Anal. Chem. 1994,66, 4034-4042
Experimental and Theoretical Study of Artifactual Peak Splitting in Capillary Electrophoresis Sergey V. Ermakov,? Michael Yu. Zhukov,* Laura Capelll, and Pier Glorglo Rlghettl’ Faculty of Pharmacy and Department of Biomedical Sciences and Technologies, University of Milano, Via Celoria 2, Milano 20133, Italy
It is shown here that the interaction of sample and background electrolyte, in addition to undergoing excessive dispersion resulting in the characteristic triangular form, may lead to a sample peak splittinginto two separate,distant peaks connected by a valley of sample substance. The first peak, in which the sample is charged, moves electrophoretically, while the second peak represents the same substance, uncharged, and moves under the impact of electroosmosis. This phenomenon, when occurring, may be misunderstood and treated wrongly, e.g., the second peak may be ascribed to the presence of another substance (impurity) in the sample, while the shift of the base level between two peaks may be interpreted as a consequence of wall adsorption. The second peak may go unnoticed if the electroosmosis is weak and the experiment is terminated before it appears a t the detector. Two mathematical models, a simplified, diffusionless one and a more sophisticated one, were developed in order to explain this phenomenon. The first model allows an analytical solution, while the second needs computers for solving the equations. Both gave good coincidence between experimental data and theoretical prediction. Qualitatively, this phenomenon may be explained using the Koblrausch regulating functions, which claim that the electrolyte solution ”remembers” its initial state and keeps it constant in time. The presence of a strong electrolyte co-ion in the buffer solution is a necessary condition for fhe existence of this effect, since its penetration in the starting zone after the electric current is applied suppresses the sample’s ionization and thus its ability to escape. Sometimes in high-performance capillary electrophoresis (HPCE), unusual experimental results are misinterpreted. For instance, a shift of the base level after the passage of a sample peak or the fact that the amount of sample in the peak reaching the detector is much lower than expected is often explained a s a consequence of sample adsorption on the wall. In other cases, additional ‘false” peaks appearing in electropherograms are treated as impurities. Various artifacts are often observed when the effect of sample overloading takes place and the conductivity of electrolyte solution is no longer constant. This could occur, for example, when a rather concentrated sample is used to achieve a necessary absorbance level or when a sample is preconcentrated to increase the separation efficiency level. The phenomena described above could be explained by a mechanism which is not connected with adsorption but is t Permanent address: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya sq. 4, Moscow 125047, Russia. t Permanent address: Rostov State University, Zorge st. 5, Rostov-na-Donu 344104, Russia.
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Analytical Chemistry, Vol. 66,No. 22, November 15, 1994
0.09
7
A
a
4
0
16
12
Time (min)
B h
0.0071
8
c!
0.005+
0
0.003
2
a
-
0.001 -0001
i 0
I
I
,
, 4
,
1
1
,
1
8
,
1
1 12
I
1
16
Time (min) Figure 1. (A) Orlglnal absorbance profile for pyridine (20 mM) In 20 mM acetic buffer titrated by NaOH (pH = 5). (6) The Increased vertical resolution (scale expansion by a factor of 10) vlsuallzes the substance between the two peaks.
entirely caused by an electromigrative interactionof the sample with background electrolyte. Previously,l a situation was described in which the interaction of two sample species resulted in unusual spreading for one of them and its splitting in two peaks. In the present work we demonstrate that, under certain conditions, the electrophoretic interaction between buffer and sample may lead to a splitting of a sample peak even in the case in which there is only one substance in the sample. We observed this phenomenon when studying the migration of a weak base (pyridine) and its interaction with the wall. The original electropherogram representing the temporal UV detector signal for sample (20 mM pyridine) in 20 mM acetic buffer titrated to pH = 5 with NaOH is shown in Figure 1A. Two distant peaks joined together with a small “bridge“ of substance are observed. The connection between the two peaks is better seen in Figure 1B, where vertical scale in the electropherogram is expanded 10times. After the second (1) Ermakov, S.V.;Mazhorova, 0.S.;Zhukov, M. Yu.Electrophoresis 1992,13, 8 3 8-848.
0003-2700/94/03664034$04.50/0
0 1994 Amerlcan Chemical Society
peak, the UV signal returns to its previous value. The shape of the electropherogram and the fact that the sample contains only one substance allow us to assume that both peaks belong to the same substance, i.e. the inserted sample. In the very first experiment, which was terminated after a 6 min run, we did not see the second peak. The small shift of base level was ascribed to sample adsorption onto the capillary wall. Afterward, when the time of experiment was increased, the second peak was discovered. Experiments with different sample concentrations showed that the distribution of mass between the peaks was also different. For instance, the second peak was higher than the first one for an initial sample concentration 20mM, whereas for lOmM sample thesituation was the opposite. At 5 mM concentration, the second peak was not observed at all. This work is devoted to the investigation of the phenomena described above. Careful examination confirmed that the second peak belonged to the sample (pyridine) and that its presence could be explained solely as the result of interaction between sample and buffer. This effect has nothing in common with wall adsorption effects. For the present study, two mathematical models were constructed, one simplified, in which the effects of diffusion and the contributions of hydrogen and hydroxyl ions in buffer conductivity were neglected, and the other more elaborated, in which all these effects .were accounted for. The analytical solution for the first model and the numerical simulation for the second one gave satisfactory coincidence with experimental data. It was found that the phenomenon may be observed for concentrated samples when the sample and buffer create either of the following systems: weak base (sample) + strong base (buffer co-ion) acid (buffer counterion) or weak acid (sample) strong acid (buffer co-ion) base (buffer counterion).
+
+
+
EXPERIMENTAL SECTION The chemicals were purchased from Merck (Darmstadt, Germany), except m-nitrophenol, which was obtained from Riedel-de Haan (Seelze-Hannover, Germany). All of them were of analytical-reagent grade. The experiments with basic sample (pyridine) were performed with 20 mM acetate buffer titrated to pH = 5 with NaOH at 25 "C. Pyridine of concentrations 5, 10, and 20 mM was used as a sample. The experiments in which an acidic sample was used (mnitrophenol) were run in 20 mM Tris buffer titrated to pH = 7.5 with hydrochloric acid. Experiments were performed on theBeckmanP/ACESystem2100 (PaloAlto,CA) running under GOLD Software (Beckman). We used untreated fused silica capillaries with an inner diameter of 75 pm (Polymicro Technologies Inc., Phoenix, AZ) and total lengths of 26.2 and 47.2 cm, with a distance between the inlet point and detection point of 19.4 and 40.5 cm, respectively. The absorbance was measured at 254 nm wavelength; most of experiments were performed in the voltage-stabilized regime with V = 5 kV for the shorter capillary and V = 9 kV for the longer one. The temperature of the liquid-cooled capillary was maintained at 25 OC. The sample was injected by application of excess pressure for 2 s. Experimental UV electropherograms were processed and recalculated to terms of concentration using a calibration curve.2 Computer modeling was performed using
an IBM AT personal computer. The details of the mathematical model will be given in the Computer Simulations section.
THEORY Preliminary Considerations. For interpreting the experimental results, a mathematical model describing transport processes in a solution of electrolytes containing two monovalent bases and one acid was constructed. The results of its investigation in the nondiffusional limit are presented in the next subsection, while the results of computer simulation accounting for diffusion effects are reported in a later subsection. However, it turned out that for a qualitative explanation of the observed effect and in order to get preliminary quantitative assessments, it is not necessary to solve the full set of equations described there. Let us consider a capillary filled with a buffer solution consisting of an acid with concentration uo and a base with concentration bo. The sample plug (base) with initial concentration co is placed in the region 0 6 x 6 xowithin the capillary where the x-axis is directed along it. It is well known that the concentrations a, b, and c are connected by the Kohlrausch regulating functions w and i13-7as follows:
b c u=a+-+-, pa
pb
Q=c+b-u
(1)
pc
where p a , pb, and pc are the ionic mobilities for substances a, b, and c. The functions w and Q are conserved in time and determined by initial data at a moment t = 0. We can thus rewrite (1) as follows:
where
(3) Using (1) and (2), it is easy to get the equations relating concentrations u(x,t) and b(x,t) with concentration c(x,t) in the form U
= UO
+ e,(C
- Co(X)), b = bo 6b(c - Co(X))
(4)
where
We now write the condition for electroneutrality of thesolution, neglecting the contribution of H+ and OH- ions to the total charge:
~~
~~
(2) Ermakov, S.V.; Righetti, P. G. J . Chromarogr. A 1994, 667, 257-270. (3) Kohlrausch, F. Ann. Phys. (Leiprig) 1897, 62, 209-216. (4) Mikkers, F. E. P.; Evcraerts, F. M.; Peek, J. A. F. J . Chromatogr. 1979,168, 293-315. (5) Mikkers, F. E. P.; Evcracrts, F. M.; Verheggen, Th. P. E. M. J . Chromorogr. 1979, 169, 1-10, (6) Babskii, V. G.; Zhukov, M. Yu.; Yudovich, V. I. Mafhemotical Theory of Elecfrophoresis;Naukova Dumka: Kiev, 1983 (in Russian; English translation by Consultants Bureau: New York, 1989). (7) Mosher, R. A.;Saville, D. A.; Thormann,W. The DynamicsofElecrrophoresis; VCH: Weinheim, 1992.
AnalflicalChem;stry, Vol. 66,No. 22, November 75, 1994
4035
14
-
12
IO
0
’
4
0
2
4
6
8
10 12 14 16 18 20
Concentration (mM) Flgwe 2. pH profile In the starting zone versus sample concentration for its two initlai values of 20 and 10 mM. Dashed lines designate the cases in which the influences of hydrogen and hydroxyl ions are accounted for.
Here H i s the concentration of hydrogen ions, a,8, and y are dissociation degrees, and Ka, Kb, and Kc are dissociation constants for substances a, b, and c, respectively. Substituting (7) into (6), we get Kaa Ka+H
E . -
Hb Kb+H
+-K Hc c+H
(8)
Using (4), we determine Hfrom (8) and derive thedependence for pH of solution (pH = -loglo H) on concentration c in the region 0 6 x 6 XO, i.e., when initially CO(X) = Eo. For our particular case, a0 = 20 mM, bo = 13.32 mM, EO = 20 mM, other data are taken as8 pKa = 4.64 (acetic acid; the original value, PKa = 4.75, is corrected to account for the buffer ionic strength), pKb = 14 (NaOH), pK, = 5.25 (pyridine); p a = 4.24 X 10-8 m2/V.s, p b = 5.19 X 10-8 mZ/V.s, pc = 5.11 X m2/V-s. This function and also the case in which EO = 10 mM are plotted in Figure 2 (solid lines). The concentration bo = 13.32 mM was chosen to get pH = 5 in pure buffer solution. The dashed lines in Figure 2 represent the case in which, instead of the reduced electroneutrality equation (8), a more accurate relation was considered, accounting for the contribution of H+ and OH- ions (Kwrefers to the ionic product of water) Kaa +-=-+Kw Hb Ka+H H Kb+H
Hc Kc+H
+H
(8’)
From Figure 2, it is clearly visible that, for an initial concentration EO = 20 mM, when concentration c changes from 20 mM to cmin= 13.32 mM, the pH value is changed slightly, pH = 5.75 to 6.25. At the point Cmin, the pH value has a jump of about 6 units. A further decrease of sample concentration at the starting zone from cmin= 13.32 to 0 mM causes a gradual increase in pH from 12 to 14 (pH = 10-12 (E)Weast, R.C.,Ed. C R C Handbook of Chemistry and Physics, 67th cd.;CRC Press Inc.: Boca Raton, FL. 1986-1987; pp D159-Dl69.
4036
when H+ and OH- are accounted for). This means that the sample will be dissociated in the region 0 Q x Q xoonly when its concentration exceeds c- (c > c,,,i,,); then the dissociation degree equals y 0.5. If the concentration is less then cmin (c < cmin), the dissociation is close to zero, y = 10-8. Thus, during the electrophoreticrun, the sample will leave the region 0 6 x Q xountil the moment when its concentration becomes approximately equal to c-, after which its further escape from this region will be blocked. Hence, only a limited amount of sample equal to (EO - cmin)x& (where S is the capillary cross section) will be able to leave the starting zone. This is exactly the phenomenon, called by us the “locking effect”, observed in the previous experiment (Figure 1). The first peak corresponds to the amount of sample (EO - C,,,in)X&, while the second stands for the locked substance, whose amount is of the order of C,inx&. The observation of both peaks was possible exlusively due to the action of electroosmosis, which delivered the second peak to the detection point, since its own velocity in effect is equal to zero. The “locking effect” in the starting zone occurs due to the sharp change of pH value (ca. 6-8 units) when the sample escapes from the zone. This in turn makes the substance in the sample zone practically neutral and its motion under the impact of electric field impossible. One can easily imagine the situation, e.g., weakelectroosmosis, in which the second peak would not be visible, Le., would never reach the detector. In this case, the loss of mass will be discovered only during the processing of experimental data. It may be misinterpreted as a consequence of sample adsorption onto the capillary wall. On the contrary, when the electroosmosis is rather strong, the second peak may be ascribed to the prescence of a different substance, which is also a wrong conclusion. Note that a rather accurate estimation for the cminmay be obtained in the following way. Suppose that the base (b) in the buffer is strong and fully dissociated, fl = 1 (as in the case of NaOH). The electroneutrality equation (8) may then be rewritten as
Analytlcal Chemisby, Vol. 66,No. 22, November 15, 1994
(8”)
This equation has a solution only if the following inequality is valid: a > b. From (1) and (2), we then get c
+b-
(I
= Eo + bo - QO, c = Eo + bo - UO c 3 Eo
+ bo-
a0
,,c
+ (a - b), (9)
For our case, this gives the same estimation, c 3 13.32 mM. In the case in which E = 10 mM, cfin = 3.32 mM (Figure 2). The amount of sampleescaping from thestarting zone is equal to (EO - Cmin)X& = 6.68x&, while the amount of sample “locked” equals cminx,$3 = 3 . 3 2 ~ ~ 8The . second peak should contain a lower total mass compared to the first one. The same considerations are valid for a weak acid sample, in which the buffer is composed of a weak base titrated with a strong acid. Further on, theoretical examinations are provided only for a basic sample; however, they are analogous for acidic compounds. Solution for Diffusionless Problem. A complete pattern of sample evolution can be derived by examination of a diffusionless mathematical model for electrophoretic transport processes. The set of equations for an electrolyte solution
containing two weak monovalent bases and one weak monovalent acid has the In reality we excluded the differential equations (10) and (1 l), substituing them by algebraic relations (23), using the Kohlrausch regulating functions. The function CP introduced above characterizes the properties of buffer solution. Note that usually for the description of zone electrophoresis, instead of (20), a more simple model equation is u ~ e d : ~ . ~
ab a + -{p@bEJ = 0 at
ax
ac a + - ( p c ~ ~ E=J0 at ax
in which the influence of buffer on sample motion is described by the constant coefficient k . In more complicated models, the zone electrophoresis process is considered on the bases of an isotachophoresis model, and (25) is substituted by a set of equations:6.9 I \ Here, in addition to the notation introduced above, a, E , j , and F a r e the conductivity, the intensity of electric field, the current density, and the Faraday constant, respectively. As known, for the set (lo)-( 15), there exist two Kohlrausch regulating functions,’ which are independent of time: a
b
w=-+-+--, pa pb
c
ao/at=o
pc
(16)
Q = c +b -0, a q a t = o (17) The existence of such functions allows one to simplify significantly the problem by excluding the equations describing the evolution of buffer species, i.e., the functions a and b. Suppose that at the moment t = 0, the concentration distribution for the buffer and sample species is known: alp0
= ao(x); blpo = bo(x);
CIpo
= co(x)
(18)
Then, obviously,
sow +-+bo(x)
w(x) = pa
pb
co(x) pc
W )= co(x) + bo(x) - ao(x)
(19)
Deriving the values a and b from eqs 16 and 17, we get eqs 4 and 5, and excluding these variables from eq 13 transforms eqs 12 and 14 to the following form:
Both cases assume serious simplifications about the motion of a zone. In (25), it is supposed that the buffer affects the motion of the sample, while the influence of sample on the buffer motion is neglected. In case of eq 26 as well as in eq 25, the effect of buffer pH variations is completely ignored. The use of simple models analogous to (25) and (26) does not explain the experimental data observed. Note that eq 25 may be modified by substituting the constant coefficient k with the function k ( x ) , which allows a proper description of the experimental phenomenon. The methods for solving equations similar to (20) are well known,6J0J1 but our particular case is more complex, since the function CP depends on the sample concentration c and its initial distribution co(x) [see (22) and (23)]. In addition, it is necessary to solve the equation of electroneutrality (21) or, more precisely, to derive the relationship between H and concentration c accounting for (23). In reality, the derivation of a solution for a special initial distribution C O ( X ) determined by relations (3) may be realized on the basis of mass balance equations for a given velocity depending on the sample concentration. In our particular case, the equation for the density of a mass flux i has the form
The velocity of the substance in the zone is determined as follows. Rewrite eq 20 in the form
+
u(co(x),c)ac = 0 at ax where u is the velocity of sample substance. Comparing (28) and (20), we get
(9) Moore, G. T.J . Chromatogr. 1975, 106, 1-16. (10) Lax. P. D.Commun. Pure Appl. Math. 1957, 10, 531-566. (1 1) Rozhdestvenskii, B. L.; Yanenko, N . N. Systems of Quasilinear Equations;
Nauka: Moscow, 1978 (in Russian).
Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
4037
Accounting for (27) and (21)-(23), this gives
L
Cmln
c,
z =Fw ($) / j -z/t
Flguro 3. (A) Sample velocity profiles v(Ig,c) In the starting zone versus concentration c. (E)The concentration in the starting zone versus new variable z = x / t = Fv(Ig,c)/).
aH ac
The relations (20)-(23) correctly describe the evolution of concentration as long as there are no discontinuities in the concentration profile. Note that such discontinuities already exist at the very first moment, t = 0 [see (3)]. In this case, it is necessary to add the mass balance relations at the points of discontinuities to the set of eqs 20-23. Let us suppose that the discontinuity point moves according to the equation x = x(t) with velocity D = dx/dt. The mass transport through the discontinuity is then equal to the product of its velocity and the concentration difference, D(cr - ci). The concentration CI is that to the left of the discontinuity, while c, is to the right. This mass flux should be compensated with the flux difference created by the electric field, i.e., i, - il. As a result, we have the following equation: D(co,c,,cI)(c,- cl) = i(co,c,) - i(co,cl)
(33)
or
In the case in which either cr = 0 or CI= 0, the expression (34) is simplified:
D(c,,O,c) = D(c,,c,O) = If the discontinuity is not moving ( D = 0), then (33) is transforming to a condition stating the equality of mass fluxes moving through a discontinuity: i(co(x - O),C,) = i(co(x + O),C,)
(36)
In fact, the evolution pattern can be found using only algebraic formulas (30)-(36), although writing the explicit expression for a solution is rather difficult, since it requires the solution of nonlinear eq 21 and the use of a function defined implicitly. For calculations, let us take the governing parameters corresponding to the experiment shown in Figure 1 . Their numerical values are specified in the preceding section, Preliminary Considerations. For these parameters, the velocity profile V(CO(X>,C) [see (30)] in the region 0 S x S XO, Le., for CO(X)= Eo, is depicted in Figure 3A. This profile determines the motion of the left (rear) boundary of the zone in the region 0 S x < XO. The substance with concentration 0 S c < cmin = 13.32 is almost stationary, while for concentrations Cmin < c 6 Eo, the substance has a finitevelocity. The transition from the stationary state to the state of motion is localized in the very small region around c = cmin. The expression relating 4038
'
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Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
the concentration c and the variables x and t is easily derived if one introduces the new variable z = x/t and supposes the following condition: z = u(c0,ca(z)), z = x/t
(37)
where c,(z) is the concentration profile at the left (rear) boundary in the zone 0 6 x S XO. Unfortunately, it is impossible to write the explicit expression for c,(z). However, the profile c&) may be depicted using Figure 3A, by redrawing it and changing the direction of the coordinate system axes (Figure 3B). It is clear from this figure that the concentration changes sharply at the left (rear) boundary in the region 0 6 x S XO. The substance with concentration c > cmin= 13.32 mM is moving out from the zone, while that with concentrations c < Cmin = 13.32 mM rests in the zone. Let us describe the evolution of the concentration profile for the sequence of time moments. Figure 4A shows the initial concentrationdistribution. Immediatelyafter theapplication of electric current, the concentration profile transforms into that depicted in Figure 4B. The specific features of the profile are characterized as follows. The left boundary of the region with concentration interval 0 < c S Cmin = 13.32 (thevery left part of the start zone) is almost stationary (see also Figure 3B). The point with concentration c = cminuniformly moves according to the law x = xm(t) with velocity U(E0,cmin). The right boundary of this region (between x, and x,) has concentration ca(x/t) and moves uniformly as x = x,(t) with velocity u(E0,Eo). In the region 0 C x Q x,, the concentration profile is determined by the implicit function (37). In the region x,(t) C x C XO,c = EO. On the discontinuity, the concentration drops down from c = Eo to c = cz,given by (36), and the right boundary in the region with concentration c = c, moves at a constant speed D(O,c,,O) according to the rule x = x,(t). The values described above are determined from formulas (35), (36), and (37):
X,
= u(Eo,Eo)t
x, = xo + D(O,c,,O)t
i(Eo,Eo) = i(O,c,), c, = 4.6 mM
A
t =o
(40)
The concentration profile shown in Figure 4B exists until the moment t = tintl, when theline xr(t) reaches the line xo(Figure 4C): X
The next step in the evolution of time interval tintl < t < tint, is shown in Figure 4D. The concentration profile Ca(Z) passes through the stationary discontinuity (shock), transforming into the profile Ea(.?). This profile as well as profile ca(z) is determined by relations analogous to (37): I, a=* , I*
I
S
T
t =t
C tnt.
.? = u(0,Ea(Z)), .? =
I
X
O
i.
- XI
= XO - u ( O , C , ) ~ ~ ~(42) ~,
XI
The concentration to the left from the shock x = xo (Le., at a point x = xo- 0) is equal to ca(xo/t); it will decrease in time up to the value Cmin. The concentration to the right from the shock x = xo + 0 is equal to ca[(xo - x*)/t]; it also decreases in time. The rightmost point in the profile moves with constant velocity u(O,c,) according to the law x r ( t ) : 2, = xo
I
x
7
+ u(o,c,)(t
- tint,)
(43)
At the moment tintlr the concentration at the point x = xo 0 will be cmin. It is easy to find that
a=#
The sample concentration at the zone 0 < x G xo (except for a very small region close to x = 0) will become Cmin = 13.32 mM. The zone that existed to the right from the discontinuity separates, leaving a small “bridge” with concentration c = Cb, which is determined by relation (36): @O?q,,in)
C b
C“dn
__----
L-
P I 5,
I .
,I
c=c. I .
t =t
,me.
G
= i(o,cb)
(45)
The moment t = tint, is presented in Figure 4E,while the following moment is shown in Figure 4F. The left boundary of the first (as it appears in the electropherogram) peak with concentration E,(?) is moving according to the law
xb = XI) + u(o,cb)(t - lint,)
(46)
Strictly speaking, the bridge xo G x G Xb(t) does not have a constant concentration, and its existence is caused by the presence of a small region in the vicinity of x = 0, where the concentration changes very quickly from 0 to cmin.In reality, it is a solution analogous to Ea(.?). Such concentration distribution will exist until the moment t = tint3,at which the boundary X , ( t ) reaches the boundary x,(t) (see Figure 4G): X
The value cz is determined from the following equation [see (36) and (27)]
(47)
Further evolution is analogous to that of usual zone electrophoresis. The first zone becomes triangular, and the base of the triangle spreads in time proportionally to O ( d ) , while its height decreases as O ( l / d ) . The front boundary moves according to the following law, determined Analytical Chemistty, Vol. 66,No. 22, November 15, 1994
4089
from the Cauchy problem (see Figure 4H): f
P
E
- x * ) / t ) ] + C,[(x - x * ) / t ]
10-
1
------------
8
Experiment Analytical solution Computer simulation
v
Computer Simulations. Computer simulations were based on a mathematical model described earlier.'J2 However, for the sake of clarity, we present here the governing equations (49)-(55) with some comments: (paaE- Um)a) = 0
(49)
U
I
1
I
l
l
I
I
'
1-
16
11
6
1
C
'r
11
6
16
Time (min)
-+-=-+Kaa Kw Ka+H
H
Hb Kb+H
Hc Kc+H
+H
(54)
Figure 5. Comparison of experlmentai electropherograms(bold lines) and profiles calculated using diffusionless model (dashed lines) and cowuter simulation results (thin solid lines) for different sample concentrations: (A) 20 mM, (6) 10 mM, (C) 5 mM.
written as (57), agreed well with experimental data: where DE(subscript [ here and below means one of chemical species a, b, c, H+,OH-)are the diffusion coefficients and U, is the velocity of electroosmotic flow. In this model the effects of diffusion are accounted for in transport equations (49)-(51) and in the equation for the current density (53), where the diffusioncurrent is added to theconductivity current. The contribution of hydrogen and hydroxyl ions is also included in total current and in the equation of electroneutrality (54). The diffusion coefficients are expressed according to the Einstein equation:
Here R = 8.314J/mol.K, the universal gas constant, and T is the absolute temperature. Unlike in previous works,1J2in the current simulations we accounted for the influence of buffer ionic strength Zon the equilibrium constants. For calculating the average activity coefficients x , we used Davies's formula,' since, as it was found,14 this formula, which in our case is (12) Ermakov, S.v.; Bello, M. S.;Righetti, P. G. J . Chromalogr. A 19% 661, 265-278. (13) Buttler, J. N. IonicEquilibrium; Addison-Wesley: New York, 1964;pp457479. (14) Giaffreda, E.; Tonani, C.; Righetti, P. G.J. Chromatogr. 1993,630,313-327.
4040
Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
log,,
x = -A
di +di
--
(1
0.2,)
(57)
Parameter A is equal to 0.509. The introduction of activity coefficients finally gave the correction for equilibrium constants. Further on, both for analytical calculations and for computer simulations, we used corrected values. The numerical method15 developed and used to solve the set of equations has already been checked for simulation of isotachophoresis1J5and zone electrophoresis12and has showed its advantages compared to the other methods.
RESULTS AND DISCUSSION The first series of experiments was performed with pyridine as a sample in acetic buffer using the shorter capillary (26.2 cm). The results of experiments are depicted by bold lines in Figure 5 for 20, 10,and 5 mM sample concentration. The first two electropherograms(Figure 5Aand B) have twoclearly visible peaks, while for mM (Figure 5c), only one peak is observed. The shapes of the peaks and the distribution of sample mass between them change when the sample concentration is varied. The lower the concentration, the lower the amount of substance at the second peak. Evidently, at 5 (15) Ermakov, S.V.;Mazhorova, 0.S.;Po-pov, Yu.P.Informarica 1992.3, 173197.
mM, the second peak is absent. The two existing peaks are connected by a continuous distribution of sample substance, as proved by the increased absorbance (Figure 1B); after the second peak the base level comes back to its original value. The effects of wall adsorption, which at first we supposed were responsible for this base level shift, did not play a significant role in the pyridine transport process. As shown in the Theory section, the second peak should be neutral, so the only possible way for it to migrate is for it to be transported to the detector by means of electroosmotic flow. Further on in analytical calculations and computer simulations, we took the retention time of the second peak to estimate the velocity of electroosmotic flow. In order to prove this hypothesis, we performed a run in which only acrylamide was injected as a neutral marker for measurement of the electroosmoticvelocity. It showed the same velocity value. These experimental runs were then simulated twice: once using the analytical solution obtained for the diffusionless model and a second time with the aid of computer modeling. The simulations were done for the following parameters: temperature, 25 OC; current, 1 1.1PA registered in experiments; the electroosmotic velocity, 2.27 X lo4 m/s in the case of Eo = 20 mM and 2.21 X lo4 m/s in others. Ionic mobilities, pK values, and initial concentrations taken for simulations are listed in the Preliminary Considerations section. The analytical solution, computer simulation, and experimental values are plotted in Figure 5 . The experimental electropherograms are represented by bold solid lines, analytical solutions by dashed lines, and computer simulation by thin solid lines. Theoretical results are in good qualitative and quantitative agreement with those of experiment. Both models give true patterns for the evolution process. They predict the presence of two sample peaks connected with a continuously spread substance for concentrations 20 and 10 mM and only one peak for 5 mM. The diffusionless model gives quantitatively close temporary characteristics and in some cases concentration values. It describes better the first peak, because its evolution is governed mainly by electrophoretic transport and sample interaction with the background electrolyte. The shape of the second peak, which is almost neutral, is determined by diffusion, so this model fails to describe it correctly. The results of computer modeling describe better the shapes of the peaks and concentration distribution. The better coincidence is observed for Eo = 10 mM. For EO = 20 mM, the second peak in the experiment is considerably lower than that in simulations and the difference in migration times for the first peak is greater. Among possible explanations, there could be imperfections in the mathematical model, errors in buffer and sample preparations, deviations in mobilities and equilibrium constants used in simulations compared to their actual values, and errors given by the electrophoretic unit and other devices. Obviously a similar situation may exist when an acidic sample is used instead of a basic. Here the buffer should contain a weak base titrated with a strong acid, and the buffer pH value should be close to the sample pKvalue. To confirm this assumption, we performed experiments with Tris (20 mM)HCl buffer titrated to pH = 7.5 with m-nitrophenol as a sample. The original electropherogram of an experiment for sample concentrations EO = 20 mM is depicted in Figure 6 . As previously, two separate peaks are observed, although the
0.12-
m-nitrophenol
-
1 OW
~
4
,
__
4
7-7-
9
14
--
19
Time (min) Figure 8. Experimental electropherogram for 20 mM mnlrophenol as sample In 20 mM Trls-HCI buffer (pH = 7.5).
distance between them is not as big as for pyridine. Here the order in which the peaks appear in the electropherogram is different, since now the charged substance moves in the direction opposite that of electroosmotic flow. In this particular case, on the one hand the separation process takes much more time, and on the other hand the electroosmotic velocity is much higher. So the sample species is delivered to the detection point well before the charged peak will move far from the starting zone, despite of the fact that here a longer capillary (total length 47.2 cm) was used. In other words, the intermediate phase of the separation process is observed, which we failed to see for pyridine. As has already been noted, such evolution of sample may lead to a wrong explanation of the experimental results. The shift of a base level after the first peak may be treated as a sample adsorption onto the capillary wall. Since the distance between the first and the second peak could be rather big, the experiment could be terminated before the second peak appears at the detector window. In this case, one might think that all the substance has already passed the detector. In fact, this will lead to wrong estimates for the mass of sample injected and hence to wrong quantitation for the sample. As it is seen from Figure 5A, “arrestedn peak could contain a substantial amount sample, so the error could be big. This could happen even in cases in which the duration of experiment is long enough but the electroosmosis is weak as, for example, in coated capillaries. It turns out that the major role in the described phenomena is played by the fact that the solution keeps a “memorynabout the initial position and concentration of the sample. As pointed out in theoretical works,”’ this is caused by the existence of Kohlrausch regulating functions slowly changing in time. A strong buffer co-ion penetrates the starting zone to replace the sample substance leaving it after application of electric field. Co-ion accumulation at the starting zone changes dramatically the pH value and suppresses the sample ionization, blocking its further motion out of the zone. To estimate the probability of this effect for a particular set of substances, it is sufficient to construct a graph giving the dependence of the pH value in the starting zone on sample concentration, as was done for pyridine (Figure 2). In cases in which the pH Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
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value changes sharply (4 or more units) at some sample concentration values, there could be a threshold value below which the concentration in the starting zone does not drop. All the sample mass below this threshold value will be locked in the injected zone via a protonation (for weak acids) or deprotonation (for weak bases) process generated by a strong electrolyte co-ion sweeping the zone. The sample escape is possible due to diffusion effects, but it takes much more time compared to electromigration processes. One could expect that at low field intensities (voltages), when the characteristic time of diffusion is comparable with that of electrophoretic migration, the second peak could be dissipated by diffusion. The characteristic time of diffusion processes, i, is usually estimated as i = L2/Dc, where Lis a characteristic length and D, is a sample diffusion coefficient. In our case, L N 2 mm, which is roughly equal to one-half of the initial sample zone length, and Dc = 1.3 X m*/s for pyridine, so i N 3.1 X l o 3 s = 51.7 min. We performed a series of experiments with different voltages during the run in order to verify this assumption. The sample concentration was 20 mM, and the voltages during the six consecutive runs were 15, 10, 5 , 3 , 2, and 1 kV. Other conditions were similar to those specified previously for the first series of experiments. The electropherograms of these experiments are plotted in Figure 7 . In order to facilitate their comparison, we used a reduced time scale. We left the original time coordinates for the experiment with 15 kV; for the others, the time coordinates were divided by the factor k, = 15/V, where Vis the voltage of a given run. Thus, for the experiment with 2 kV, the second peak appeared at a reduced time of approximately 4 min, but the real time was 4 X 15/2 = 4 X 7.5 = 30 min. For the experiment run at 1 kV, one could expect the second peak at a time -60 min. But according to the above estimations, this time exceeds the characteristic diffusion time. The experiment proved our hypothesis; in fact, in this experiment the second peak was not observed (Figure 7). In everyday practice, it is important to know how to avoid the peak splitting. For monovalent substances, it is sufficient to make the estimations presented above in the Preliminary Considerations section. As a general guideline for other substances (ampholytes, proteins, etc.), we suggest preparation of buffers with pH values distant 1 pH unit or more from isoelectric points or characteristic pK values. In cases when it is impossible or undesirable to do so, one should increase the molarity or ionic strength of the buffer. CONCLUSIONS Electrophoretic phenomena leading to a splitting of a single substance peak in two peaks separated by a long valley of sample substance were experimentally discovered and theoretically explained. It was shown that the mechanism responsible for the zonesplitting is connected with the existence of Kohlrausch regulating functions and electrolyte “memory”, which “remembers” the state of the electrolyte system existing
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Analytical Chemistry, Vol. 66,No. 22, November 15, 1994
0.06--I I
1
:I’
0.057
s 8 5 e
0.03
1 kV
0.02
25: 0.01 0.00
-0.01
>u L 15kV
----
0
1
2
3
4
5
6
Reduced time (min) Figure 7. Experimental electropherograms for different operating voltages. Injected sample, pyridine at 20 mM concentratlon. The reduced time scale is used. For a partlcular voltage V, in order to recalculate the reduced scale to a normal one, lt is sufficient to multiply the time by a factor k, = 15/V. For explanations, see text.
before the electric current was applied. The origin of this phenomenon is possible when the sample overloading takes place. The other necessary condition is an appropriate buffer composition, which for a weakly basic sample involves a weak acid titrated by a strong base, with the opposite situation for a weakly acidic sample. Two mathematical models, a simplified diffusionless system and a more complicated one, were tried to examine the phenomenon, and both of them were successful in describing the basic features of sample evolution. Good qualitative and quantitative agreement with experimental data was obtained. The diffusionless model allowed the analytical solution for the problem, while the solution for the more complicated one demanded the use of numerical methods. This phenomenon, if not understood, may lead to a wrong interpretation of experimental results, ascribing the existence of two peaks to the existence of two different substances in a sample, while the shift of the base level may be treated as a consequence of wall adsorption. As demonstrated, both of these features are explained solely by electrophoretic interaction of sample species with the background electrolyte. ACKNOWLEDGMENT Supported in part by a grant from CNR, Comitato Chimica Fine (Rome, Italy), by Agenzia Spaziale Italiana (ASI, Rome, Italy), and by the European Community (Human Genome Analysis, N. GENE-CT93-0018). L. Capelli is the winner of a fellowship from the CNR. S. Ermakov thanks the Ministry of Foreign Affairs of Italy for a fellowship enabling him to carry over this project at the University of Milano. Received for review May 24, 1994. Accepted August 1, 1994.” Abstract published in Aduance ACS Abstracts, September 15, 1994.
