Influence of solvent on temperature and thermal peak ...

Electrophoresis 2003, 24, 1553–1564

1

Institute for Analytical Chemistry, University of Vienna, Vienna, Austria 2 Department of Physical Chemistry, Faculty of Science, Charles University, Prague, Czech Republic

Influence of solvent on temperature and thermal peak broadening in capillary zone electrophoresis The present paper deals with the role of the solvent on thermal peak broadening. One main solvent property that determines the magnitude of the temperature gradient due to the generation of Joule heat in capillary zone electrophoresis is the thermal conductivity. As organic solvents have lower thermal conductivity than water (methanol and acetonitrile, e.g., nearly by a factor of 3) it can be hypothesized that the temperature gradient inside the capillary is more pronounced in organic solvents compared to an aqueous solution. On the other hand, the temperature dependence of the ion mobility (which is responsible for the velocity profile and thus for thermal peak broadening) is smaller in organic solvents. To get insight into the thermal effect of the solvent, first the temperature of a solution in a cylindrical tube was calculated utilizing the heat balance equation. It was shown that the two theoretical models most common in the literature (based on the analytical solution or on an assumption of the parabolic temperature profile in the tube, respectively) give the same results. The latter model was chosen for the further calculations, adding a quadratic term to express the electric conductivity as a function of the temperature. The temperature at the inner capillary wall and center as function of the capillary dimensions and the electric power was computed for electrolytes with a given conductivity at 25.07C with water, methanol, and acetonitrile as solvents. Capillary cooling systems used were circulating liquid cooling, enforced air-cooling, and natural convection in still air. The mean temperature (averaged over the cross section) resulting from Joule heating was compared with experimentally determined temperatures established upon application of an electric field; the latter temperature was derived from the measurement of the electric conductance of the background electrolyte solution and its (measured) temperature dependence. All investigations were carried out with solutions of the same initial electric conductivity (about 0.5 S?m21 at 25.07C). Agreement is found for natural convection conditions, and the deviation between theoretical and experimental results for the forced air and circulated liquid cooling systems can be related to the poorly defined thermal conditions of the capillaries in commercial instrumentation (with a part in a thermostated cassette and a part outside). For given conditions the temperature gradients in the organic solvents exceed largely those in water, independent of the type of cooling. As a consequence, the thermal plate height is significantly larger in organic solvents, at least under conditions where the deviation from the Nernst-Einstein limiting case is not too high. However, even for the maximum applicable field strengths the thermal plate height contributions are negligible compared to longitudinal diffusion in all solvents. Keywords: Capillary zone electrophoresis / Efficiency / Nonaqueous solvents / Organic solvents / Peak dispersion / Plate height / Thermal effects DOI 10.1002/elps.200305437

1 Introduction Peak dispersion in capillary zone electrophoresis (CZE) stems from a number of sources. In addition to extracolCorrespondence: Prof. Ernst Kenndler, Institute for Analytical Chemistry, University of Vienna, Währingerstrasse 38, A-1090 Vienna, Austria E-mail: [email protected] Fax: +43-1-4277-9523 Abbreviations: MeCN, acetonitrile; TPACl, tetrapropylammonium chloride

 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

umn effects (that are not depending on the movement of the sample zone), other dispersion processes are occurring during migration. They are caused from longitudinal diffusion of the solutes, from the difference of analyte and co-ion mobility of the background electrolyte (electromigration dispersion), from adsorption of the analytes, from laminar flow, from coiling of the capillary, from thermal effects arising due to Joule heating of the solution. All dispersion effects have been described theoretically (see, e.g., recent reviews [1, 2]), and the influence and relevance of the many parameters that determine the separation efficiency are well understood. 0173-0835/03/1005–1553 $17.501.50/0

CE and CEC

Simo P. Porras1 Ettore Marziali1 Bohuslav Gasˇ2 Ernst Kenndler1

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The knowledge of the physical basis of these dispersion phenomena can be utilized to evaluate the efficiency of CZE in nonaqueous solvents, which have attracted the interest of the CE practitioner since the beginning [3, 4]. So far much more effort has been devoted to the aspects of selectivity upon change of the solvent than to efficiency. However, we have previously shown – based on the theory of conductance and of diffusion in dilute electrolyte solutions – that the contribution of longitudinal diffusion to zone broadening is significantly higher in most organic solvents compared to water [5]. This has fundamental reasons and is independent of the chosen experimental conditions. The loss of efficiency with increase in ionic strength of the background electrolyte (BGE) is caused by the reduction of mi/Di, the ratio of the mobility, mi, and the diffusion coefficient, Di, of ion, i. In many organic solvents this ratio decreases more strongly with ionic strength than in water. This is due to differences in viscosity and relative permittivity of the solvents. The basic difference in maximum achievable plate numbers at finite ionic strengths leads to the question about the potential influence of the solvent on the thermal contribution to peak broadening in CZE. The basis of the problem lays in the fact that the Joule heat is generated in the whole volume of the electrolyte inside the capillary upon passage of the electric current, while the heat loss is accomplished by the capillary wall. After a certain time a balance is reached and the radial temperature profile attains a steady state. The entire electrolyte solution in the capillary exhibits a higher temperature than the environment and, moreover, it has a nonuniform radial temperature profile. This profile leads to an axial dispersion of the analytes due to the dependence of mobility on temperature. Heat transfer from the capillary column in electromigration methods has been a subject of a number of papers. Fundamental work has been made by Virtanen [6] and Grushka et al. [7, 8], who formulated a model (named further the classical model) of thermal effects responsible for increase in temperature and peak broadening. This model is based on the assumption that the temperature profile in the capillary has a parabolic shape. Knox and McCormack [9, 10] added a discussion of the excess temperature and the temperature gradient in the bore of the capillary of various dimensions using this model. Gobie and Ivory [11], on the other hand, derived an analytical solution of the heat transfer problem giving a complete steady state temperature profile in the solution, capillary wall and possible capillary coating. Their model (we name it the analytical model) supposes a linear dependence of electric conductivity on temperature. The

Electrophoresis 2003, 24, 1553–1564 radial temperature profile in the solution is given by the Bessel functions. Under CE conditions it is, however, very close to the parabolic one [8]. The assumption of the linear dependence of conductivity on temperature was utilized by Bello and Righetti [12, 13], who brought the mathematical solution of the unsteady heat transfer in the cylindrical geometry. To prove the predictions of the theoretical calculations, experiments have been performed to determine the temperature of the solution, and the temperature profile. For these purposes thermochromic solutions [14], Raman spectroscopy [15, 16], or NMR thermometry [17] have been utilized. In most cases a good accordance between theory and experiments was observed, though some discrepancies remained. As in the theory of heat transfer in electrophoretic capillaries several solvent-specific parameters occur, we anticipate in the present work a dependence of the magnitude of the thermal plate height contribution on the solvent. Predominantly the thermal conductivity of the solvent, kL, is known to be significantly smaller for organic solvents (see Table 1) than for water. The preliminary hypothesis seems thus reasonable, namely that, e.g., in methanol (MeOH) or acetonitrile (MeCN) thermal peak dispersion should be more pronounced than in water due to a lower heat transfer. It is known, on the other hand, that the electric conductivity of electrolytes has a different dependence on temperature in the various solvents. The critical reflection of these facts and an experimental proof of the temperature effects in organic solvents in comparison to water is the topic of this work. The effects of the solvents will be rationalized concerning three types of cooling: forced circulation of a liquid, forced air-cooling, and still air (with natural convection). These types are chosen not only for theoretical reasons (circulating liquid has the highest, free convection of air the lowest thermostating performance), but also due to their practical importance. Commercial instrumentation applies a forced circulation of either air or a liquid, whereas homemade apparatus often consist simply of a capillary positioned in air without further cooling devices. Moreover, it must be pointed out that in all commercial instruments a section of the separation capillary is kept in still air, namely the part outside the thermostated cassette. This causes a thermal effect that should be considered in relation to the entire thermostating system. This is one more reason why still air conditions are included in the present investigation. The present paper deals with the most common organic solvents in CZE, namely MeOH and MeCN, and compares the effects with those in aqueous solutions.


Solvent effects on temperature in CZE

Table 1. List of relevant parameters Symbol

Parameter

Dimension

Experimental values

B

Constant in equation for dynamic viscosity as function of T: Z = C exp (B/T)

K

Water 1600; MeOH 1235; MeCN 870.5. Calculated from data taken from [22, 23]

D

Diffusion coefficient of the analyte

m2 s21

E

Intensity of the electric field

Vm21

G

Total (or overall) electric conductance of solution

S

Hdiff

Plate height from longitudinal diffusion

m

Htherm

Thermal plate height

m

hO

Overall heat transfer coefficient

Wm22K21

hS

Surface heat transfer coefficient

Wm22K21

Still air 130; forced air 4000; circulating liquid 10 000 [18]

I

Current

A

0–361024

kP

Thermal conductivity of outside coating

0–80 000

21

21

0.155 (polyimide)

21

21

Water 0.61; MeOH 0.2; MeCN 0.188 [20]

WK m

kL

Thermal conductivity of solvent

WK m

kW

Thermal conductivity of capillary wall

WK21m21

1.5 (silica)

rL

Inner radius of capillary, lumen radius

m

12.561026 – 10061026

rW

Outer radius of capillary without coating, capillary wall radius

m

167.561026

rP

Outer radius of capillary with coating, coating radius

m

187.561026

Ts

Temperature of surrounding coolant

K

298.15

TL

Temperature at inner wall of capillary

K

Tcenter

Temperature at lumen center, temperature at center of capillary

K

Tdiff

Temperature difference between lumen center and inner wall of capillary

K

Tmean

Mean temperature of solution (averaged over cross section)

K

U

Voltage

V

0–30 000 21

vi

Mean velocity of the analyte

m?s

z

Charge number

2

a

Slope of linear function k = f(T)

T21

Z

Dynamic viscosity

kg?m21s21

kTmean

Electric conductivity at mean temperature Tmean

S?m21

kTs

Electric conductivity of solution at temperature Ts

S?m21

0.517 (TPACl in MeOH and MeCN; LiCl in water) See Eqs. (8) and (10)

l

Autothermal parameter

2

mi

Ionic mobility of the analyte

m2V21s21

y

Dimensionless temperature

y = (T – Ts) /nTref, for nTref see Eq. (9)

r

Dimensionless radius

r = r/rL

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2 Materials and methods 2.1 Chemicals All chemicals, except water, were used without further purification. HPLC-grade acetonitrile was from Mallinckrodt Baker (Deventer, Holland) with an initial water content of 0.01% (specified by the manufacturer) and with a measured conductivity of 361025 S?m21 at 25.07C. Methanol was from E. Merck (Darmstadt, Germany) with a maximal initial water content of 0.02% (specified by the manufacturer) and with a measured conductivity of 761025 S?m21 at 25.07C. Water was double-distilled and had a conductivity of 1.561024 –2.061024 S?m21 at 25.07C. Lithium chloride was from E. Merck, tetrapropylammonium chloride (TPACI) from Fluka (Buchs, Switzerland).

2.2 Instruments The following CE instruments were used: (A) a CE system (CAPEL-105; Lumex, St. Petersburg, Russia) with liquid cooling for the capillary cassette and a power supply capable of delivering maximum 25 kV and 200 mA; (B) a 3D CE (Hewlett-Packard, Waldbronn, Germany) instrument with forced air cooling for the capillary cassette and a power supply capable for maximum 30 kV and 300 mA, (C) a laboratory-made system consisting of a power supply (Heinzinger, Germany) delivering maximum 20 kV and 200 mA. In the latter system the capillary was placed in still air. With the instruments (A) and (B), the temperature of the capillary cassette was set to 25.07C. Instrument (C) was operated in a laboratory room with controlled temperature of 25.060.57C. Uncoated fused-silica capillaries with 34.0 cm total length and ID/OD of 50/375, 75/375, 100/375, and 180/350 mm were purchased from Composite Metal Services (Hallow, UK).

2.3 Procedures Due to solubility reasons two different salts had to be used as BGEs. The conductivity of the solutions of TPACl (in MeCN and MeOH) and of LiCl (in water) was adjusted to 0.517 6 0.001 S?m21 at 25.07C. Concentrations of salts were 50 mmol?L21 of TPACl in MeCN, ,96 mmol?L21 of TPACl in MeOH, and ,55 mmol?L21 of LiCl in water. Approximated concentrations of methanolic and aqueous BGEs were calculated after dilution of more concentrated stock solutions. Conductivities of the pure solvents and BGEs were measured with a laboratory conductivity meter (inoLab Cond Level 2) and TetraCon 325 standard conductivity cell (both from WTW Mess- und Analysengeräte, Vienna, Austria) in a temperature-controlled water bath. The temperature dependency of the conductivity was measured in the temperature range of 20.0–60.07C

Electrophoresis 2003, 24, 1553–1564 (20.0–50.07C for MeOH solution). The length-averaged temperature inside the capillary was measured indirectly from the conductance of the BGE calculated from Ohm’s law. For this purpose the voltage, V, applied over the capillary length was increased stepwise and the resulting current, I, was recorded. From the conductance, G, determined from the ratio I/U, and from the capillary dimensions, the temperature could be obtained with the aid of the calibration curve of conductivity vs. temperature measured with the external conductimeter (note the relation of G and conductivity, k: G = kL/A, with L being the capillary length and A the cross sectional area of the capillary).

3 Results and discussion 3.1 Different theoretical models for calculating temperature increase It seems to be generally accepted that only the analytical solution presented first by Gobie and Ivory [11] is able to yield correct theoretical values of the temperature stemming from basic laws of the heat transport. We will demonstrate in this section that the approach supposing a parabolic radial temperature profile in the lumen according to Grushka et al. [7, 8] (denoted as the classical approach) gives almost identical results as the analytical approach. We suppose that the geometry of the capillary column is cylindrical, with following parameters: rL, rW, and rP are the lumen, capillary wall, and coating radii; kL, kW, and kP are the thermal conductivities of the solvent, capillary wall, and coating, respectively (for abbreviations and symbols, see also Table 1). The heat balance is reached when Joule heat generated in the solution inside the capillary due to the electric energy equals the heat loss due to the transport across the capillary wall. While the Joule heat is easy to calculate, the correct calculation of the heat loss is rather tedious. A measure of the heat loss is the overall heat transfer coefficient, which is given by 1 lnr W =r L lnr P =r W 1 (1) ho r L kW kP r P hs where hS is the surface heat transfer coefficient, which depends on the cooling conditions of the capillary. It has radiative and convective components. The radiative component is nonlinearly dependent on the temperature. As it is relatively small in comparison with the convective component, approximating values can be taken for certain typical cooling conditions: for a horizontal cylindrical capillary in still air hS = 130 Wm22K21, for a capillary intensively cooled by circulating liquid hS = 10 000 Wm22K21; for intensively forced air-cooling (with a velocity of the air flow of about 10 m/s) we take a value of 4000 Wm22K21 (data from [18]).



It is seen from Eq. (1) that the heat loss depends on the capillary dimensions, the capillary material, and the type of cooling. It is independent of the solvent used for the BGE. Solvent-dependent, in contrast, is the heat generated at a certain applied electric field strength, because the temperature dependence of the electric conductivity of an electrolyte solution differs for the particular solvents. We will denote the calculated mean temperature and mean electric conductivity of the solution averaged over the cross section as Tmean and kmean, respectively. TS , TL , and Tcenter are the temperatures of the surrounding coolant, the inner wall of the capillary, and the lumen center, respectively. kS is the electric conductivity of the solution at TS.

3.1.1 The classical approach If a parabolic radial temperature profile in the capillary lumen is supposed [7, 8], the governing equations are: E 2 kmean ho Tdiff

2 TL rL

TS

(2)

E 2 kmean r 2L 4k L

(3)

Tcenter TL Tdiff

(4)

Tmean TL Tdiff =2

(5)

kmean kS 1 aTmean

Ts

3.1.2 The analytical approach According to Bello and Righetti [12, 13, 19] the dimensionless temperature y = (T 2 Ts) /Tref and its dependence on the dimensionless radius r = r/rL in the capillary lumen is

A

1=l2

1

BiOA l2 BiOA J0 l lJ1 l E 2 kmean r 2L kL

is the characteristic temperature rise, and p l aDT ref

3.1.3 Steady-state temperature derived by both models Comparison of the results of both models was made for a variety of experimental parameters. A simple numerical example gives a more concrete impression about the calculated temperature and can also serve as a reference for a comparison with other models published by other authors. The input data for a silica capillary cooled by still air with temperature TS = 298 K and with natural convection (hS = 130 Wm22K21) are: rL = 50 mm, rW = 177.5 mm, rP = 187.5 mm, kL = 0.61 Wm21K21, kW = 1.5 Wm21K21, kP = 0.155 Wm21K21, k = 0.5 S?m21, a = 0.02 K21, E = 30 000 Vm21. The temperatures in the capillary center and the inner wall predicted by both, the classical approach (Section 3.1.1) and the analytical approach (Section 3.1.2), are Tcenter = 344.53 K and TL = 343.64 K. It is found that the calculated temperatures agree within five significant figures (two decimals in absolute temperature).

3.2 Calculated temperature inside the capillary as function of solvent There is one drawback of the analytical approach for calculating temperatures: the governing equations were derived when supposing a linear dependence of electric conductivity on temperature. This assumption is not entirely fulfilled, as the deviation from linearity could be substantial, especially over a temperature range of several tens of degrees. As we intended to show the thermal effects in the broad range of temperatures, and for different solvents, we adopt a more accurate function. Thus, a quadratic term is added to the dependence, so that the electric conductivity is expressed as h i (11) kmean kS 1 P1 Tmean TS P2 Tmean TS 2

(7) (8)

where DTref

is the autothermal parameter; BiOA = hOrL / kL is the overall Biot number, J0 and J1 are the Bessel functions of the first kind and the zero-th and first order, respectively. For r = 0 and r = rL the Eqs. (7), (8) give the temperatures Tcenter and TL, respectively. Tdiff is then Tdiff = Tcenter 2 TL.

(6)

where E is the electric field strength and Tdiff is the temperature difference between the lumen center and the inner capillary wall. a is the slope in the linear dependence of conductivity on temperature. Equations (2)–(6) form a simple system of algebraic equations for TL, Tcenter , Tdiff , Tmean , and kmean , which is linear in the voltage-stabilized mode (E = const.), and which is easy to solve.

y AJ0 lr

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(9)

(10)

Parameters P1 and P2 can be derived using values of the conductivity at different temperatures from the literature. However, for consistency between the calculated conductivity values of the solutions and those experimentally determined in the present work, we use the data for the conductivity of the solutions in dependence on the temperature measured by the aid of an external conductimeter. The resulting parameters P1 and P2 are given in Table 2. Note that the terms differ significantly for the par-

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Table 2. Parameters P1 and P2 for Eq. (11) describing the dependence of the electric conductivity of the BGE on the temperature Solvent

P1 , K21

P2, K22

Water MeOH MeCN

0.02074 0.01261 0.00799

4.361025 ,55 mmol?L21 LiCl 2.861025 ,96 mmol?L21 TPACl 28.261026 50 mmol?L21 TPACl

BGE

ticular solvents. For the given BGE, e.g., the coefficient P1 depicting the linear term is about 0.021 for water, 0.013 for MeOH and 0.008 for MeCN. It is seen that in water this parameter is by a factor of 1.6 higher than in MeOH, and nearly 3 times larger than in MeCN. The classical approach can be easily utilized even in cases when the conductivity is nonlinearly dependent on temperature. We used it therefore for our calculation, the only change is the replacement of Eq. (6) by Eq. (11).

We considered electrolyte solutions with identical electric conductivity (0.500 S?m21at 25.07C) in the absence of an electric current in all solvents. This data is taken as we use about the same conductivity for experimental prove (0.517 S?m21, see below). Under these experimental conditions the electric power is the same for all solvent systems at the reference temperature of 25.07C. This arrangement was preferred over that with identical concentrations of the electrolytes, because the different mobilities of a particular electrolyte in the different solvents (see Walden’s rule) would otherwise lead to deviant initial power conditions. The temperature as a function of the applied field strength was computed by the aid of Eqs. (2)–(5) and (11). As one of the equations is a quadratic one, a commercial computer solver software can serve for this purpose or the explicit solution can be easily found. The resulting mean temperatures Tmean are depicted in Fig. 1 for water, MeOH and MeCN for the three cooling modes. Outer diameter

Figure 1. Effect of the solvent on the mean temperature, Tmean , of the BGE solution in dependence on the electric field strength, calculated from the heat balance equation. Data are calculated for 25 and 200 mm ID of the capillary (375 mm OD), for three different cooling systems: circulating liquid, forced air and natural convection in still air. Conductivity of the BGE at 25.07C: 0.500 S?m21; values for P1 and P2 see Table 2. Each data point is the result of iterative calculation.

Electrophoresis 2003, 24, 1553–1564 of the capillary including the polyimide layer is 375 mm, the thickness of the outer polyimide layer is 20 mm. We give here the results only for two inner diameters, namely those that may confine the practical range: 25 and 200 mm, respectively. Capillaries with IDs smaller than 25 mm or larger than 200 mm are rarely used in practice. It can be seen that the temperature rises only by a few degrees with circulated liquid cooling, and slightly more with forced air. Even in the wide-bore capillary the temperature increases by not more than 87C at the high field strength of 30 000 V/m. Higher field strengths are not routinely used in practice due to the limited currents the power supplies can normally deliver. Indeed, in the 200 mm ID capillary this current would be exceeded with the electrolyte solution under consideration. It is obvious that temperature increase is most pronounced when the capillary is in still air with natural convection; under this condition the solvent is heated to such a large extent that boiling would occur before the highest possible voltage in usual instrumentation (30 kV) could be applied (the boiling points of MeOH and MeCN are 64.57C and 81.67C [20], respectively). The calculated temperatures for aqueous solutions are in accordance with literature data [7, 9, 10, 21]; the effect of organic solvents however, has not been quantified so far. We can draw the following conclusions concerning the effect of the solvent. In the circulating liquid and air-cooling systems, the temperature increase is higher for a given field strength with the organic solvents compared to water. Interestingly, a reversal of the effect is observed under still air conditions, where water shows the higher temperature at a given field strength than the organic solvents.

3.3 Experimentally obtained temperature, measured from increase of electric conductance In order to prove the above computed data, the temperature in the capillary after application of an electric current is measured indirectly. This is done from the change in total conductance of the electrolyte solution across the whole capillary length in the CE setup. The total electric conductance, G, of the solution is derived from the resulting current at a certain applied voltage, simply by Ohm’s law: G = I/U. If temperature does not change, this ratio should be constant for a given experimental setup. Deviations from the constant value are related to temperature changes, which are expressed by the quadratic Eq. (11) for the conductivity with the corresponding values for P1 and P2 given in Table 2. The reference value of the electric conductance (corresponding to the conductivity kS) of the BGE in the capillary at the temperature of the surrounding


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(25.07C) is determined from the ratio I/U at a voltage low enough that generation of heat is negligible. In the present case this was done by applying a voltage of ,1 kV. It should be pointed out that the experiments arranged in this way give the temperature averaged over the capillary length, whereas the calculation delivers the mean temperature, Tmean , averaged over the cross section of the capillary. These two temperatures should be strictly distinguished. The BGE was a solution of TPACl in MeOH (,95 mmol?L21) and MeCN (50 mmol?L21), and of LiCl (,55 mmol?L21) in water with identical conductivity at 25.07C (0.517 S?m21, measured with an external conductimeter, see experimental part). The temperature inside the capillary obtained experimentally is depicted for water, MeOH and MeCN in dependence on the applied voltage for capillaries with different ID in Fig. 2 (full symbols). These temperature data were indirectly derived from the electric conductance data related to the current at a given voltage according to G = I/U (see above). In the same figures the dependence of the temperature on the voltage calculated by Eqs. (2)–(5) and (11), based on the classical approach, is given as well (empty symbols), with the same conductivity (0.517 S?m21 at 25.07C) as in experiment as input parameters for calculation.

3.3.1 Natural convection in still air Under still air condition a large increase in temperature is found even at relatively low voltage (Fig. 2). It should be pointed out again that the upper applicable voltage is given by two limitations: (i) by the boiling point of the liquid which is reached due to excessive heating, and (ii) by the limited current, especially in wide-bore capillaries, given by the capacity of the power supply unit. In all cases the measured temperatures are in rather good accordance with the theoretical points calculated from the classical approach by iteration. Concerning the influence of the solvents we find that the aqueous solution shows the largest temperature increase in still air for a given voltage compared to the organic solvents. This experimental finding is in agreement with the results of theoretical calculation, confirming to the effectiveness of the theoretical model used. The only exception is in 180 mm ID capillaries, where no difference between the solvents can be found. Here all three solvents show the same temperature increase.

3.3.2 Forced circulating air or liquid The cooling systems with circulating liquid or air lead to about the same temperature increase at a given voltage (Fig. 2), which means that the cooling effects are compar-

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Figure 2. Temperature of the electrolyte solution in the capillary averaged over the capillary length (full symbols), derived from measured conductivities, and mean temperature, Tmean , averaged over the capillary cross section (empty symbols). Tmean was calculated from the heat balance equation by iteration (see text). Initial conductivity of the BGE: 0.517 S?m21 at 25.07C. OD 375 mm, except for 180 mm ID capillary. For BGEs, see Table 2. Solvents: (u, j) MeCN, (n, m) MeOH; (s, d) water.

Electrophoresis 2003, 24, 1553–1564 able. Not surprisingly, the increase is much lower compared to still air. However, a considerably large deviation from the predicted values is observed. A source for this remarkable deviation in practice is given by a constructive item of the commercial instruments: a part of the capillary is neither dipping into the BGE solution, nor placed in the cooled cartridge. Indeed, in one of the instruments (with forced air-cooling) the part of the capillary between the thermostated region in the cartridge and the BGE solution is about 0.095 m at the injector side, and 0.070 m at the detector side, which in total is nearly 50% of the 0.340 m tube. In this part the cooling conditions are poorly defined, but they may resemble natural convection in still air. As the overall temperature effect is composed from those in the particular serial of capillary segments having different temperatures, it is clear that the resulting temperature averaged over the length is significantly higher than the temperature in the thermostated cartridge. The result indicates that this contribution from the noncooled parts of the capillary is often underestimated in practice. The solvent influences the temperature for the liquid cooling system as predicted by theory: temperature is higher (at a given voltage) in the organic solvents than in water. In the forced air-cooling systems this sequence is not as clear. The reason might be in the inverse of the sequence under still air compared to the circulating fluid systems (see above). As in practical instrumentation both systems are present, a mixed effect might finally occur.


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Figure 3. Temperature difference, Tdiff, between the center of the capillary and the capillary inner wall for water, MeOH and MeCN as solvents. Data calculated from the heat balance equation for 25 and 200 mm ID, respectively. Capillary length, 0.340 m; OD, 375 mm. Conductivity of the BGE, 0.500 S?m21 at 25.07C. In the case of still air in 200 mm ID capillary, the boiling point of the solvent is reached before application of 10 kV.

MeCN. About the same experimental conditions as above are considered: fused-silica capillary, length 0.340 m, IDT 25 and 200 mm, OD 375 mm; electric conductivity of the BGE solution, 0.500 S?m21. For simplicity, we depict only the values for the system with circulating liquid cooling and still air, because the forced aircooling system delivers values that are similar to that which uses a liquid.

3.4 Plate height contributions Although temperature can increase drastically, it must be clarified to what extent these changes influence peak dispersion stemming from thermal effects, and whether the influence differs for the particular solvents. Thus, the plate height contribution from the thermal effects has to be elucidated for the different experimental setups. This is done in the following with the BGEs for which the temperature measurements were carried out.

3.4.1 Temperature gradients in the different solvents We suppose that a parabolic temperature gradient is formed due to the difference Tdiff between the center of the cylinder with temperature, Tcenter, and the inner wall with temperature, TL. Note that in the present cases Tdiff derived from the classical approach is almost identical with that stemming from the analytical solution. The difference between Tcenter and TL, as calculated by Eqs. (2)–(5), and (11) is shown in Fig. 3 for capillaries with different IDs in dependence on the applied voltage in water, MeOH and

It can be seen that for both capillaries the temperature difference is significantly higher in the organic solvents than in water, regardless which type of cooling is used. The difference Tdiff is smaller than 17C in the narrow tube even at highest applied voltage, but it reaches several degrees in the wide-bore capillary. This relatively small difference is in sharp contrast to the overall temperature increase, which could lead to boiling of the liquids under extreme conditions. However, not the overall increase influences peak broadening, it is the temperature gradient inside the capillary. This consequently forms the gradient of the ionic mobility and thus the different electrophoretic migration velocity as a function of the radial ordinate, resulting in thermal peak dispersion.

3.4.2 Plate height: thermal peak broadening in relation to diffusional one The plate height caused by the temperature gradient inside the capillary has restricted meaning when not set into relation to other dispersive effects. All broadening effects can be potentially suppressed except the inevita-

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ble longitudinal diffusion. We compare therefore the thermal plate height with that caused by longitudinal diffusion, the latter given by Hdiff

2Di vi

(12)

In order to facilitate the comparison, we consider the situation where no electroosmotic flow occurs. In this case the analyte velocity, vi, is determined by its mobility: vi = miE. Grushka et al. [7] derived the following expression for the plate height Htherm using the according mass balance equations Htherm

r 6c E 4 k2Tmean B2 24Di 8T 2W k S E 2 kTmean r 2c B2

vi

(13)

B is the coefficient in the exponential term of the dependence of the dynamic viscosity, Z, of the solvent on absolute temperature according to Z = C exp (B/T), where C is a certain constant. The coefficient B in Eq. (13) relies on the Walden’s rule stating that the electric conductivity is indirectly proportional to the viscosity of the solvent. We have fitted the experimental data of the temperature dependence of the viscosity for the three solvents used (data taken from [22, 23]), and obtained the following coefficients B: 1600 for water, 1235 for MeOH, and 870.5 for MeCN. Upon combination of Eqs. (12) and (13) (without the second term in parentheses, which can be neglected for not too large radii and not too high field strengths), we obtain the ratio between Htherm and Hdiff Htherm r 6c E 4 k2Tmean B2 2 r 6c E 6 k2Tmean B2 mi 2 v (14) i Hdiff Di 3072T 4W k 2S D2i 3072T 4W k 2S At infinite dilution (indicated by suffix 0) the ratio mi/Di is independent of the solvent, and is given by the NernstEinstein relation mo,i /D0,i = zi F/RT (F is the Faraday constant, R is the gas constant, zi is the charge number of the ion). For zero ionic strength, Eq. (14) reads Htherm r 6c E 6 k2Tmean B2 F 2 2 zi Hdiff 3072T 6W k 2S R2

(15)

We take in a first step m/D as constant and equal to m0,i /D0,i (the influence of the ionic strength in the different solvents will be discussed later). The resulting ratio of thermal to diffusional plate height is given in Fig. 4 for a monocharged analyte (zi = 1) with the BGEs used in the present work for water, MeOH and MeCN. The 75 mm ID capillary is chosen because it was found that in practical experiments the highest temperature gradients were reached here (in capillaries with smaller ID the voltage limit of the power supply was reached before such high temperature gradients were established; vice versa, in tubes with

Figure 4. Thermal plate height, Htherm, related to that caused from longitudinal diffusion, Hdiff, for water, MeOH and MeCN as solvents, and with two different types of cooling. Data calculated according to Eq. (15) for analyte charge number z = 1, using the Nernst-Einstein relation between mobility and diffusion coefficient. Conditions: capillary length, 0.340 m, 75 mm ID, 375 mm OD; no electroosmotic flow. For BGEs, see Table 2. Note that above these maximum voltages the solvents start boiling.

larger inner diameter the applicable current limited the conditions). It can be seen that for a given voltage indeed the thermal plate height contribution is much higher for the organic solvents compared to water, confirming our initial hypothesis. However, it can also be seen that thermal peak broadening is negligible compared to diffusional dispersion; Htherm is maximally only about 15% of Hdiff at 25 kV in MeOH and MeCN. Such low thermal plate height contribution was also calculated for other capillaries with larger diameter (data not shown), at least under the practical conditions normally used (capillary length several ten centimetres, applied voltage 30 kV maximum, limited power of the instrument). This result is in agreement with data recently found in nonaqueous propanolbased BGE [24]. For conditions with finite ionic strength, we have to consider that (mi/Di) in Eq. (14) is not constant, and that the deviation from constancy depends on the solvent. We have pointed out in a previous paper [5] about the solvent effect on the plate height stemming from longitudinal diffusion that this ratio decreases to a different extent with increasing ionic strength in water, MeOH and MeCN. Indeed, the decrease of the ratio m/D is less manifest in water, and much steeper in most of the organic solvents. This follows from the different retardation mechanism of an ion in the presence of an ion cloud: in electromigration the electrophoretic and the relaxation effect retains the ion, whereas self-diffusional transport is only reduced by the relaxation effect (see, e.g. [25, 26]). Although (for a monocharged ion) the reduction of the ratio mi/Di for BGE ionic strengths of say 50 mmol?L21 is only few % in water,

Electrophoresis 2003, 24, 1553–1564 it might reach 40–50% in MeOH and MeCN. What is the consequence of this difference for the thermal peak dispersion in comparison to the diffusional one? As mi /Di for a given ionic strength is smaller in the organic solvents than in water, (mi /Di)2 and thus the ratio of Htherm to Hdiff (Eq. 14) is smaller at finite ionic strength than at infinite dilution. This allows the conclusion that thermal peak dispersion is even less evident compared to diffusional dispersion under the conditions described so far. The steeper decrease of the ratio Htherm to Hdiff, with increasing ionic strength, as predicted for organic solvents, on the other hand, could lead to an elimination of the more pronounced thermal peak broadening contribution compared to water. For more extreme conditions (higher field strengths, shorter capillaries, wider IDs, high charge number of the analyte) thermal peak broadening could be more distinct. It should be noted, however, that modification of the CE instrument is needed for more extreme electric field strengths than those typically used in CE (see, e.g., [27, 28]). Also, in order to use capillaries wider than about 200 mm ID, either restrictors [21], larger BGE vials [21] or other special arrangements (see, e.g., [29, 30]) are needed to suppress Taylor peak dispersion of a hydrodynamic flow.

4 Concluding remarks (i) The application of the analytical approach (according to Bello and Righetti, [12, 13]) leads to the same calculated temperature effects as the classical model (given by Grushka et al., [7, 8]). The latter, simpler model is thus adequate to compute the temperature increase and the radial temperature gradients for different solvents. (ii) Organic solvents have a lower thermal conductivity than water. As a consequence, the temperature gradient and the thermal plate height are larger in MeOH and MeCN than in water for a given capillary and electric conductivity (at the reference temperature) of the BGE. (iii) In organic solvents like MeOH and MeCN self-heating is more pronounced compared to water, at least in enforced cooling systems. Significant temperature increases inside the capillary due to Joule heating can be observed. Such temperature variations certainly lead to changes of the analyte mobility and therefore the migration time in CZE, affecting thus both, the accuracy and the reproducibility of the migration parameters. It is also conceivable that such temperature changes (reaching several ten degrees under natural convection in still air) could influence other parameters decisive for electrophoretic movement like pH and/or equilibrium constants, e.g., pKa values or complexation constants.


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(iv) In commercial instruments with thermostated cartridges, the measured temperature increase is much higher than that derived from the heat balance equation based on the conditions within the cartridge. The reason for this deviation is that a relevant fraction of the capillary is placed outside the cartridge, which is in part exposed only to natural convection in still air. This fraction contributes markedly to the length-averaged temperature of the solution. (v) In general thermal plate height contribution is higher for the organic solvents than for water for BGEs with the same electric conductivity (at the reference temperature), at least for not too high ionic strengths of the BGE. However, under most practical conditions the effect is so small that thermal heating does not lead to a significant loss in peak efficiency compared to longitudinal diffusion, although temperature can rise extraordinarily. This influence on peak efficiency is negligible for all solvents under consideration. It could gain, on the other hand, more relevance when wider capillaries are used or larger electric power is applied (or when the analyte has a large charge number). (vi) In this context a statement frequently found in the literature must be questioned critically, namely that BGEs with organic solvents allow application of higher field strengths in comparison to aqueous solutions. In general, this argument is not justified. It is based on the lower conductivity the BGEs often have in organic solvents. The reason for this is not caused by the generally lower mobility of the ions in organic solvents. In contrary, in MeCN most ions have higher mobilities than in water. Lower conductivity of BGEs in organic solvents is often the result of lower ion concentrations selected, either from lower total electrolyte concentrations, or from lower ionic dissociation. Donation of the Ernst Mach fellowship for S. P. P. from the Austrian Academic Exchange Service is acknowledged. Received March 12, 2003

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