Biotechnol. Prog. 1990, 6, 485-493

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Theoretical Evaluation of Capillary Electrophoresis Performance? Ravindra Datta Department of Chemical and Biochemical Engineering, The University of Iowa, Iowa City, Iowa 52242

An analytical model (Datta and Kotamarthi, 1990) for the electrokinetic dispersion coefficient in capillary electrophoresis (CE), for the case of low {-potential, that accounts for the effects of pressure-driven and/or electroosmotic flow of the elutant, is utilized here to theoretically explore the performance of CE in terms of plate height, plate number, peak resolution, resolving power, and the time of analysis. Practical operating conditions for the voltage gradient and Poiseuille flow fraction, v, are explored that optimize CE column performance. The implications of the results in the rational design of CE columns are discussed. I t is also shown that the superposition of Poiseuille flow on the natural electroosmotic flow, while allowing greater freedom in the choice of elutant velocity, does not always cause increased dispersion.

Introduction Since its introduction about a decade ago (Mikkers et al., 1979; Jorgenson and Lukacs, 1981), capillary electrophoresis (CE), also called capillary zone electrophoresis (CZE) or high-performance capillary electrophoresis (HPCE), is fast developing into a powerful analytical instrument for rapid resolution of complex biochemical mixtures (Ewing et al., 1989). Unlike conventional electrophoresis methods, CE is amenable to automation and can provide higher resolution and greater speed than HPLC. This is due to the application of very high electric field strengths made possible by the use of very small diameter capillaries, typically around 50 pm, that effectively dissipate the electrical heat generated. The high electric field strength used also produces a strong electroosmotic flow of the electrolyte that is typically utilized in CE for the zone elution instead of conventional pumping. Since the electroosmotic velocity profile is considerably flatter than that in Poiseuille flow, it also produces less hydrodynamic dispersion. The small capillary size also requires very small sample size, which is a key advantage for some applications, e.g., in the analysis of the cytoplasmic fluid of a single cell (Ewing et al., 1989,1990). However, this also creates practical problems in ultrasmall sample injection and detector sensitivity. On-line detection methods so far have been limited mostly to fluorescence and UV absorption. Other detection methods need to be developed. The applications of CE so far include separation of amino acids, proteins, peptides, nucleotides, oligonucleotides, benzoic acids, chiral compounds, and virus and colloidal particles. The problems presented by protein separation have not yet been fully addressed. The main difficulty is that since proteins contain various functional groups such as cationic, anionic, hydrophobic, and polar, they tend to strongly adsorb on a variety of capillary surfaces (Lauer and McManigill, 1986; Jorgenson, 1987). Approaches developed to overcome this problem include modification of the capillary surface, operating at a pH when both the proteins and surface are negatively charged, resulting in electrostatic repulsion, or raising the concentration of competing ions in the buffer to reduce the interaction of proteins with the surface. Another practical problem is This work was presented at the 199th ACS National Meeting, Boston, MA, April 1990.

that, unlike in conventional instruments such as HPLC, there is no easy way to control the electroosmotic flow in CE, apart from capillary surface modifications or by changing the electrolyte composition. CE has also been extended to other modes of electrophoresis such as isotachophoresis, capillary gel electrophoresis, micellar electrokinetic capillary chromatography, and capillary isoelectric focusing (Grossman et al., 1990). In summary, while it is apparent that CE is a very versatile and efficient technique, many practical problems are yet to be resolved. It is evident that an understanding of the basic theory of separation and resolution in CE is essential to its development and optimal operation. However, this has so far not been addressed in adequate detail. A comprehensive discussion of the various possible causes of zone spreading in conventional electrophoresis is provided by Wieme (1975). These include axial diffusion, the temperature profile due to Joule heating, sample overloading, sample injection plug length, any sorptive interaction of the solute with the surface, microheterogeneity, electrodiffusion, and the elutant velocity profile. In the related field of chromatography,performance and resolution are quantified by utilizing the concepts of plate height, H , and the number of theoretical plates, N . Giddings (1969) extended this approach to electrophoresis. Jorgenson and Lukacs (1981) applied the approach of Giddings (1969, 1982) to CE. Terabe et al. (1985) extended this approach to the case of CE involving partitioning in micelles. However, these investigators as well as practically all of the subsequent ones have incorrectly assumed that since electroosmotic flow velocity is largely flat, it does not contribute to dispersion. In fact, diffusion is generally considered to be the only peak-broadening mechanism in CE, except in the work of Huang et al. (1989) and in that of Foret et al. (1988). In related work, Martin and Guiochon (1984) and Martin et al. (1985) have provided plate height expressions for open-capillary liquid chromatography for the case when elutant flow is electroosmotic. They approximated the electroosmotic velocity profile by empirical expressions in their analysis. The objective of this paper is to develop expressions for plate height, plate number, zone resolution, column length, and time of analysis in capillary electrophoresis that are based on the expression for electrokinetic dispersion coefficient in CE, Ki,developed recently by Datta and

8756-7938/90/3006-0485$02.50/00 1990 American Chemical Society and American Institute of Chemical Engineers

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Kotamarthi (1990). The general case involving pressuredriven and/or electroosmotic elutant flow is considered, since the utilization of pressure-driven flow, in addition to the ubiquitous electroosmotic flow, allows greater operational freedom. It has also been shown (Datta and Kotamarthi, 1990) that a judicious combination of electroosmotic and Poiseuille flows can actually provide lower dispersion. Optimization of the CE performance is also discussed here within the framework of this model. The results should be useful in the rational design and operation of CE columns.

Theory Consider a straight cylindrical capillary of internal diameter d, and a total length LT connected between two reservoirs of electrolyte buffer containing the electrodes. A sample containing species 1and 2 is introduced as a pulse a t z = 0 toward the anode end. The species migrate at different rates under the influence of the applied electric field and are convected by a combination of the electroosmotic and pressure-driven elutant flow. The species are detected by a detector located a t z = L and are subsequently eluted at the cathode end. It may be noted that, in general, the length of the capillary from the injection point to the detection point, L , is smaller than the total capillary length, LT. The need for this distinction is pointed out by Huang et al. (1989). The applied voltage gradient -dV/dz = (VI - V I I ) / L T = (VO- VL)/L,and the applied pressure gradient -dn/dz = (HI - IIII)/LT = (no - IIL)L. Here the dynamic pressure n H p - pg,z and accounts for both the applied pressure and gravity. The residence time of the elutant from the injection to the detection point is to and those of species 1 and 2 are tl and t2, respectively. It is assumed that species 1arrives a t the detector earlier than species 2. The assumptions involved in the theoretical analysis are listed in detail by Datta and Kotamarthi (1990) and include uniform temperature inside the capillary (even though the capillary temperature may be higher than the ambient), constant electrolyte and solute properties, small (-potential, and no solute-surface interaction. One assumption that is obviously limiting is that the capillary temperature is assumed not to change with voltage gradient. (a) Zone Concentration Profile. The cross-sectional area averaged concentration profile of species i within the CE column a t time t after a pulse input of mi moles of i is assumed to be Gaussian, i.e.,

for a run of a fixed time t. In elution capillary electrophoresis, on the other hand, the distance of migration to the detector of all peaks is the same and is equal to L, while the detection times are different. The shape of the peaks on the time axis is much like that in the space coordinate and is also assumed to be Gaussian. The detection time of the center of gravity of the peak of species i from eq 3 is ti = L/(U,i) (4) and the temporal standard deviation of the peak is equal to the time needed to wash the zone a distance q,the spatial standard deviation, through the column: ut, = “i/ (U,i)

(5)

(b) Species Velocity. The capillary cross-sectional area averaged species velocity, ( u z i ) , in eq 3 is

+ uei

(6) where the area-averaged elutant velocity, ( u,), is written as a sum of its two components: (uzi) = (uz)

(u,) = (up)

+ (u,)

(7)

where the Poiseuille elutant flow component is

and the electroosmoticflow component is (Rice and Whitehead, 1965) (9) (u,) = ueo(l - d In this expression, u,, is given by the Helmholtz-Smoluchoski equation:

dV

u,, = -u

dz The function 7 in eq 9 for the Debye-Huckel approximation is (Rice and Whitehead, 1965) eo

where cp is the dimensionless capillary radius, d,/2X. Equation 11 is of the form of the effectiveness factor for first-order reaction in a cylindrical catalyst particle and 1.0 as 4 0 and 2/4 as has the limiting forms Q 4 a. The electrophoretic velocity of species i in eq 6 is given by

-

-

-

-+

2)

( C i ) = 4mi/7rd,2 exp( rJi( 2*)”2 20: This is strictly valid only when the sample is injected as a pulse. If the sample plug length is considerable,the peaks may no longer be Gaussian (Huang et al., 1989). In eq 1, the variance of the peak of species i a t a time t after sample injection is ai2 = 2Kit

(2) where Ki is the electrokinetic dispersion coefficient, the expression for which is given in the next section. In eq l , z 1 is the axial distance from the center of gravity of the peak. f i , Le., z1 = z - t i , and the location of the center of gravity of a peak moving with the mean velocity, ( u z i ) ,at a time t after sample injection is (3) The above equations are useful in open-column electrophoresis where the bands migrate to different distances 2i = ( U Z i ) t

uei = -ui dV/dz

(12)

There is a paucity of data on free-solution electrophoretic mobilities in the literature, although theoretical expressions are available for estimation of ui (Wiersema et al., 1966; Edward, 1966). Grossman et al. (1989) have recently provided a semiempiricalexpression for estimating electrophoretic mobilities of peptides. (c) Electrokinetic Dispersion Coefficient, Ki. The approach developed by Westhaver (1947), Taylor (1953), and Aris (1956) was followed by Datta and Kotamarthi (1990) to obtain the following expression for the electrokinetic dispersion coefficient, Ki,for the general case involving pressure driven and electroosmotic elutant flow: d2

K, = Di + L{(u 1920, with the functions

P

)2

+ 62(up)(u,) + 6,(~,)~]

(13)

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Biotechnol. frog., 1990, Vol. 6, No. 6

By using eqs 6 and 19 in eq 18, Hi may be written in the alternative form involving ki: 6, = 48

(&)'(

1

2 1 +---

(15)

$2 7)42 71242 In these expressions, 7) is given in terms of 4 by eq 11. Thus, 82 and 83 are functions only of the dimensionless capillary radius, 6. The limiting form 71 = 214 may alternatively be used in the equations for 6 2 and 63 when 4 is relatively large. Equation 13 gives Ki in terms of the area-averaged Poiseuille and electroosmotic flow velocities, (up) and ( ue). When ( u e ) = 0, eq 13 reduces to the so-called "TaylorAris" expression for the dispersion coefficient for laminar flow in tubes (Westhaver, 1947; Aris, 1956). The case of electroosmotic flow only, common in CE, is obtained by setting (up) = 0 in eq 13. Further, since (up) or ( u e ) could possibly assume negative values if these components were in the reverse direction, the second term in the braces of eq 13 would then become negative, resulting in a reduction of the dispersion coefficient. In fact, Datta and Kotamarthi (1990) showed that the presence of a small reverse pressure/gravity-driven flow relative to the electroosmotic flow could substantially reduce the dispersion coefficient. (d) Plate Height and Number. Plate height and plate number are common terms used in the chromatographic literature [e.g., Karger et al. (197311 to characterize the efficiency of a separation column and are also utilized here. The basic interrelationship between the height equivalent of a theoretical plate for species i, Hi,and the number of theoretical plates for species, i, Ni,is

L = NiHi where the plate number is

Ni= ( t i / U t i )

(16) 2

(17) Use of eqs 2, 4, 5, 16, and 17 yields the plate height for species i: (18) Hi = 2Ki/ ( v2i) A smaller Hi,or a larger Ni,indicates a more efficient column. The parameter ki, for species i, is next defined:

ki u e i / ( u z ) (19) Note that this definition is different from the conventional definition of capacity factor in chromatography, which is = (ti - to)/to. Further, unlike in chromatography, ki can be positive or negative depending upon the sign of uei. ki may also be expressed in terms of system parameters by using eqs 7, 9, 10, and 12 in eq 19: ki = Ki(1 - U ) / ( l - 71) (20) where the parameter ~i is defined as the ratio of the electrophoretic mobility, ui, and the electroosmotic flow mobility, ueo: (21) and the operational parameter v is defined as the ratio of the Poiseuille flow to the total elutant flow: X i E Ui/Ueo

iJ = ( u p ) / ( u z ) (22) It may be noted that the parameter v is not necessarily limited to values between 0 and 1but can, in theory, vary from --oo to +-oo or, more restrictvely, within the limits of laminar flow. Thus, for a given CE and a separation system, ki varies only with v.

Hi = H/(1 + ki) where the plate height for an uncharged solute is

(Ui

(23) = 0)

(24) H = 2Ki/ ( u z ) In the chromatographic literature, it is commonly assumed that the numerical values of Hi and Nj are the same for all species. In reality, they vary somewhat from one species to another. For this reason, for two species 1and 2 with juxtaposed peaks, an average plate number, N,may be defined for a hypothetical solute (Karger, 1967) with an average residence time 2 = ( t l + t2)/2 and an average temporal standard deviation ut = (atl + ut2)/2 by

N = (t/.,)2 and the corresponding plate height H assumed to be given by

(25) = L / N , where H is

R = H/(1 + K)

(26) where k E (kl + k2)/2, which may, with the use of eq 20, be written in the form l+a l-v = KZ(T)(i.;;)

in which the selectivity, a , is

k l / k , = K ~ / K=~u1/u2 (28) Use has been made of eqs 20 and 21 in arriving at the second and third equalities in eq 28. Note that a can be a positive or negative quantity depending upon the signs of u1 and up. Finally, by using eq 13 for Ki in eq 24, the expression for the reduced plate height for an uncharged solute, h, becomes a

H--+-

hrd,-ReSc

(ReSc)(v2

96

+ 8,u(l-

u)

+ 8,(l

- v)']

(29)

In this expression, the product of the Reynold and Schmidt numbers ReSc = d,(u,)/Di. This expression was used by Datta and Kotamarthi (1990) for a comparison with experimental results available in the literature for plate height of uncharged solutes as a function of the electroosmotic flow rate. They found good agreement with the data with 4 used as a fitted parameter. The values of the corresponding double-layer thickness, A, while seemingly large, agreed with those calculated by Martin and Guiochon (1984)in their numerical simulation of the data. It is useful to write ReSc in eq 29 in terms of the parameter v and the dimensionless voltage gradient dsZ/d[ by using eqs 7, 9, 10, and 22: (30) where the dimensionless voltage gradient is related to the actual voltage gradient, dV/dz, by -dfl = - - 1 dV

E, dz where the characteristic electric field strength, E,, is defined as d(

E , Di/d,ue, (32) When eq 30 is used in eq 29, an expression for the plate height of an uncharged species in a given CE system is obtained only in terms of the operational parameters v and

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400

Table I. Parameters Used in Simulation Darameter value 50 cm 50 pm

the dimensionless voltage gradient dQ/d[:

25 (A = 1 pm) 5.0 X lo4 cm2 V-' s-l -3.0 X cm2 V-l s-l 0.975 5.0 X lo4 cm2 6-l

6,(1 - U T I (33) For the important special case of electroosmotic flow only, substitution of v = 0 in the above equation yields

For charged species, similar expressions for the plate height can be obtained by using eqs 33 and 20 in eq 23. The plate number may then be obtained by using eq 16. (e) Peak Resolution. The resolution between the peaks of species 1and 2 is defined as the ratio of the difference in elution times of the two peaks and their average base width [e.g., Karger (1967)l: (35) Assuming Wti i= 4qi, and using eqs 5,6,19, and 25 in eq 35 (36) Using N = L / H in eq 36, where H is assumed to be given by eq 26

where h is given by eq 33 in terms of the dimensionless voltage gradient and v. Thus, resolution may be calculated for a given CE system ( L and d,) and a given solute pair (a and K Z ) from eq 37 in terms of only the operational parameters v and the dimensionless voltage gradient, dQ/ d[. If, as is commonly done in CE, only electroosmotic elutant flow is utilized, u = 0, and then the resolution for a given system becomes only a function of the applied voltage gradient. Note also from eq 37 that R, 0 as a 1 or as R 0. (f) CE Column Design. The L / d c required for a CE column of a given diameter, d,, to give a required R, for given operational parameters u and dQ/d[ can be obtained simply by a rearrangement of eq 37:

-

-

-

Thus, for instance, the column length between the injection and detection points, L , necessary for a given separation increases 4 times if the resolution were to be doubled. (g) Time of Elution. The objective, normally, is to achieve a desired resolution (e.g., R, = 1.5 for baseline resolution) as rapidly as practicable. The elution time is given by eq 4. By using eq 19 in eq 4,the elution time of species i in a dimensionless form is Ti

=4

1 (") d , ReSc(1 + ki)

(39)

Substituting for L / d c from eq 38 into eq 39, the dimensionless average elution time for species 1and 2 becomes

where from eqs 29 and 30

Discussion of Results The principal performance characteristics of a capillary electrophoresis column are next discussed in the light of the theoretical model described in the preceding section. The questions of interest in this regard are mainly of the following two types: (1)For a given CE column ( L and d,), what are the optimum operating conditions to obtain the best performance? (2) For a desired performance under a given set of operating conditions, what is the optimum design ( L and d,) for a CE column? The first problem, however, in addressingthese questions is to define an appropriate performance index. In chromatography, the following performance criteria are typically used: (a) plate height or plate number, (b) peak resolution, (c) resolving power, and (d) time of elution. We will discuss these in turn for CE. The major CE design parameters are the capillary diameter, d,, and the length-to-diameter ratio, Lld,. The species parameters are u2 and a. The main operational variables are the dimensionless voltage gradient applied, dQ/d[, and any ratio of pressure drop/gravity-driven flow to the total elutant flow, u. Other parameters that may possibly be altered include buffer composition, ionic strength, pH, and any chemical modifications of the capillary surface aimed at controlling the [-potential and, hence, the electroosmotic flow. The performance objectives usually are the maximization of resolution, R,, and the minimization of the average peak elution time, .; A composite performance objective incorporating both these goals may, therefore, be formulated as the minimization of ;/Rs2. The set of parameters employed for the following discussion is given in Table I, which represents the case of a fairly difficult separation in a capillary column of typical dimensions. (a) Plate Height and Numljer. Equation 33 for the dimensionless plate height of an uncharged solute, h, is plotted in Figure 1 as h versus -dQ/d[ for a variety of v values. It should be noted that as ( u e ) increases with increasing voltage gradient, there is a corresponding increase in (up) for a constant value of u. It is seen that h goes through a minimum with dQ/d[, the location of which depends upon the value of u. When u = 0.5, corresponding to half pressure-driven and half electroosmotic flow, h is considerably larger than for the case of only electroosmotic flow ( u = 0), except at rather low electric field strengths. As expected, the curve for u = 0.75 is higher than that for u = 0.5. However, the curve for u = -0.5, corresponding to the case of pressure-driven flow opposing the electroosmoticflow, is intermediate between the curves for u = 0.5 and u = 0. The case of u = -0.2 is

489

Biotechnol. Prog., 1990,Vol. 6, No. 6 10

v = 0.75

1

h

h 1

."I

50

0

100 - dnlag

150

200

Figure 1. Dimensionless plate height for an uncharged solute versus dimensionless voltage gradient for d = 25 and for a variety of u values.

-1.5

-2.0

-1.0

-0.5

0.0

0.5

1.0

V

Figure 2. Dimensionless plate height versus u for 4 = 25 and -dQ/dt = 100. 200000,

particularly interesting since the predicted h is even lower than that for the case of electroosmotic flow only, except at low voltage gradients. For a constant u, the optimum -dQ/d[, (-dQ/dg)o,corresponding to the minimum in h, can be obtained by differentiating eq 33 with respect to dQ/d[ and setting it equal to zero. Then

(-G)=d5

0

8 (1- 7) [ ' / 3 ( u 2

+ 6,u(1

(1 - 4

- u ) + 6,(1-

u ) 211112

-

100000

-

/

K2 =

- 0.6

1

1

v = - 0.2

(42)

which is a function of 4 and u only. For electro-osmotic flow only ( u = 0), this equation reduces to (-dQ/d[)o = 8/((1 - 7)(63/3)'J2). Frequently, it may be desirable to operate at a voltage gradient higher than that indicated by eq 42 for the purpose of speedy analysis. The effect of the parameter u on h is more obvious in Figure 2, in which h is plotted as a function of u for a fixed voltage gradient -dQ/d[ = 100. A minimum in h corresponding to v -0.2 is observed. The location of the minimum, which is a function of 4 only, can also be obtained mathematically by differentiating eq 33 and setting dh/av = 0. As u 1, h a,since at a given dQ/ d[, u 1 implies an extremely large ( u p ) . Figure 3 is a plot of the average plate number, N, attained in a given column with L/dc = 10 000 and 4 = 25, as the function of the dimensionless voltage gradient. The three lower curves are for a solute pair with a = 0.975 and ~2 = -0.6, whereas the upper curve is for an uncharged solute and for u = 0. N in Figure 3 is calculated by using eqs 26 and 27 in N = L / H , resulting in

-

R

150000

I

- -

0

50

100

- dWd6

150

200

Figure 3. Average plate number versus the dimensionless voltage gradient for different values of u. The other parameters are given in the figure. 150000 o

Benzyl alcohol (uncharged)

50000v

Di = 1.OE-5 cmWs @ = 18 ( X = 1.4 pm)

DaraofFazioetal. (1990)

where h is given by-eq 33. For the case of electroosmotic flow only (v_ = 0), N rises with the voltage gradient to a maximum N = 67 000 plates at -dQ/d[ = 80, and then decreases as voltage gradient increases further. For u = -0.2, higher N is obtained at higher voltage gradients. u = 0.5, on the other hand, yields a much poorer plate number. For an uncharged solute, since ~i = 0 in eq 43, a much higher plate number is obtained. Equation 43 also implies that with cationic species, since ~i > 0, an even higher plate number would result, provided, of course, that there is little coulombic interaction with the capillary walls. Figure 4 provides a comparison of the theoretical model with the experimental results of Fazio et al. (1990) for plate numbers as a function of the elutant electroosmotic velocity obtained in a CE column of d , = 50 pm and L = 45 cm for the uncharged solute benzyl alcohol. The diffusion

0 ~ " ' ~ " ~ " " ' ' " ' ' ~ ' ' ~ 0.0 0.5 1.o 1.5 cvz> (mmls)

2.0

Figure 4. A comparison of theory and experimental data of Fazio et al. (1990) for the plate number obtained for the uncharged solute benzyl alcohol in 50 mM phosphate buffer, pH = 7. The other paran-etersare given in the figure. 4 = 18 is used as a fitted parameter.

coefficient for benzyl alcohol was estimated to be Di = 1.0 X cm2/s at 30 OC (Reid et al., 1977). A value of 4 = 18, used as a fitted parameter, provides a very good fit between theory and experiments. This value of 4 implies a double-layer thickness X = 1.4 pm. This compares with the value of X = 1.8 pm calculated by Martin and Guiochon (1984) as well as by Datta and Kotamarthi (1990) for the data of Tsuda et al. (1982) obtained under conditions similar to those of Fazio et al. (1990). (b) Peak Resolution. The resolution between the

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Biotechnol. Prog., 1990, Vol. 6 , No. 6 1

81

Rs

'Iy

0

0

50

100

150

-

200

250

dnld5

0 80 100

10

RS

Figure 5. Resolution attainable versus dimensionless voltage gradient for different v values for the CE system described in Table I.

exiting peaks of the two species 1 and 2, as given by eq 37, is plotted in Figure 5 for the case of a CE column with parameters given in Table I. For u = 0, the resolution attained rises sharply with a maximum of 2.95 a t a dimensionless voltage gradient, -dfl/d[ = 80. The resolution then declines rather gradually at higher voltage gradients. For u = 0.2, the maximum resolution obtained is 1.5 a t dfl/d[ = 35. For the case of Y = -0.2, the increase in resolution is more gradual, peaking at R, = 6 at a d 0 / d[ = 180. Figure 5, thus, not only demonstrates the considerable resolving power of capillary electrophoresis for the given example, which represents a difficult separation, but also shows that with a judiciously selected pressure-driven flow ratio, Y , a further substantial improvement in the resolution can result. (e) Resolving Power. The resolving power of a given CE column, CY,for a given R,, u , and dfl/dt, can be calculated from eq 37. Figure 6 provides the selectivity of a pair of species versus resolution for three different values of Y for the CE system described in Table I and for -dfl/d[ = 100. Thus, if the minimum acceptable resolution is unity, the largest cy resolvable under the given conditions is 0.9775 for u = 0.2, cy = 0.99 for v = 0, and CY = 0.995 for Y = -0.2. (a) Time of Elution. The operational objective frequently is to minimize the elution time for an acceptable resolution. The dimensionless average time of elution of the peaks of species 1 and 2 is given by eq 40 along with eq 41. Figure 7 is a plot of the resolution attained and the average elution time of the two peaks, t (in minutes), as a function of the actual voltage gradient, -dV/dz (in volts per centimeter), for the example described in Table I and for electroosmotic flow only ( u = 0). At small voltage gradients, the resolution obtained rises rapidly, peaking at R, = 2.95 at -dV/dz = 160 V/cm, and then declines gradually. On t h e other hand, t h e elution time is inordinately high a t low voltage gradients. There is then a precipitous decline in ? followed by a more gradual decline at higher -dV/dz. Thus, at higher -dV/dz, any decrease in t comes a t the expense of decreased resolution. In plotting Figure 7 , it was assumed that E , in eq 31 is a constant. In fact, E, varies with the voltage gradient owing to its temperature dependence. The temperature variation of E , can be obtained from eq 32 if the variation of u, and Di with temperature is known. The issue of temperature rise in the capillary due to voltage gradient increase and its effect on CE performance is planned to be addressed in a subsequent paper. The interrelationship of R, and t for this example can

Figure 6. Selectivity versus resolution for the CE system described in Table I and for -dO/d[ = 100.

1 @=25 ; v=o a = 0.975

; XZ = L/dc = 10,OOO

10000

- 0.6

1 : 100

0 0 1 .

"

100

200

'

"

"

400

500

300

-

'

'5

I1 600

dV/dz (V/cm)

Figure 7. Resolution obtained and the average time of elution versus voltage gradient for the system described in Table I, with v = 0 and constant E,. be more directly observed in Figure 8. As dV/& increases, ? reduces sharply while R, increases. As t decreases further, however, R, peaks and then begins to decrease, with t for a given resolution becoming essentially constant. The last observation can be more readily discerned in Figure 9, which is a plot of the dimensionless average elution time for a given resolution and a given selectivity, versus the dimensionless voltage gradient. The form of the ordinate in Figure 9 results from using eq 27 in eq 40 and rearranging to obtain

where h/ReSc is given by eq 41. Figure 9 is a plot of the left-hand side of eq 44 versus -dfl/d[ for 4 = 25, KZ = -0.6, and a variety of Y values. It is seen that this quantity is initially large but drops rapidly a t low -dfl/d[ and then becomes rather constant at higher -dfl/dt. Thus, little improvement in elution time for a given resolution is obtained at very high voltage gradients. Clearly,even when thermally permissible, very high voltage gradients are not likely to be useful from this standpoint. The asymptotic behavior of eq 44 for large -dfl/d[ shown in Figure 9 can be understood by examining eq 41, in which the first term on the right-hand side becomes negligibly small a t large voltage gradients. The left-hand side of eq 44 is also plotted versus the parameter Y in Figure 10 for 4 = 25, KZ = -0.6, and -dfl/d[ = 100. This figure shows that, as expected, the average dimensionless elution time for a given selectivity and a resolution is a minimum around v = -0.2.

49 1

Biotechnol. Prog., 1990,Vol. 6, No. 6

2 -

. U

U

I2

a = 0.975 ;

K2 = -0.6

100

10

1000

Figure 8. Interrelationship of resolution and the average elution time for the example of Figure 7.

v = 0.75

V = - 0.75

I

.01 I

100

50

- df2IdS

150

200

Figure 9. Average dimensionless elution time for a given selectivity and a desired resolution versus dimensionlessvoltage gradient for 4 = 25 and K Z = -0.6. 10

.01 -2.0

-1.5

-1.0

-0.5

0.5

0.0

V

Figure 10. Average dimensionless elution time for a given selectivity and a desired resolution versus u for 4 = 25, ~2 = -0.6, and -dQ/dE = 100.

(e) CE Column Design. The design of a CE column primarily involves the selection of appropriate injectorto-detector distance, L , and the capillary diameter, d,, for a desired separation under a given set of operating conditions. The minimum dimensionless injection-todetection length, L f d,, required for unit resolution versus species selectivity, a , is shown in Figure 11. The parameters employed in the figure are 4 = 25, K Z = -0.6, -dQ/ dE = 100, and v = 0. As a 1, the separation becomes increasingly difficult, and the required L / d c m .

-

0.80

0.85

0.90

0.95

1 .oo

a

f (min)

0

I

0.75

-

Figure 11. The dimensionless CE column length from the injection point to detection point required for unit resolution versus species selectivity. The other parameters are specified in the figure.

Figure 12 plots the resolution obtainable and the average elution time, t,of the two species versus capillary injectionto-detection length, L , for a given total voltage drop of 10 kV across the injection-to-detection length and for the case of v = 0. The other parameters are specified in Table I. The total voltage drop may, for instance, be limited by the available power supply. Thus, the voltage gradient decreases as L increases for a constant overall voltage drop, VO- VL. At small L, t is low but so is the resolution, due to low separation, At, between peaks. t increases steeply with L due to the dual effect of increased column length, and hence elution time, and the smaller elutant velocity due to lower dVfdz. Thus, small increases in R, beyond a certain L are at the expense of large increases in t. The issue of the selection of an appropriate capillary diameter, d,, is explored in Figure 13, in which H (in micrometers) and the resolution attained is plotted versus the capillary diameter for a constant injection-todetection length, L = 50 cm, and for -dQ f d[ = 100. The other parameters are listed in Table I. Interestingly, H initially increases with d , and then decreases before finally increasing. Although h declines monotonically with d,, H = h d , increases initially with d,. The initial sharp drop in resolution is followed by a more graduate decline. It, therefore, appears that although, for this set of parameters, d , < 20 pm is desirable for high resolution, the drop in resolution beyond about 50 pm is relatively small. It was assumed in this analysis that the range of capillary diameters considered is sufficiently small so that heat transfer considerations are not limiting.

Conclusions In this paper, an analytical expression developed by Datta and Kotamarthi (1990) for the electrokinetic dispersion coefficient in capillary electrophoresis, Ki,for the case of low l-potential, is used to theoretically evaluate the performance of a CE column in terms of the plate height, plate number, zone resolution, resolving power, and time of analysis. There is good agreement between the model and experimental data reported in the literature for the number of theoretical plates for an uncharged solute, provided that the double-layer thickness, A, is used as a fitted parameter. The value of X thus assumed, however, is of the same order as that calculated by others for a similar system. The performance evaluation of a CE column in terms of plate height and resolution shows an optimum with

492

Biotechnol. Prog., 1990,Vol. 6, No. 6

electrokinetic dispersion coefficient of species i, m2 S-1

$=25 ; v=O a = 0.975 ; k2 = - 0.6

80

60

.E

0 25

50

0

75

100

L (cm)

Figure 12. Resolution and average elution time obtained versus CE column length from the injection point to detection point required for a constant total voltage drop, Vo - VL. h=lpm;v=O ;-dn/d~=lOO a = 0.975 ; ~2 = - 0.6

- 6

3 -

v)

' a i 2 -

4

1 -

7

Boltzmann constant, 1.38054 X 10-23 J K-1 dimensionless parameter defined by eq 19 or eq 20 arithmetic average of k l and kz = (kl + k2)/2 (eq 27) capillary length between sample injection and detection points, m total capillary length, m total moles of species i injected, mol number of theoretical plates for species i = L / H , average plate number of species 1 and 2 (eq 25) number of positive/negative electrolyte ions per unit volume, m-3 pressure, Pa Reynold number = dc(uz)p/h, dimensionless peak resolution (eq 35) product of Reynold a n d Schmidt numbers = d,( uz)/Di,dimensionless Schmidt number = h/pDi, dimensionless time elapsed since sample injection, s characteristic time d,2/4Di, s residence time of species i from injection point to detection point (eq 4, s residence time of elutant from injection point to detection point, s residence time from injection point to detection point of species detected first, s residence time from injection point to detection point of species detected later, s average residence time of species 1 and 2 = (tl + t 2 ) 12

O0

20

40

60

80

1 0L 0

de

120

140

160

180

2 0 0O

(w)

Figure 13. Plate height and resolution versus capillary diameter.

respect t o b o t h t h e voltage g r a d i e n t a n d t h e r a t i o of pressure-driven flow t o total elutant flow, v. Further, while t h e elution time decreases monotonically with t h e voltage gradient, there is a n a t t e n d a n t drop in resolution beyond a certain value. Thus, very high voltage gradients are not necessarily desirable. T h e effect of CE column length and diameter on t h e performance is also studied.

Notation (Ci) Di dc

E, e g2

capillary cross-sectional area averaged concentration of i, mol m-3 diffusion coefficient of solute i in liquid elutant, m2 s-1 capillary internal diameter, m characteristic electric field strength = Di/dcueo, V/m charge on an electron/proton = 1.60210 X l O - l 9 C z-component of the acceleration due to gravity, m 5-2

H Hi

A h

he hi

height equivalent of a theoretical plate for an uncharged solute (eq 24), m height equivalent of a theoretical plate for a solute i, (eqs 18 and 23), m average plate height of species 1 and 2 = L / N , m dimensionless height of a theoretical plate for an uncharged solute = H / d c dimensionless plate height of an uncharged solute for only electroosmotic flow (U = 0 ) dimensionless height of a theoretical plate for species i = H J d ,

electroosmotic mobility of elutant = + / h ,

m2 V-1

S-1

electrophoretic mobility of species i, m2 V-1 s-l applied electric potential, V; subscripts 0 and L correspond to V a t z = 0 and L,respectively, and subscripts I and I1 refer to the anode end and the cathode end of the capillary, respectively maximum possible electroosmotic velocity (eq lo), m s-1 cross-sectional area averaged electroosmotic velocity, m s-l electrophoretic velocity of species i (eq 12), m s-l cross-sectional area averaged Poiseuille velocity (eq 81, m s-l cross-sectional area averaged elutant velocity (eq 71, m s-1 cross-sectional area averaged and velocity of species i (eq 6), m s-l baseline band width of peak 1, s baseline band width of peak 2, s charge number of the symmetric electrolyte buffer axial coordinate, m axial distance with respect to a plane moving with a velocity (u,i) = z - Z i , m location of the center of gravity of the peak of species i a t a time t , m Greek Symbols ff selectivity of solutes 1 and 2 = uI/u2 (eq 28) function defined by eq 14, dimensionless 82 function defined by eq 15, dimensionless 63 t permittivity of elutant = trto, C V-l m-l 61 relative permittivity, dimensionless permittivity of vacuum = 8.854 X 10-l2C V-1 m-1 60

Biotechnol. Prog., 1990,Vol. 6, No. 6

P

zeta potential of capillary surface (at the plane of "shear"), V function of C#J (eq 11) ratio, ui/u,, (eq 21) Debye length = [&T/2n(Ze)2]1/2, m elutant viscosity, Pa s ratio of Poiseuille flow t o total elutant flow, (up)/

F

dimensionless axial distance = z/dc combined applied and hydrostatic pressure = p pgzz, Pa s; subscripts 0 and L correspond t o n at z = 0 and L, respectively, and subscripts I and I1 refer t o t h e anode end and the cathode end of the capillary, respectively elutant density, kg m-3 spatial standard deviation of the Gaussian peak of i, m temporal standard deviation of the Gaussian peak of i, s average temporal standard deviation = (uti + ut2)/2 dimensionless detection time of species i = ti/tc dimensionless average detection time for species 1 and 2 = t / t c dimensionless capillary radius = dc/2X dimensionless electric potential = V / (Di/ue,,)

(UZ)

n

P Qi uti

9 D

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Grossman, P. D.; Lauer, H. H.; Moring, S. E.; Mead, D. E.; Oldham, M. F.; Nickel, J. H.; Groudberg, J. R. P.; Krever, A.; Ransom, D. H.; Colburn, J. C. A Practical Introduction to the Free Solution Capillary Electrophoresis of Proteins and Peptides. Am. Biotechnol. Lab. 1990, 8 (2), 35-43. Huang, X.; Coleman, W. F.; Zare, R. N.; Analysis of Factors Causing Peak Broadening in Capillary Zone Electrophoresis. J. Chromatogr. 1989,480, 95-110. Jorgenson, J. W. Capillary Zone Electrophoresis. In New Directions in Electrophoretic Methods; Jorgenson, J. W., Phillips, M., Eds.; ACS Symposium Series 335; American Chemical Society: Washington, DC, 1987; pp 182-198. Jorgenson, J. W.; Lukacs, K. D. Zone Electrophoresis in OpenTubular Glass Capillaries. Anal. Chem. 1981, 53, 12981302. Karger, B. L. A Critical Examination of Resolution Equations for Gas-Liquid Chromatography. J. Gas Chromatog. 1967, 5,161-169. Karger, B. L.; Snyder, L. R.; Horvath, C. An Introduction to Separation Science; John Wiley: New York, 1973. Lauer, H. H.; McManigill, D. Capillary Zone Electrophoresis of Proteins in Untreated Fused Silica Tubing. Anal. Chem. 1986, 58, 166-170. Martin, M.; Guiochon, G. Axial Dispersion in Open-Tubular Capillary Liquid Chromatography with Electroosmotic Flow. Anal. Chem. 1984,56,614-620. Martin, M.; Guiochon, G.; Walbroehl, Y.; Jorgenson, J. W. Peak Broadening in Open-Tubular Liquid Chromatography with Electroosmotic Flow. Anal. Chem. 1985,57, 559-561. Mikkers, F. E. P.; Everaerts, F. M.; Verheggan, Th. P. E. M. HighPerformance Zone Electrophoresis. J. Chromatogr. 1979,169, 11-20. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977; p 557. Rice, C. L.; Whitehead, R. Electrokinetic Flow in a Narrow Cylindrical Capillary. J. Phys. Chem. 1965, 69, 4017-4024. Taylor, G. I. Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube. Proc. R. SOC. London 1953, A219,186203. Terabe, S.;Otsuka, K.; Audo, T. Electrokinetic Chromatography with Micellar Solution and Open-Tubular Capillary. Anal. Chem. 1985,57,834-841. Tsuda, T.; Nomura, K.; Nakagowa, G. Open-tubular Microcapillary Liquid Chromatography with Electro-Osmosis Flow using a UV Detector. J. Chromatogr. 1982,248, 241-247. Westhaver, J. W. Concentration of Potassium39by Countercurrent Electro-migration: Some Theoretical Aspects of the Operation. J. Res. Natl. Bur. Stand. (U.S.) 1947, 38, 169183. Wieme, R. J. Theory of Electrophoresis. In Chromatography: A Laboratory Handbook of Chromatographic and Electrophoretic Methods, 3rd ed.; Heftmann, E., Ed.; Van Nostrand Reinhold Co.: New York, 1975; pp 228-281. Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. Calculation of the Electrophoretic Mobility of a Spherical Colloid Particle. J. Colloid Interface Sci. 1966, 22, 78-99. Accepted September 18, 1990.