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Anal. Chem. 1989, 6 1 , 241-246
Effect of Temperature Gradients on the Efficiency of Capillary Zone Electrophoresis Separations Eli Grushka,*J R. M. McCormick, and J. J. Kirkland
E . I . du Pont de Nemours & Company, Central Research and Development Department, Experimental Station, B-228, Wilmington, Delaware 19898
A flat electrophoretlc migration velocity profiie Is assumed to be characteristic of current capillary zone electrophoresis (CZE) separations. However, this electrophoretlc velocity profile may not actually exlst In some experiments due to heat generated within the capillary. The actual migration velocity stream within capillary electrophoresis tubes probably has a parabolic proflie. Using a parabolic model, we have developed an expresslon that quantitatively relates the plate height to experimental parameters. The plate height In CZE is a strong function of the capillary radius, the strength of the applied field, and the buffer concentration. The newly derived plate height equation suggests the following: (a) large operating voltages may broaden solute peaks; (b) capillaries should be carefully thermostated; (c) thermostating at e i s vated temperatures may lead to improved plate heights; and (d) decreasing the electrolyte concentration In the buffer allows the use of wider capiliarles. Wider capillaries, In turn, would permit more convenient operation and large sample volumes.
INTRODUCTION There is considerable current interest in capillary zone electrophoresis (CZE) because of its potential for attaining very high plate numbers or separation efficiencies. However, in practice, the solute peaks in CZE are broader than anticipated; predicted high theoretical plate numbers have not been observed. Some workers maintain that temperature effects are among the factors that contribute to excessive peak width (1).
Temperature effects are a result of Joule heating due to current passed through the buffer solution within the capillary. Loss of heat from the tube wall to the surrounding environment causa a radial temperature gradient within the capillary that results in a viscosity gradient in the buffer within the tube. This viscosity gradient, in turn, causes radial variations in the electrophoretic migration velocity profiles of solutes. These perturbed velocity profiles give rise to mass-transfer effects that broaden peaks and cause increased plate heights and lower separation efficiency. Several groups investigated the shape of the temperature gradient in CZE. H j e r t h (2) assumed that the resistivity of the buffer in the capillary is temperature independent and suggested that the temperature profile within the capillary is parabolic. Coxon and Binder (3) claimed that the radial temperature distribution in capillary isotachophoresis is not parabolic but involves a Bessel function. The argument of the Bessel function depends on the radial position and on the rate of heat production within the capillary. However, while the Coxon and Binder model is valid for well-thermostated systems, it does not hold rigorously for nonthermostated cases because of a mathematical discontinuity in the equation.
Therefore, the Coxon-Binder equation should be modified for nonthermostated systems. For capillary dimensions normally used in CZE, the modified form of the Coxon-Binder equation gives a temperature profiie that is indistinguishable from a parabolic function (4). Hinckley and co-workers ( 5 , 6 )also showed that the temperature gradient within a capillary is not parabolic, but a function of a Bessel series. However, since Hinckley et al. used an approach similar to that of Coxon and Binder, the equation describing the temperature profile is not applicable in many real situations. In this communication we critically examine the effect of heat generated within tubes on the efficiency of CZE systems. We assume only parabolic temperature profiles since, in narrow capillaries, such profiles seem to describe adequately the temperature gradients (4). We propose certain conditions for operating the CZE system that minimize the undesirable effects of thermal gradients.
THEORY Temperature Profile. The parabolic profile of the temperature difference across a capillary is defined by solving the heat conduction equation
where T is the temperature (see the glossary for dimensions), r is the radial position, G is the heat generation rate, and kl is the thermal conductivity of the buffer solution. Using a simple integration and the appropriate boundary conditions gives the following solution to eq 1:
"( 5)
T = T1 + 4kl
1-
where R1 is the internal radius of the capillary. Solving a differential equation similar to eq 1 (with the appropriate boundary conditions) gives an expression for the capillary wall temperature T p For a fused silica capillary with a polyimide coating, the wall temperature is given by
where R2 and R, are the capillary radii without and with the polymide coating, respectively, k2 and k , are the thermal conductivities of the fused silica and polyimide coating, T , is the ambient temperature, and h is the heat transfer coefficient to the surroundings. Shape of Electrophoretic Velocity Profile. The electrophoretic velocity, u(r),within a capillary can be written as
u(r) = q,cr\kE/q Permanent address: Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel. 0003-2700/89/0381-0241$01.50/0
(4)
where E is the field strength, Q is the permittivity of a vacuum, 0 1989 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 3, FEBRUARY 1, 1989
tr is the dielectric constant, i(l is the {potential, and 7 is the viscosity of the buffer. Normally, the electrophoretic velocity profile in CZE is assumed to be flat. However, it will be shown here that the migration velocity can be a function of the radial position. This is the reason for the notation u(r). The temperature dependence of the velocity arises through viscosity changes
infinitesimally narrow plug, the boundary and initial values are C(O,x,r) = 0
C(O,O,r) = c,
C(t,m,r) = 0 u ( R i ) = u1 dC(t,r,R,)/dr = 0
(5)
77 = A e x p ( B / n
where A and B are constants. Therefore, the velocity can be written as
u(r) = (tot,\EEexp(-B/T))/A
(6)
Expanding the exponential part of eq 6 as a Taylor series around T = T1and retaining the first two terms yields
dC(t,x,O)/dr = 0 We used the approach of Gill and his co-workers (e.g., ref 7) to develop an expression for the dispersion coefficient of the concentration profile described by eq 12. Availability of this dispersion coefficient permits the development of an expression for the plate height (detailed derivation of the plate height, H, equation can be found in the Appendix)
RI6E4Cb2B2A2~
20
Equation 2 allows the temperature difference T - T1to be expressed in terms of the radial position, so that the electrophoretic migration velocity expression becomes
-(
u(r) = ul[ 1 + GBR12 1 4klTl2
I)$-
(8)
where u1is the velocity a t the wall (at Tl). Equation 8 shows that two additive terms contribute to the overall electrophoretic velocity: a flat velocity term, ul,and a parabolic velocity profile that arises from temperature variations across the tube. Equation 8 is similar to the parabolic equation derived by Hjert6n (2) in a somewhat different way. The heat generated within a capillary can be described as
G = VI/*Rl2L
(9)
where I is the current and V is the voltage drop across the length, L, of the capillary. Note that V/L is the field strength E. From Ohm's law and from the definition of the equivalent conductance, eq 9 can be rewritten as
G = ACbE2
(10)
where c b is the concentration of the buffer electrolyte solution and A is its equivalent conductance. The expression for the electrophoretic migration velocity becomes
Thus, eq 11 relates the electrophoretic velocity to many of the experimental parameters such as the concentration of the buffer, the field strength, and the capillary radius. Contribution of Electrophoretic Migration Velocity Profile to Plate Height. This study examines only the effects of temperature gradients on the plate height of CZE separations. For this analysis, we assume that solutes do not adsorb on the capillary wall or that they do not interact with the buffer components. With these constraints, contributions to the plate height are due only to molecular diffusion of the solute molecules and to the electrophoretic migration velocity profile arising from thermal effects. Thus, the mass balance can be described as at
= D[:
$(r
$)] +
D sa2-cu ( r ) ac (12) ax
where C is the concentration of the solute, D is the diffusion coefficient of the solute in the buffer, and x is position along the longitudinal axis. Assuming the solute is injected as an
H=-+ u
24D(8k1Tl2- E2ACfi:B)2
(13)
where u is the cross-sectional average of the electrophoretic migration velocity.
DISCUSSION Equation 13 explicitly shows the functionality between the resistance-to-mass-transferterm of the plate height and experimental conditions. The mass-transfer term takes into account the radius of the capillary, physical properties of the buffer (A and k J , concentration of the buffer (Cb), field strength ( E ) , and temperature. The expression in eq 13, relating the radial temperature gradient to plate height, is much simpler than that developed by Cox et al. (8),using the procedure of Aris (9). When the capillary radius is small and the applied field is not large (below 20 kV in most typical CZE systems), then the following relationship exists: AE2C&12B
