83 8

S. V. Ermakov, 0. S. Mazhorova and M. Y . Zhukov

Sergey V. Ermakorl Olga S. Mazhorova' Michael Y. Zhukov2 'Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow *Rostov State University, Rostov-Don

Elecrrophoresis 1992, 13. 838-848

Computer simulation of transient states in capillary zone electrophoresis and isotachophoresis Transient states in the evolution of electrophoretic systems comprising aqueous solutions of weak monovalent acids and bases are simulated. The mathematical model is based on the system of nonstationary partial differential equations, expressing the mass and charge conservation laws while assuming local chemical equilibrium. It was implemented using a high resolution finite-difference algorithm, which correctly predicted the behavior of the concentration, pH and conductivity fields at low computational expense. Both the regular and the irregular modes of separation in capillary zone electrophoresis and isotachophoresis are considered. It is shown that the results of separation, particularly zone order, strongly depend on pH distribution. Simulation data as well as simple analytical assessments may help to predict and correctly interpret the experimental results.

1 Introduction In the last decade computer simulations have become an integral part of theoretical investigations in all branches of science.This is relevant to the study of electrophoretic separation when attempting to predict the time evolution of electrolytes. The appearance of high performance computers gave the opportunity of solving many problems, which were regarded earlier as too complicated. This stimulated the creation and the development of mathematical models describing electrophoresis,more complicated than the first, formulated by Kohlrausch [I]. Currently there are many models, regarding various aspects of electrophoresis, of which we will mention only a few. One of the most general models [2] is characterized by the unified approach in describing the basic electrophoretic techniques for substances charged by means of a dissociation mechanism. It is formulated in terms of continuous media physics and accounts for most of the effects governing the evolution of electrophoretic systems. Simultaneously, another report [3] appeared, which was followed by a series of mathematical models [4, 51, developing the initial model. On the basis of these models, many phenomena accompanying the sample transport, by means of electromigration and diffusion in the main electrophoretic techniques, were simulated [6]. Note that, when speaking of particular separation techniques, capillary zone electrophoresis (CZE) and isotachophoresis (ITP) are currently simulated extensively due to their increasing popularity. Good reproducibility of experimental results, the absence or very small influence of bulk solvent motion, the effective automatic detection and computerized data processing on one hand and relatively simple mathematical models, required for description, on the other make CZE ideal for comparison of theory and experiment. If the process is carried out in a thin capillary with open ends, electrophoretic migration could be studied with minimal interferences caused by thermal convection and electroosmosis. It also means that a one-dimensional treatment ofthe problem often turns out to be sufficient for considering the electrophoretic transport. Correspondence: Dr. S . V . Ermakov, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya sq. 4, Moscow, 125047, Russia

Abbreviations: CZE. capillary zone electrophoresis; ITP, isotachophoresis; Tris, trishydroxymethylaminomethane

0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1992

Theoretical studies concerning CZE and ITP can presumably be divided into two groups.The first one is devoted to the investigation of fundamental features of electrophoretic transport, the influence of initial and boundary conditions [7-191. Since most of these reports do not state the specific capillary geometry, they may be applied to column, free flow and other apparatus. Usually only isothermal conditions and the absence of solvent bulk flow are assumed. The steady state or time evolution of concentration fields, electric conductivity and pH distribution under the impact of various transport mechanisms are studied here. Part of these works examine the steady state in ITP [7,8], considering electromigration as the only means of transport. ITP separation dynamics [9] and the electromigration dispersion of a singular zone in zone electrophoresis 12,101are described in diffusionless approximation. With the help of computers, some studies [ 11-13] simulate the behavior of fully ionized electrolytes.The effect of various initial conditions on ITP separation, when the leading or terminating electrolyte includes the impurity ions, are examined in [12-131. In [13],togetherwith ITP and CZE simulation, special attention is paid to optimization of numerical algorithms for reducing the computational efforts. However, the ITP of weak electrolytes exhibits a larger variety of situations than the strong electrolytes. The ITP separation of fully ioni7ed substances usually occurs according to their ionic mobilities, while this is not the case for weak electrolytes. The order of so-called pure zones and the formation of steady state mixed zones is predicted theoretically in 114-151. Computer simulations 1161proved the existence of an ITP mixed zone for weak bases with intersecting mobility curves. The separability problem is closely connected to the behavior of ITP boundaries between weak electrolytes, which has its specific features. It was the subject of numerical modeling in [17], where three possible migration modes of ITP boundaries were regarded. They arise when electrolytes of different strengths and mobilities have various mutual arrangements referring to electrodes. Boundary evolution in other electrophoretic techniques is simulated in [3, 191. The interaction of ampholytes with ITP boundaries in cases in which their isoelectric point (pl) lies within a pH drop of the leader-terminator interface is reported in [19]. The second group of works accounts for specific features inherent to capillary electrophoresis. They consider the impact of electromigration dispersion, Joule heating, solvent convective flow, and some other factors on resolution (see

Computer simulation of transient states

Electrophoresis 1992, 13, 838-848

for example [20-26]).These studies are aimed mainly at the optimization of experimental procedures and apparatus, such as regime parameters [20-221, detection [23], sample broadening due to thermal disturbances [24],as well as electroosmotic and Poiseuille flow interaction [25].A complex investigation of main dispersive mechanisms is carried out in [26]. A more comprehensive review of papers dealing with ITP and CZE may be found in [27,28].

In this paper the numerical simulation of different separation modes in capillary ITP and CZE is based on the unified approach, stated in a general mathematical model [2,3].The results of computer simulation confirmed the predictions of possible migration modes, as formulated in [14, 151 for ITP. It is demonstrated that they also hold true in case of CZE. The high precision of numerical results was proved by their comparison with analytical data. For CZE the optimal pH value in buffer needs to be found to optimize the separation process. It is shown that the choice of selectivity as a characteristic of resolution for monovalent anionic and cationic species results in four cases, depending on the relationship between their ionic mobilities and dissociation constants. In two cases, maximum selectivity is achieved at high pH values; in the other two, at low pH. An improper choice of pH value, when the concentration of sample species is comparable with those of buffer, may severely distort the separating zones as well as the usual electromigration dispersion.

considered to be in mechanical equilibrium. Therefore, with respect to a moving system, the sample transport is the sum of electrophoretic migration and diffusion. Given these assumptions, one can further assume that all parameters are constant over any capillary cross section; the problem may therefore be reduced to a one-dimensional simulation. The transport processes in capillary electrophoresis are characterized by at least three different time scales, corresponding to physical mechanisms running at different rates. The fastest transport mechanism is connected with association/ dissociation reactions, in our case being:

* A;

t H + , n = 1,2. . N

(1)

HfBm+BmtH+, m=1,2..M

(2)

H 2 0 + H + +OH-

(3 1

HA,

where HA, and B, are weak acids and bases; A-,,, anions; H'B,, cations; H,O, water; and H', OH-, hydrogen and hydroxyl ions. Since the rate of reactions is several orders higher than that of electromigration and diffusion transport, we will assume the existence of local chemical equilibrium at every time moment. This means that the concentration of chemical species is defined by the following equations: (4)

2 Theory The purpose of the present research is the simulation of a separation process in free fluid capillary zone electrophoresis (CZE) and isotachophoresis (ITP). It will provide information on separation patterns in the form of concentration profiles, conductivity and pH distribution along a capillary. We shall simulate a transition phase of the separation process up to the moment in which a steady state (if it exists as in ITP) is reached. For situations without a steady-state behavior, the simulation is confined to the moment in which separation occurs or further evolution becomes clear. Let us consider a thin capillary filled with an aqueous solution of N monovalent acids and Mmonovalent bases. Usually in free fluid electrophoresis the sample transport is a combination of convection (when it has traveled due to bulk fluid motion), electrophoretic migration under the impact of the electric field, and diffusion. We shall assume that the capillary is thin enough to be thermally stabilized by the cooling system, so the effects of thermal convection may be neglected. This also means that the transport coefficients of electrolytes. (e.g. ion mobilities, diffusion coefficients, solvent viscosity and some others) are believed to be constant. On the other hand we assume the inner diameter of the capillary to be much greater than the Debye radius. In this case, for capillaries with open ends, as used in many experimental installations, the electroosmotic flow has a pistonlike velocity profile. It has no influence on the separation dynamics and the sample distortion. Actually, its effect is reduced to a uniform drift of the separation pattern because, due to the continuity principle, the electroosmotic velocity V,, is constant along the capillary. By introducing a coordinate system that moves with the same velocity with respect to the capillary, we eliminate the bulk flow; a solution is then

839

[Bm I W+I [H+Brn1

= K k , m = 1,2,..M

[H+][OH-] = K O . [ H 2 0 ]fK i

=K ,

(6)

where square brackets denote the molar concentration; K",, x",, and KOare equilibrium constants, and K, is the ionic product of water. It is more convenient to describe the transport processes in terms of analytical concentrations for acids and bases, which are expressed as a n = [HA,

1 + [A;]

n = l,Z,.. . N

bm = [H+Bm I t [Bm I m = 1,2,. . .M

(7)

These are concentrations as can actually be observed in rapid chemical reactions. It is also suitable to introduce degrees of dissociation:

These depend only on the hydrogen concentration [H']. The behavior of a multicomponent medium with chemical reactions is described by the set of equations given in [2]. Implying the simplification stated above and neglecting the description of the short initial period, when the local chemical equilibrium is establishing in solution, it may be written in the terms of the slow variables a, and 6,

840

S. V. Errnakov, 0. S. Mazhorova and M. Y. Zhukov

f aa, + v . z;=o at

Electruphoresis 1992, 13, 838-848

n = 1,2.. N (9)

rn

=

obtained a diffusionless model. In this case the contribution of hydrogen and hydroxyl ions to the total current through the capillary (Eq. 1l), as well as their contribution to electric neutrality (Eq. 13), are usually omitted. With these assumptions, two conservation laws, known as Kohlrausch regulating functions, are valid:

1,2. .M

They are independent of time. For isotachophoresis, by using these functions and the initial distributions a,(x) and b,(x), the final stage of the separation process can be calculated easily. In the case of stepwise initial concentration profiles the diffusionless problem is reduced to the set of algebraic Eqs. (I l)-( 12), together with the Rankine-Hugoniot conditions for concentration shocks 191:

4

=F(

2 zk&,,brn

8.{ a , } t

IH'1-Z

m

z~a,a,-[OH-] n

Here D,,Db,, D,,, Do,, pan,pbm, p,, poH, fn, and zbmare diffusion coefficients, ion mobilities, and electric charges fo_r acid, base, hydrogen and hydroxyl, respectively. Vectors i J , E denote molar mass flux, current density and intensity of electric field, while p is its potential, q and F stand for total electric charge and Faraday constant. Equations (9-10) represent mass conservation for every acid and base; Eq. (1 1) describes the generalized Ohm law, accounting for diffusion current; Eq. (12) is the electric current continuity equation; Eq. (13) states the local electrical neutrality condition; and Eq. (14) states the condition for the existence of electric field potential.

As was stated earlier, to study the transport process in capillary electrophoresis, it is reasonable to confine this treatment to a one-dimensional approach. The set of Eqs. (9)-( 14) was solved numerically, using an IBM personal computer. A special finite-difference scheme (FDS) with artificial dispersion has been developed to solve the partial differential Eqs. (9) and (10) [29]. Our FDS allows a reduction in computational efforts while retaining the high accuracy of computations.

3 Results and discussion 3.1 Diffusionless approach The system of Eqs. (9)-(10) describes the sample constituent transport by means of electrophoretic migration and diffusion. Since the diffusion coefficients are rather small, we neglected diffusion as a first approach. The electrophoretic migration was therefore the only transport mechanism. Formally assigning zero to all D, in Eqs. (9)-(10) we

= { z,a,a,~)

n = 1,2,.. ..N

9 . { b m = { z r n P m b m E }m = 1 , 2 , . . . .M

(15)

where{f)=f(x(t)+O,t)-f(x(t)-0,t) is thejump offunctionf at point x(t),and u is the speed ofthe moving concentration shock (or zone boundary). Relations (15) are used instead of differential Eqs. (9)-(10).

In the study of isotachophoresis, the important questions are whether the stack of separated zones is a stable formalion, and what the conditions of stability are. For this purpose let us consider a boundary between two separated zones, containing the different monovalent anionic substances 1and 2 with corresponding values of [H+,],K",,,u",,E,, a,,(i= 1,2). Suppose that substance 1 moves in front of substance 2. The velocity of the boundary between them equais

Now suppose that due to fluctuations the 1st substance appears in the 2nd zone. Its velocity in this zone will equal:

where the number in round brackets denotes the zone number. For the boundary to be stable, the 1st substance has to accelerate and return to its (1st) zone, i.e.: 9, (2) > 9, (1)

(17)

By analogy, the 2nd substance appearing in the 1st zone must be retarded: 9,(1)

< Q2(2)

(18)

Computer simulation of transient states

Electrophoresis 1992, 13, 838-848

841

When combining (17) and (18) with (16), two analogous relations for [H+J and [H+2]are derived:

These may be replaced by one, for arbitrary [H-'1

KLalK;(P; +;I

> [H+I(P;K;-P?K'i)

(19)

This stability condition is a function of parameters pd,and K", (i= 1,2),which are related to the substances underobservation and may be considered as constant. It also includes the hydrogen ion concentration [H'], and for particular substances (i.e. for fixed pa,and K",) it is natural to deduce the stability condition depending on the [H'] value. Let us analyze it forvarious p",and K", (i= 1,2). Ifpa,>pa2and pa,K",> pd2KZ, then (19) is transformed to:

which is always satisfied because [H'] > 0. If pd,> pd2but pa,K",< pd2K",,i.e., the weaker acid 1 has the more mobile ion, then

In this case the boundary would be stable only for those H' concentrations that are lower than the definite value H,. The third variant is, in fact, equivalent to the second one with transposed subscripts, but for convenience we pick it out as a separate item. The assumption that pal< pa2and p",K", > p d 2 P Z yields:

then

Similar speculations were reported in [14] and [15]. The treatment of inequalities (20)-(25) can be extended if we analyze them from another point of view. In fact, these relationships state the condition for which the net mobility of the first substance is greater than that of the second. Being applied to a mixture of two substances, they turn out to be conditions of separation. In ITP, relationships (20)(25) have to be considered for the mixed zone, arising in the region initially occupied by the leader [14].In CZE they are to be applied to the region where the concentration profiles of both substances overlap. If, for every moment in time, the pH value in the mixed zone satisfies one of the corresponding conditions (20)-(25), substance 1 would separate first (i.e., it would move in front of substance 2). If this were not the case, then substance 2 would separate first. Since for parameters pl and K, (i = 1,2) given in (20) and (23), the separation conditions are always satisfied, these situations may be spoken of as unconditional or regular separations. In cases (21)-(22) and (24)-(25), we speak about irregular separation, assuming that the separation pattern, particularly the final zone order, depends on initial data. Conditions (20)-(25) may be generalized for the system containing the arbitrary number of sample constituent, but, unlike CZE, in ITP separations the leader and the terminator substances are also to be taken into consideration. 3.2 Numerical simulation

For this situation the solution must have a lower pH value to stabilize the boundary. If, for a given pair of parameters, paland R, (i= 1,2),the appropriate stability condition is violated, the existence of a sharp boundary between these substances would be impossible. In this case the rarefaction waves spread from the boundary of initially separated substances. The same analysis may be conducted for cationic species. We restrict ourselves by writing the stability condition for various pb,and x",,assuming that now the cathode is to the right:

then

We studied the separation process in capillary ITP and CZE for a sample consisting of two monovalent anionic or cationic substances. In the course of numerical simulation all three separation modes, corresponding to various stability conditions, (20)-(22) or (23)-(25), were considered. The input parameters included dissociation constants for all substances, their ionic mobilities, the initial concentration distribution of the substances along the capillary, the total current, and the geometric parameters of the capillary. Since we worked with capillaries with open ends, the fixed concentration of all substances was given at the capillary ends. Most computations were performed for capillaries with an inner diameter, d, of 100 pm. The length of the computational domain depended on sample volume and specifics of separation and usually did not exceed 10 cm. The sample volume was 20 or 40 nL, which corresponds to approximately 0.25 cm and 0.5 cm sample plug length in the capillary. The input data for the substances used in the simulations are summarized in Table 1.

842

S V krrnakov, 0 S Mazhorova and M Y Zhukov

Ekcrrophoresfs 1992, 13, 838-848

are depicted for four moments in time. The bottom panel corresponds to time t=O.O, the next two show the intermediate states of separation, and the upper panel exhibits t h c final steady stable pattern. The leading electrolyte is HCI, the terminating electrolyte is cacodylic acid, the sample consists of formic and acetic acids and the common counterion is Tris. The initial concentration distribution for the given separation and other variants are collected in Table 2.

Table 1. Input data (or computer simulation

4.24 2.31 7.91 4.12 5.64 3.65 2.41 36.3 20.5

4.16 6.21 -7.0 2.85 3 .IS 3.86 8.30

Acetate Cacodylate Chloride Chloroacetate Formate Lactate Tris H+ OH-

~

3.2.1 ITP simulation The first case of ITP simulation considers the regular separation, when condition (20) is satisfied and the order of separated zones is independent of the initial concentration of components. Here we imply that (20) is valid not only for sample constituents, but for all pairs of anionic substances, including leader and terminator. It is obvious that this mode of separation is preferable in common practice. The dynamics of sample separation is presented in Fig. 1,where the evolution of concentration profiles (left), as well as the pH and conductivity distribution (right) along the capillary

I

t = 163.6

-I

7.0

An excess of Tris in a solution leads to almost full dissociation of anionic substances and their behavior is similar to strong electrolytes. The pH value is approximately 8.0; the influence of H’ and OH- ions is therefore negligible.These initial conditions were specially chosen to satisfy the simplifications adopted in [9] because it made possible the comparison between our numerical solution and the analytical one [9]. Simultaneously, it also enabled us to verify the numerical code. Two types of characteristics were taken for comparison, (i) steady state parameters: concentrations and lengths of separated zones at the final stage (Fig. 1, t = 163 s) and (ii) dynamic parameters: separation time t,,,, the moment t , when the formic acid leaves the starting region (see Fig. 1, t=45.4), and moment t2,when acetic acid leaves this region. Here we designate “the start region” a capillary segment, occupied by the sample in t = 0, i.e. the space be-

,

I

9.0

7.0

t = 116.3

_I

,_ -

- - I

L

I

U

I

5.0

8.0 I--

3.0

_______________________

I l , l = i 0.0

1.o

Capillary length (cm)

2.0

1 .o

7.0

0.0

1.o Capillary length (crn)

2.0

Figure 1. ITP regular separation dynamics. Here and also in Figs. 2 , 3 , 4 , 7 on the left panels: concentration profiles (solid lines -sample species; long dash line -counter ion; short dash lines - leader and terminator). One division on vertical axis corresponds to a concentration of 0.1 mol/L. Time is in seconds. On the right panels: conductivity (dashed line) and pH (solid line) profiles. The conductivity scale is on the left border, the pH scale on the right border. They are equal for all fragments (in the 2nd and lhe 3rd fragments axes ticks are not labeled). Anode is to the left.

Computer simulation of transient states

Electrophoresis 1992, 13, 838-848

tween 2.0 a n d 2.25 cm. The results of the comparison are summarized in Table 3. Some simulation data are presented in approximate form, when the range of values are given. Unlike the analytical solution, where the boundaries between different zones are absolutely sharp, the numerical model describes the diffusion distortion of the zone boundary. It is therefore impossible to determine exactly when the species leaves the starting position. We define the moment t, as the interval in which the concentration of formic acid is reduced from 1/2 of its value in a mixed zone to

zero at point x=2 cm. The moment tl is defined in the same way for acetic acid. The separation time was defined as the moment when the concentration profiles came to a steady state. All the data compared here exhibit good agreement, which means, on the one hand, that correct simulation was given by a numerical code, and, on the other hand, that the diffusional transport had only little influence on separation dynamics. The detailed study shows that diffusion plays a significant role only within narrow regions of the substance interfaces.

Table 2. Initial conditions and regime parameters for ITP and ZE simulations

Concentration distribution (mol/L) Leader HC1-0.06 Tris-0.2

HC1-0.1 Tris-0.15

HCI-0.15 Tris-0.1

HCI-0.18 Tris-0.17

Terminator

Sample

Cacodylic acid-0.1

Formic acid-00.4

Tris-0.2

Acetic acid-0.05 Tris-0.2

Cacodylic acid-0.1

Formic acid-0.04

Tris-0.2

Chloroacetic acid-0.06 Tris-0.2

Cacodylic acid-0.1

Formic acid-0.04

Tris-0.2

Chloroacetic acid-0.06 Tris-0.2

Cacodylic acid-0.1

Formic acid-0.04

Tris-0.2

Chloroacetic acid-0.06 Tris-0.2

Concentration distribution (mol/L) Buffer

Current PA

Sample volume

5

L-’

Figure and technique

20

1 ITP

40

5

2

ITP

10

40

3

ITP

25

4

40

ITP Current I.LA

Sample volume

Sample

Figure and technique

L-9

HC1-0.15 Tris-0.1

Acetic acid-0.005 Lactic acid-0.004 Tris-0.1

10

20

5b) ZE

HCI-0.1 Tris-0.2

Acetic acid-0.005 Lactic acid-0.004 Tris-0.2

10

20

5c) ZE

HCI-0.09 Tris-0.1

Acetic acid-0.05 Lactic acid-0.04 Tris-0.1

10

20

I ZE

Table 3. Comparison of analytical [9] and numerical results Data

Analytical results Numerical results a) Formic acid b) Acetic acid

Steady parameters Concentration Zone length (rnol/L)*lO-* (cm) Fd’ Ab) F A

843

Dynamic parameters tsep S

tl S

t2

44.4 43.2-45.4

52.6 50.0-52.3

5.485

4.939

0.186

0.258

164.5

5.489

4.940

0.188

0.250

163.6

S

844

Elrctrophoresi.r 1992, 13, 838-848

S. V. Ermalwv. 0 . S . Marllorova and M . Y. Zhukov

The next three figures represent the simulation results for another sample, consisting offormic and chloroacetic acids, for which condition (20) is violated. For identification purposes, the formic acid concentration profile is marked with asterisks on the ligures. Here the result of the separation, i.e., the zone order at final state, depends on the initial concentration distribution, particularly on the leader composition. For a given pair of substances the H 8o value is positive and pH",, = 4.1. The separation dynamics will therefore be determined by the pH value in the mixed zone displaced behind the leader. The sample separation at high pH values throughout the capillary is shown in Fig. 2. It corresponds to condition (21), where p ~ 'and , P , denote the formic acid, while p'I2 and KY.'? stand for chloroacetic acid. The excess of Tris in solution provides a p H range within 7-9 in which the sample species are almost fully dissociated, and hence their net mobilities are close to the ionic mobilities. Since the formate ion is more mobile, the net mobility of formic acid is higher than that of chloroacetic acid. As a result, after separation, the formic acid zone follows the leader. Quite the opposite is observed when an excess of chloride ion is created in the leader electrolyte. This corresponds to (22), assuming that substance 1 is chloroacetic acid and substance 2 is formic acid.At low pH the dissociation of formic acid is essentially suppressed, and therefore its net mobility is lower than thilt of chloroacetic acid. Thus chloroacetic acid will directly follow the leading ion (Fig. 3, t = 125 s).It is interesting here to watch the separation dynamics. At the starting time (Fig. 3, t = 0.0), the pH value in the sample zone is higher t:han pH",, = 4.1 (this value is depicted by small circles on the figures). Hence the net mobility of formic acid is higher and, initially, it moves faster everywhere,

1 ; ; ;; .- - - - - - - - .. - - - - - - - - - - -

I

except in a small zone around the leader. When leaving the starting region, it is retarded because o f a pH drop, whereas the chloroacetic acid accelerates. In the mixed zone the pH value is lower than pH",; simultaneously, in that part of the sample which is still situated in the starting region, the pH value is higher than pH". It leads to an unusual -W- shaped profile of chloroacetic acid concentration (Fig. 3, t = 34.1 s). When the sample leaves the starting region (Fig. 3, t =56.8 s), the pH value in the mixture becomes lower, pHdo. Hence condition (22) is satisfied and, after the separation, chloroacetic acid will reach the detector first (Fig. 3, t = 125 s). If we start increasing the pH value in the leader and consequentlyin the mixed zone, the difference in net mobilities of sample species would decrease. In the case when the pH of the mixed zone is close to pH",,,this difference is close to zero and the existence of a steady mixed zone is possible. It is proved by the results of the simulation displayed in Fig. 4 (t=295). At this moment a steady state migration pattern is attained, where the formic acid zone is mixed with part of the chloroacetic acid zone. Unlike the previous examples, the conductivity profile is nonmonotonic. It has a clearly visible rectangular gap in a pure zone location. This was already observed in experiments [15] for cationic species. 3.2.2 CZE simulation As stated in Section 3.1, conditions (20)-(25) are also applicable to the treatment of CZE separations; in particular, they may be used in the prediction of zone order after separation. As in the case of ITP, three modes of CZE

T;-;---l;I6' L - _

4.0

2.0

0.0

1 .o Capillary length (cm)

2.0

i' 7.0

0.0

1 .o Capillary length (cm)

2.0

Fi,yurr 2. ITP separation at high pH, when relation (21) is satisfied. Formic acid profile is marked with asterisks. The horizontal axis for the upper fragments ( t = 397.7) is shifted. The negative sign in capillary length denotes coordinates to the left of the reference point. For further details see Fig. 1.

Elecfrophoresis 1992, 13, 838-848

Computer simulation of transient states

separations were simulated for weak acids. Only two o f them, corresponding to irregular separations, whose sample parameters satisfy either Eqs. (21) or (22), are exhibited here. As an example, let us take a mixture containing acetic and lactic acids. For this pair, pH", = 5.48 and, according to Eqs. (21)-(22), when the buffer has a pH < pH',,, lactic acid will migrate ahead as a more mobile component. On the contrary, at a pH > pH",, the acetic acid will move in front of lactic acid after the separation. The corresponding simulation results are shown in Fig. 5 : (a) the upper panel shows the initial concentration profiles, (b) the middle one represents buffer at a pH = 4.05 < pH", and (c) the lower panel corresponds to buffer at pH 8.3 > p H , . That moment in time is chosen, when the separation is nearly perfect. Here the acetic acid is plotted with bold tracing. The concentration profiles of buffer components, as well as pH and conductivity distribution, are not presented here because they are nearly constant along the capillary. As is usual for analytical CZE, the concentration of sample species is much lower than that ofbuffer; the pH value and othercharacteristics are therefore not influenced significantly.

Capillary length (cm)

845

Note that the separation time and separation distance are different for these two simulations; they are several times higher in the case of p H = 8.3 (Fig. 5c). Simultaneously, the driving force, i.e. electric field intensity, is almost equal. This phenomenon may be easily explained considering the function S = Au/O, called selectivity [30], which describes the relative difference in net mobilities and may be used as a characteristic trait of separability. In our case, when we consider the migration of two anionic substances with velocities u , and u2 it has the form: - 2 . ( A + B [H+]) 91-82 S = S([H+]&, K f ) = 0.5(9, + 9,) - C + ID [H+]

(26) where A = F , F , ( p a ,- pCL2), IB = pU",K", - pa2K",,C = K",K", (pa,+ p d 2 )and , ID =pd,K",+p",K",. We would treat Sas a function of [H'], while pa,and K", (i = 1,2) would be its parameters. Suppose for certainty that p a , > pa2;S = S([H']) is a monotone increasing or decreasing function, S(0)= 2 A/C =So and S 2 IB/D = S* when [ H i ] + 0"; it therefore rea-

Capillary length (crn)

+

-+

Figure 3. ITP separation at low pH when relation (22) is satisfied. Formic acid is plotted with marked line. The horizontal dotted line on the right panels marks pHao= 4.1. For further details see Fig. 1.

846

Ekctruphuresis 1992, 13, 838-848

S V Ermakov, 0. S. Mazhorova and M. Y . Zhukor

Capillary length (cm)

Capillary length (cm) -2.00

-1

.oo

0.00 t

=

-2.00

-1

.oo

295.4 30.0 ~

g

.

O

20.0

10.0

!

0.0

A

I

i.0

I

-U

I

30.0

20.0

I

............. .....

10.0

L

5.0

-_____ I_-

1 .o

0 .0.0 3 -

1 .o

0.0

2.0

Capillary length (cm)

Figure 4. Formation of a steady mixed zone in ITP separation. Formic acid is plotted with marked line. A nonmonotonic conductivity profile is clearly visible ( t = 295.4). For further details see Fig. 1.

rO.010

t = 0 sec

_I

-0.W5

z -

-0.OM

ches its maximum and minimum values when [H'] = 0 or [H'I- + 03. The question is: what is greater, Soor IST? (Sois always assumed to be positive). In general, when parameters pa,and Fk", ( i = 1,2) are arbitrary, four cases arise: (i)

P;K;

> &K;

and K;

> K;

(ii)

piK;

> p;K;

and K j

< K;

(iii)

p!K;

< p;K;

and ( p i I&)'

> (K;/K;)

(iv)

P X i < P;K;

and (P; IP9'

< (K; IKi 1

(27)

pH = 4.05

t = 159 sec

pH = 8.3

t = 631 sec

Figure 5. CZE separation of lactic acid and acetic acid (marked line) for two pH values of buffer: (a) initial concentration distribution; (b) buffer, pH = 4.05; (c) buffer, pH = 8.3.

Computer simulation of transient states

ElrctrophoreJis 1992, 13, 838-848

They are schematically presented in Fig. 6a where S([H'J) versus [H'] is depicted for these cases. It turns out that S* (ii),lS*(iii) I&.This means that maximum selectivity is obtained at [H'] 0 (or pH + -) in cases (ii) and (iii) and at [H+]++M(orpH+-w) in cases (i) and (iv).For the parameters given in (iii) and (iv) the curves S = S([H']) cross the O[H'] axis at [H'] = Hao,where Haois defined by formula (20). When S([H']) becomes negative it means that substance 2 is moving faster than substance 1. Returning to simulations, the results of which are presented in Fig. 5, one can easily calculate that this corresponds to case (iv) and S0=0.15,ssF= 1.49,pW0=5.48,S(pH = 4.05)/S(pH = 8.3) = 6.77, which correlates with computed data. The above calculations show that the selectivity at low pH value ([H'] -,+ m) is approximately 10 times higher than that at high pH value ([H'] 0). +

I

\

a)

847

+

+

The same analysis may be carried out for cationic separations. Function S = S([H'], p h , , P , ) is analogous to that defined by (26), but its coefficients are different: A = p b , p 2 - p b 2 P l ,[B =pb,-ph2,C = p b l P 2 + p b , P , , [D = p b ,+,ub2.Here

I / bl

Figure 6. Selectivity S versus hydrogen concentration [H'] in (i)-(iv) cases for (a) anionic and (b) cationic species.

).O

-

t = 227.2 9.0

-

0.0

-

7.0

--,

6.0

-

_

..............

-

t";----t

=

-

8.0

\

..-

-

_*

- 3.0 -

2.0

5.0

56.8

L

5 1 L

-0

I

t = 28.4

1

--

-

t = 0.0 9.0 8.0

-

- 6.0

-................. . I---

7.0 6.0

-?.O

-

5.0

-4.0

_

0 .5.0 1 -

0.0

L

_

1.o 2.0 Capillary length (crn)

Figure 7. CZE separation in buffer with low capacitywhen pHis close to pHao;pH", =5.48 is denoted by horizontal dotted line. Acetic acid is plotted with marked line. For further details see Fig. 1.

848

Electrophoresis 1992. 13, 838-848

S. V. Ermakov, 0. S. Mazhorova and M.Y. Zhukov

we are also faced with four cases, when the relation between & and S* is changing:

They are schematically plotted in Fig. 6b. As previously, we imply that ph, > p b z . Here S,(ii), I S,(iii) I S*. For cases (iii) and (iv) S(Hh,) = 0, where Hb,,is given by formula (25). The above briel' analysis shows that the creation 01' an appropriate pH value in a buffer solution makes it possible to govern the separation process to some extent. When trying to achieve the highest selectivity and, hence, resolution, one must use buffer systems with either low or high pH values, depending on which case, (i)-(iv), is realized for the sample species. !Since small sample concentrations are commonly being used in CZE, so that the pH value remains approximately constant, these conclusions turn out to be correct and predictable. In the preceding simulations the pH value in solution was nowhere equal to pH',,; it was either higher or lower. But sometimes, especially in a buffer with poor capacity, the following situation might occur: in part of the solution, e.g. within the sample zone, pH < pH",, and in another part the pH > pH",. This case was simulated and the results are shown in Fig. 7 .At the initial time moments the lactic acid moves faster than the acetic acid because the pH value in the sample starting position is lower than pH",. As the sample species leave the starting region, they enter the solution where the p€I value is higher than pH", and S changes its sign. This leads to an unusual concentration profile of acetic acid. A part of it concentrates on the leading edge of the lactic acid zone, the other part stays in the start region and the third part mixes with lactic acid, spreading between the first two. The Sinversion sign causes severe sample distortion when it spreads over a large area. Finally, of course, the sample species will separate, but this requires more time and a longer capi1lary.Togetherwith this unusual zone spreading, detection capabilities deteriorate. So, when preparing for a separation, this situation must be avoided.

4 Concluding remarks The correct treatment of experimental results in electrophoresis requires a careful theoretical analysis of major transport mechanisms through all transient stages of the separation process, leading to the final state. Separation dynamics can b e understood by mathematical modeling, using advanced models with minor simplifications. This is only possible with the aid of computer simulation, as was attempted in the present paper. The evolution of a free fluid electrophoretic system was studied for two common electrophoretic techniques, ITP and CZE. Although some specific features of capillary electrophoresis were accounted for,

the main conclusions could be applied to systems with other chamber geometries (column and free flow chamber), allowing for a one-dimensional treatment. For solutions of weak monovalent acids and bases, two possible modes of separation were considered. It was shown that the pH value in a mixed zone was of key importance, especially in the case o f t h e irregular separation mode. In this case the order of separated zones depended on its magnitude both in CZE and ITP. For CZE the resolution vs. pH value was analyzed in terms of selectivity. In the case of large buffer capacity, when the pH is nearly constant, the results of separation are easily treated and may be used in a process of optimization. It was also demonstrated that for an irregular separation the proper choice of buffer pH helped to avoid the excessive sample distortion, which normally ruins the final separation pattern. We greatly thank Pro$ P. G. Righetti, University of Milano,for valuable criticism and advice. Received April 27, 1992

5 References [I] Kohlrausch, F., Ann. Phys. (Leipzig) 1897, 62, 209-216. [2] Babskii, V. G., Zhukov, M. Yu. and Yudovich, V. I.. Mathemotical Theory of Electrophoresis, Naukova dumka, Kiev 1983, pp. 7-196. [3] Bier, M., Palusinsky, 0. A,, Mosher, R. A . and Saville, D. A,. Science 1983, 219. 1281-1287. [4] Saville, D. A. and Palusinsky, 0. A., AIChE J. 1986, 32, 207-214. [S] Mosher, R. A , , Dewey,D.,Thormann,W.. Saville, D.A. and Bier, M., Anal. Chem. 1989, 61,362-366. [6] Thormann, W. and Mosher, R. A , , Adv. Electrophoresis 1988, 2, 45108. [7] Everaerts, F. M., Beckers, J . L. and Verheggen, T. P. E. M., Isoracliophoresis: Theory, Instrumentation and Application, Elsevier, Arnsterdam 1976. [8] Shimao, K., Electrophuresis 1986, 7, 121-128. [9] Zhukov, M . Yu., Zhurnal Vychislitel'noy Matemtrtiki i Matematicheskuy Fiziki 1984,24, 549-565. [ 101 Mikkers, F. E. P., Everaerts, F. M.and Verheggen,T. P. E. M., J . Chromatogr. 1979, 169, 1-10. [ I I] Fidler, V., Vacik, J . and Fidler, Z., J. Chromatogr. 1985,320, 167-174. [12] Fidler, Z., Fidler,V. and Vacik, J.,J. Chromatogr. 1985,320,175-183. [13] Dose, E.V. and Guiochon, G . A . , A n a l . Chem.. 1991,63,1063-1072. [14] Mikkers, F. E. P., Everaerts, F. M. and Peek, J. A. F., J. Chromatogr. 1979, 168,293-332. [15] Gebauer. P. and BoCek, P., J. Chromatogr. 1983, 267, 49-65. [16] Gag, B.,Vacik, J. and Zelensky, I., J. Chromatogr. 1991,545,225-237. [I71 Mosher, R. A.,Thorrnann, W. and Bier, M.,J. Chromatogr. 1985,320, 23-32. 1181 Palusinsky, 0. A., Graham,A., M0sher.R. A. and Bier, M., AlChEJ. 1986, 32,215-223. [19] Mosher, R. A. and Thormann, W., Electrophoresis 1986, 7,395-400. [20] Mikkers, F. E. P., Everaerts, F. M. and Verheggen, T. P. E. M., J. Chromatogc 1979, 169, 11-20. [21] Jorgenson, J. W. and Lukacs, K. D., Science 1983, 222, 266-272. [22] Rasmussen, H. T. and McNair, H. D., J . Chromatogr. 1990, 516, 223-231. [23] SustaEek, V., Foret, F. and BoEek, P., J. Chromutogr. 1991. 545,239248. [24] Gohie, W. A . and Ivory, C. F., J. Chromatogr. 1990, 516, 191-210. [25] Roberts, G . O., Rhodes, P. H. and Snyder, R. S., J. Chromatogr. 1989, 480, 35-67. [26] Datta, R. and Kotamarthi, V. R., AlChE J. 1990, 36, 916-926. [27] Foret, F. and Botek, P.. Adv. Electrophoresis 1989, 3, 45-108. [28] BoEek, P., Deml, M., Gebauer, P. and Dolnik,V., Analytical Isotachophoresis, VCH Verlagsgesellschaft, Weinheim 1988. [29] Ermakov, S.V., Mazhorova, 0. S. and Popov,Y. P., Informatic'a 1992, in press. [30] Giddings, J. C., Separ. Sci. 1969, 4, 181-189.