are replaced by the ionic ones. Because the effects of H+ and OH- are neglected as well, the electroneutrality condition simplifies, e.g., for zone A to cBH,A ) cA1,A + cA2,A. According to eq 15, to obtain an explicit formula for the velocity slope as a function of known variables, the velocity νX,AX given by (see eq 2)
]
uX uA1 + uBH 1 - a1 uX uA2 + uBH + / uBH uA1 - uX a1 uBH uA2 - uX
uA1uX uA1 + uBH 1 - a1 uA2uX uA2 + uBH + uBH uA1 - uX a1 uBH uA2 - uX
(A27)
uX2 uA1 + uBH 1 - a1 uX2 uA2 + uBH + ) k1uX (A28) uBH uA1 - uX a1 uBH uA2 - uX k7 ) k1k3 - k2k5
(A29)
k8 ) k1(k4 - k3) - k5 - k2(k6 - k5)
(A30)
The derivative of νX,AX with respect to aX is then
)
aX cA2,AX uA2 + uX + cBH,AXuBH (A17) 1 - aX uA1 uX + uBH uX uA1 + uBH cA1,AX ) cBH,AXa1 - cA1,A (A18) uBH uA1 - uX uBH uA1 - uX
dνX,AX iuX ) {(1 - k2 - 2aX)[k7 + k8aX + (k5 - k6)aX2] daX FcA1,A [k2 + (1 - k2)aX - aX2][k8 + 2(k5 - k6)aX]}/ [k7 + k8aX + (k5 - k6)aX2]2 (A31) Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
3405
By setting aX ) 0, we obtain the expression for the velocity slope (see eq 1)
SX )
iuX k7 - k2(k7 + k8) FcA1,A k2
express the velocity slope as shown by eq 4. For a1 ) 1, we have a single co-ion BGE with A1 as the only co-ion, and the equation simplifies to the form shown by eq 3.
(A32)
7
Because it can be shown that k7 + k8 ) -k7uX/k2uBH, we can finally
3406 Analytical Chemistry, Vol. 70, No. 16, August 15, 1998
Received for review March 16, 1998. Accepted May 15, 1998. AC980301F
