Temperature-Insensitive Electrokinetic Behavior in Capillary Zone

insensitive behavior is caused by the cancellation of the temperature effects as stated in Walden's rule. A series of capillary zone electrophoresis e...
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J. Phys. Chem. B 2004, 108, 17685-17693

17685

Temperature-Insensitive Electrokinetic Behavior in Capillary Zone Electrophoresis Chieh-Kuang Wang and Heng-Kwong Tsao* Department of Chemical and Materials Engineering, National Central UniVersity, Jhongli, Taiwan 320, R.O.C. ReceiVed: June 27, 2004; In Final Form: August 27, 2004

Capillary zone electrophoresis generally suffers the Joule effect at relatively high driven voltage owing to the temperature control problem. The electro-osmotic and electrophoretic velocities are greater than those predicted by the well-known Smoluchowski equation when the Joule heating is significant. It is attributed to the reduction of the solvent viscosity caused by the temperature increase. A simple analytical theory is derived to show that when the electric current is steady, the electroosmotic flow remains pluglike regardless of the temperature profile inside the capillary due to Joule heating and nonuniform surrounding convection. Moreover, the electroosmotic (or electrophoretic) velocity is always linearly proportional to the electric current. This temperatureinsensitive behavior is caused by the cancellation of the temperature effects as stated in Walden’s rule. A series of capillary zone electrophoresis experiments has been performed to confirm the temperature-insensitive electrokinetic behavior. Our results suggest that the Joule effect can be circumvented by controlling the electric current even under nonuniform convective cooling.

I. Introduction Capillary zone electrophoresis (CZE) is a rapid, high resolution method for separating a variety of compounds. Under typical CZE conditions, electro-osmotic flow (EOF) always takes place because the capillary wall is charged. During recent years, EOF has been widely utilized as the driving force to transport and control nanovolume liquid samples in microfluidic systems used for chemical analysis and medical diagnosis. However, the high voltage applied to the microcapillary during electrophoresis (or electro-osmosis) causes significant Joule heating, which has been known to limit the efficiency of the electrophoretic separation for quite some time. With free-air convective cooling from the outer surface of the capillary to the surrounding medium, the liquid temperature rises above ambient and may even approach the boiling point of water. As a consequence, the thermal behavior of CZE and EOF has attracted much attention, both theoretical and experimental.1-7 Joule heating in CZE is the inherent byproduct of the electric work. The heat is generated by ohmic resistance of the electrolyte solution due to the passing electric current. From the microscopic viewpoint, the frequent collisions between migrated ions and solvent molecules convert some of the kinetic energy done by the electric field into the heat. This scenario is similar to the electrons moving through metal atoms. Under typical CZE conditions, the voltage gradient is 100-400 V/cm, and the power dissipation density may reach up to 1 kW/cm3. The experimental study based on Raman microthermometry2,3 showed a 20-40 °C local temperature rise inside a capillary cooled by free-air convection when the average power density was about 0.7 kW/cm3. Moreover, differences between local temperatures and average capillary temperatures were observed at all operating conditions.3 This result implied the presence of axial temperature distribution in the capillary. Nevertheless, it is generally recognized that the radial temperature gradient is responsible for the heat transfer out of the capillary. Under isothermal conditions, the velocity of EOF in CZE, uE, is given by the Smoluchowski equation8 * To whom correspondence should be addressed. E-mail address: [email protected].

uE ) -

r0ψs V η L

(1)

where 0 is the permittivity of a vacuum, r the dielectric constant, ψs the ζ potential, and η the viscosity of the running buffer. V is the applied voltage and L the capillary length. Equation 1 is valid when the double layer thickness κ-1 is small compared to the characteristic length of the microchannel such as the capillary radius R, i.e., κR . 1. Under the influence of an electric field E ) V/L, the electrophoretic velocity of a charged molecule with valency z, uEP, is given by8

uEP )

ze V 6πηa L

(2)

where a denotes the hydrodynamic radius of the charged solute and e is the fundamental charge. Equation 2 is valid when the solute size is small compared to the double layer thickness, i.e., κa , 1. When the cooling system surrounding the capillary is not effective enough, the Joule heating effect becomes significant and thereby the internal temperature is raised. The liquid viscosity generally declines with increasing temperature. The resistances to the fluid flow (EOF) and the charged particle motion (electrophoresis) are proportional to the viscosity and thus are reduced. As a consequence, the elevated temperature resulting from Joule heating leads to an increase in the velocity of EOF or electrophoresis, and a nonlinear dependence of the migration velocity on the electric field is observed experimentally. The heat generated by Joule heating per unit volume per unit time, q, is the product of the flux (electric current density i) and the driving force (electric field E)

q ) i·E

(3)

The current density is linearly related to the electric field by Ohm’s law

i ) K eE

(4)

where Ke represents the electric conductivity and is proportional to the ion mobility and the electrolyte concentration. For CZE

10.1021/jp0472156 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/14/2004

17686 J. Phys. Chem. B, Vol. 108, No. 45, 2004 at steady state operation, the power input, which is equal to the product of applied voltage and electric current, is consumed by the heat generated, which can also be obtained by integrating eq 3 over the capillary volume. According to the NernstEinstein and Stokes-Einstein equations, the ion mobility (or diffusivity) rises with increasing temperature. Because of the temperature-dependent nature associated with the electric conductivity, a self-heating process, called the autothermal effect,8 might take place at high electrolyte concentrations and lead to unsteady state operations. Both theoretical and experimental studies indicate that the Joule heating is significant under typical CZE conditions and results in temperature gradient in both radial and axial directions in the capillary. Consequently, quantitative analysis of the electrokinetic behavior in CZE may suffer difficulties associated with nonuniform temperature distributions caused by the surrounding cooling system, particularly free-air cooling. In this paper, on the basis of a simple analytical theory, we shall show that the Joule heating effect can be circumvented by expressing the migration velocity in terms of the electric current. Even under nonuniform surrounding convection, electro-osmosis in CZE exhibits a pluglike flow as long as the electric current can be maintained at a constant. This temperature-insensitive behavior is attributed to the cancellation of the temperature effects. For example, as stated by Walden’s rule, the product of solvent viscosity and electric conductivity is essentially constant regardless of the temperature. The series of experiments has been conducted to verify this temperature-insensitive electrokinetic behavior predicted by our theory.

Wang and Tsao phosphate-2-hydrate NaH2PO4‚2H2O (RdH, Germany) for buffer concentrations 5-30 mM. We used 0.5 M NaOH to keep the phosphate buffer solution at the same pH value (pH = 7.0). One of the simplest and most natural methods for determining the EOF velocity is to observe the migration peak of a neutral marker. We used 50 mM formamide in background electrolyte solutions as a neutral marker sample. For measuring the electrophoretic velocity, we used 5 mM anionic surfactant sodium dodecyl 4-benzene sulfonate C12H25C6H4SO3Na (Fluka, Switzerland) in background electrolyte solutions as an electrophoresis sample. Its critical micelle concentration is about 7.5 mM9 or slightly less owing to dilute salt solutions. Therefore, the surfactant should undergo electrophoresis as individual molecules. We were able to observe two migration peaks and calculated EOF and electrophoretic velocities. Note that the running buffer refers to the aqueous backgrouhd electrolyte solution and NaOH. III. Simple Analytical Theory First, we analyze the temperature profile inside the capillary owing to both Joule heating in CZE and nonuniform heat convection from the capillary wall to the ambient surrounding. Then, on the basis of Walden’s rule and Kohlrausch’s law, the electro-osmotic velocity profile is obtained for nonuniform capillary wall temperature when the electric current can be maintained at steady state. A. Temperature Profile. At steady state, the temperature distribution is determined from the conservation equation of energy

II. Materials and Methods Apparatus and Procedures. All CZE experiments were carried out using CE-L1 capillary electrophoresis system (CE Resources Pte Ltd, Singapore) equipped with a UV-vis detector (Ecom spol. S.R.O. CCD 2083). Fused-silica capillaries (Polymicro Technologies, Phoenix, AZ) coated with polyimide were used with inner diameters of 75 µm and outer diameters of 360 µm. The capillary lengths were measured 50 cm from the injection end to the detector window and 62 cm in total length. After first being rinsed with 1 M NaOH (20 min, 20 psi) and then with deionized water (10 min, 20 psi), new capillaries were used immediately. Each experiment was preceded with a rinse of 0.1 M NaOH (3 min, 20 psi) and deionized water (2 min, 20 psi), and followed by background electrolyte (3 min, 20 psi). The sample was introduced using a 0.4 psi pressure injection for 4 s. Constant voltage runs were performed at applied voltages of 5.0-25.0 kV under an air-cooling system. When the applied voltage was low enough, the capillary temperature could be effectively maintained at 25 °C. The temperature at the outer surface of the capillary was measured by the DTM 305A thermocouple (TECPEL, Taiwan). The electric current and migration time were recorded to obtain the electric resistance and migration velocity. The linear relationship between the electric resistance and the conductivity associated with the running buffer was verified by measuring both quantities independently. A electrical conductivity meter (MPC227) and an InLab 730 electrode, both from Mettler-Toledo (Switzerland), were used to obtain the electric conductivity of the background electrolyte solutions. The temperature was control using a FIRSTEK B-402L circulating water bath. The exponential temperature dependence of the conductivity was determined over the range 20-60 °C. Chemicals. The aqueous background electrolyte solution comprised a variety of concentrations of sodium dihydrogen

Fcp(u·∇T) ) k∇2T + KeE‚E

(5)

where cp and k denote the specific heat capacity and thermal conductivity of the running buffer. The term on the left-hand side stands for the heat transfer by electro-osmotic convection. The terms on the right-hand side represent heat conduction and Joule heating with Ke designating the electric conductivity of the buffer. At the low Reynolds number condition, the viscous dissipation is small and thus ignored. Evidently, the heat generation in the capillary with the cylindrical coordinate (z, r) is transferred out either by axial convection Fcpuz∂T/∂z or by radial conduction k(1/r)(∂/∂r)(r(∂T/∂r))

Fcpuz

∂T 1 ∂ ∂T r ) KeEz2 -k ∂z r ∂r ∂r

( )

(6)

The former mechanism may lead to an increase in temperature along the capillary T(z) while the latter one may give a radial temperature distribution T(r). The relative importance of the two mechanisms can be shown through the Peclet number

Pe )

2 axial convection Fcpuz∂T/∂z Fcpuz R ) ≈ radial conduction kL 1 ∂ ∂T k r r ∂r ∂r

( )

(7)

In our experimental system, the physical parameters are F = 1000 kg/m3, k = 0.6 W/m °C, cp = 4.2 kJ/kg °C, L ) 62 cm, and R ) 37.5 µm. The migration velocity at high voltage can be as fast as uz ∼ (0.01) m/s. Under such conditions, the Peclet number can be as large as 2 × 10-4, which is small compared to unity. That is, Pe , 1. The significance of radial conduction can also be comprehended by considering the case that the Joule heating is delivered out by axial convection only. That is,

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FcpuzπR2[T(z ) L) - T(z ) 0)] ) (KeEz2)πR2L. Solving this algebraic equation gives the temperature difference between capillary exit and entrance as high as 4150 °C, which is far beyond the boiling point of water. In other words, less than 2.4% of the heat generation is transferred outside the capillary by axial convection. Equation 7 also indicates that the transient length associated with the axial temperature change is less than O(1) mm in order to reach Pe ∼ O(1). Under uniform wall temperature condition, the axial temperature gradient is significant only at the position very close to the capillary entrance, i.e., z‚O(1) mm. In summary, the Joule heating for EOF in CZE is transferred out by radial conduction instead of axial convection. Our above scaling analysis is confirmed by both numerical solution5 and experimental measurement.2 Now the temperature distribution inside a capillary is simply a balance between the heat generation and heat conduction at any position z away from the entrance. Assume that the radial temperature difference is small6 so that the heat generation can be taken as a constant, independent of r. This assumption will be justified later. Neglecting the convection term and integrating eq 6 with respect to the radial direction yields the parabolic profile6

[ (Rr ) ]

T(r) ) Tw + ∆T1 1 -

2

KeEz2 2 R 4k

(9)

The experimental condition of 20 mM phosphate buffer solution at 20 kV displays significant Joule heating but only gives ∆T1 = 0.16 °C according to eq 9. This result indicates that it is reasonable to treat the physical properties at position z as radialindependent quantities because of small ∆T1 (T0 = Tw). Consequently, the assumption of constant heat generation due to uniform conductivity Ke is justified, and the parabolic temperature profile is observed experimentally.2 The thermal conduction through the fused silica with the outer radius Rg yields the radial temperature drop ∆T2

∆T2 ) Tw - Tg )

[ ]

KeEz2 2 Rg R ln 2kg R

(10)

where Tg is the temperature at the glass surface, r ) Rg. Since the thermal conductivity of the silica glass wall is kg = 1.5 W/m °C and Rg ) 160 µm, one has ∆T2 = 0.19 °C. A similar analysis for the thermal conduction through the polyimide coating with the thickness 20 µm and the conductivity kp = 0.155 W/m °C gives ∆T3 ) Tp - Tg = 0.15 °C, where Tp represents the temperature located at the surface the polymer coating (r ) Rp). The aforementioned analysis shows that the temperature drops from the lumen center (r ) 0) to the outer surface (r ) Rp) is only about T0 - Tp ) ∆T1 + ∆T2 + ∆T3 = 0.5 °C. However, the temperature inside the capillary is not known yet and has to be determined by the free or forced convection around the capillary

(2πRL)h(Tp - T∞) ) KeEz2(πR2L)

0 ) ∇·[η(T)∇u] + FeE

(8)

where Tw is the temperature at the inner capillary wall and ∆T1 denotes the temperature difference between the lumen center T0 and the wall Tw

∆T1 )

one has h = 25 W/m2 °C 10 and the temperature rise is Tp - T∞ = 45 °C. When forced-convection is involved, the heat transfer coefficient is h and 100 W/m2 °C,10 and therefore, the temperature rise is less than about 10 °C. The Joule heating causes the temperature of the running buffer to rise significantly above the ambient temperature T∞, but the radial temperature gradient within the capillary can be neglected in general. Equation 11 also suggests that Tp (or Tw) may vary with the axial position z owing to the nonuniform distribution of the heat transfer coefficient, h(z). For example, the front section is under forcedair convection while the back section is under free-air convection. As a matter of fact, it is generally difficult to maintain a uniform capillary wall temperature in most air-cooling systems. Consequently, how the axial temperature profile Tw(z) influences the electrokinetic behavior in capillary electrophoresis is an important issue in constructing a microfluidic system. B. Electro-Osmotic Velocity. The electroosmotic flow is driven by the electric force (body force) in the vicinity of the charged wall because of nonzero local net charge. Since the Reynolds number is small, Re ) FuzR/η‚10-5, the inertial term in the equation of motion can be neglected. For an incompressible flow, one has

(11)

where h is the heat transfer coefficient and T∞ the ambient air temperature. For free-convection around a horizontal cylinder,

(12)

where Fe and η represent the local net charge density and the viscosity of the electrolyte solution, respectively. Note that the local charge density is opposite to the wall charge within the so-called electric double layer and becomes zero away from the wall because of electroneutrality. For an electrolyte solution with concentration exceeding about 1 mM, the Debye length κ-1 is less than 10 nm and the electric body force exists only within a few Debye lengths from the wall, i.e., a few nanometers. In other words, we have a thin double layer condition, i.e., κR ∼ O(103) . 1. The local charge density Fe is related to the electric potential ψ by the Poisson equation

∇·[r(T)0∇ψ] ) - Fe

(13)

Under the thin double layer condition, the physical properties related to the electro-osmotic velocity can be considered at the wall temperature Tw albeit there is a radial temperature distribution within the capillary. The temperature drop within the thin double layer is O(∆T1/κR) small. Furthermore, the fluid flow can be separated into two distinct regions (similar to the boundary layer flow problem). In the vicinity of the charged wall, one has

[

] [

]

∂uz ∂ ∂ψ ∂ η(Tw) )  (T ) E ∂x ∂x ∂x r w 0 ∂x z

(14)

where x ) R - r. Integrating the above equation with respect to x yields the well-known result, eq 1

u z ) u∞ ) -

r0ψs η(Tw)

Ez for κx . 1

(15)

where ψs is the electrostatic potential at the surface of shear x ) 0 up to which the fluid stays at rest. Away from the wall, the bulk fluid flow can be described by

0)

[

] [

]

∂uz ∂uz ∂ 1 ∂ rη(T) + η(T) r ∂r ∂r ∂z ∂z

(16)

where the electro-osmotic flow is manifested through the slip

17688 J. Phys. Chem. B, Vol. 108, No. 45, 2004

Wang and Tsao

boundary condition uz(r f R) ) u∞. Obviously, as long as the axial velocity gradient ∂uz/∂z is absent, the radial velocity gradient disappears as well because of the finite velocity at the lumen center or ∂uz/∂r ) 0 at r ) 0. That is, regardless of the radial temperature distribution, EOF in CZE still exhibits a pluglike flow, uz(r) ) u∞ for κr . 1. Generally, three EOF cases can take place in CZE experiments: (i) T is constant everywhere, (ii) T(r) is uniform in z-direction but varies in r-direction, (iii) T(z, r) varies in both z- and r- directions due to nonuniform cooling. When the temperature is uniform everywhere in the capillary, the voltage drop across the tube length can be maintained at a constant value. Since the slip velocity is uniform, the axial velocity gradient is absent and the electro-osmotic velocity is simply given by eq 1. When the electric field is high enough, the Joule heating becomes significant. If the heat transfer rate is not fast enough, the temperature within the capillary is increased and the radial temperature gradient is established. However, eq 15 indicates that the slip velocity is determined by the wall temperature Tw(>T∞), which is uniform in case ii. As a result, the electro-osmotic velocity is still given by eq 1 with η(Tw), and it is no longer linearly increased with the voltage drop (electric field). In the first two cases, a pluglike flow is observed irrespective of the radial temperature profile. Now we return to the essential experimental problem, nonuniform wall temperature. Once Tw(z) is specified, u∞ seems to be z-dependent and eq 16 suggests the electro-osmotic flow being a full two-dimensional problem, i.e., uz(r, z). Moreover, the radial velocity ur(r, z) must be taken into account due to the continuity equation (conservation of mass)

∂uz 1 ∂ (rur) + )0 r ∂r ∂z As a result, the sophisticated numerical technique must be invoked to solve such a complicated problem. It is worth noting that the temperature distribution along the axial direction leads to the change in electric conductivity Ke(z) (resistance) and thereby the electric field Ez(z). This consequence seems to complicate the problem even more, and both momentum and energy equations have to be solved simultaneously. However, one is able to find an interesting relation between the viscosity η and the electric field Ez when eq 15 is carefully examined. A simple electro-osmotic flow is therefore the result for case iii. The sufficient condition for the absence of an axial velocity gradient is the uniform slip velocity condition for eq 16. If the slip velocity u∞ is no longer axial-direction independent under the nonuniform wall temperature distribution, the axial velocity gradient must be present. Hence, the radial velocity has to be introduced, and the plug flow cannot exist for EOF in CZE. Now the question is whether the nonuniform wall temperature profile necessarily leads to z-dependent slip velocity or not. In fact, if the current can be maintained steadily, the product Ez(Tw)/ η(Tw) in eq 15 becomes temperature-insensitive and a uniform slip velocity boundary condition can still be observed for nonuniform cooling experiments. The effect of the temperature on r0ψs in eq 15 is generally unimportant, and a possible explanation will be given later. Under the steady-state condition, conservation of ions gives ∇‚i ) ∇2ψ ) 0, where i ) -Ke∇ψ. Because of the boundary conditions ψ(z ) 0) ) V and ψ(z ) R) ) 0, one has ∂i/∂z ) 0. Hence, the electric current density i must be z-independent regardless of the axial-direction temperature distribution

i ) Ke[T(z)]Ez[T(z)]

(17)

Even though the total voltage drop between entrance and exit is maintained at V, both electric resistance and electric field may vary with the position z due to a nonuniform axial temperature distribution. In dilute electrolyte solutions, the electric current is simply the sum of the current carried by individual ions if the ion-ion interaction is ignored. It is known as Kohlrausch’s law of independent migration of ions. Therefore, the electric conductivity of the electrolyte solution is related to the concentration of species ci and its mobility µi by

Ke(T) ) Σizieciµi(T)

(18)

where zi denotes the valency of species i. Empirical observations indicate that the product of the limiting conductivity and the solvent viscosity (ηKe) is very approximately constant for the same ions under different solvent conditions. This is known as Walden’s rule and can be expressed as11,12

µi(T) 1 1 ) ∝ |zi|e fi η(T)

(19)

where the friction coefficient (resistivity) fi is the force required to drag the particle through the liquid at unit speed. Equation 19 can be rationalized in terms of the well-known Stokes law, which states that the friction on a rigid sphere of radius a in creeping flow is f ∝ ηa. Inserting eqs 18 and 19 into 17 yields

i∝

Ez[T(z)] η[T(z)]

∝

∑ i zi2e2ci L

V

(20)

This result indicates that the ratio of the electric field to the viscosity is essentially independent of the axial temperature distribution and proportional to the ionic strength (electrolyte concentration) at constant current condition. Note that although the electric field varies with temperature, its average is V/L. Another possible temperature-dependent factor in eq 15 is r(T)0ψs(T). In terms of the surface charge density σ of the capillary wall, the ζ potential is given by ψs ≈ σ/κr0. As a result, one arrives at u∞ ∝ (σ/κ)(Ez/η), where the inverse Debye screening length in aqueous solutions is weakly increased with increasing temperature. The surface charge density might rise slightly as well because more thermal energy is provided to overcome the energy barrier of dissociation. Consequently, the temperature-dependent factors in eq 15 cancel out each other and thus provide a uniform slip velocity condition for the bulk flow in eq 16. A uniform axial velocity profile (plug flow) is therefore obtained, and an intriguing conclusion is found: under steady electric current, the electro-osmotic flow is essentially insensitive to the axial temperature profile caused by Joule heating and nonuniform heat convection from the capillary surfaces. Although the electro-osmosis in a capillary remains a pluglike flow regardless of the nonuniform temperature distribution, the Joule heating leads to an increase in average temperature and thus raises the electric current density and the electro-osmotic mobility directly through the reduction in solvent viscosity. The current density i can be determined from the average resistance 〈R〉 by Ohm’s law

i)

1 1 V 〈Ez〉 ) πR2 〈R〉 〈Ke-1〉

(21)

where the average conductivity and resistance are defined by

Capillary Zone Electrophoresis

〈R〉 )

∫0LR[T(z)]dz L

)

1 πR2

J. Phys. Chem. B, Vol. 108, No. 45, 2004 17689

∫0LKdz(T) ) πRL 2 〈Ke-1〉

(22)

e

and the average electric field is given by

〈Ez〉 )

∫0LEzdz/L ) V/L

According to eqs 18 and 19, one has 〈R〉∝‚η[T(z)]〉. Since the solvent viscosity declines with increasing temperature, the electric conductivity rises and the resistance decreases. Therefore, when the Joule heating becomes substantial, the electric current is no longer linearly proportional to the total voltage drop. In fact, the higher the applied voltage, the higher the resulting temperature is, and thus, the higher the generated current is. Under constant current conditions, instead of a constant voltage drop across the tube length, the electroosmotic velocity can be expressed in terms of current

uE )

0r(T)ψs(T) η(T)Ke(T)

i

(23)

Note that the temperature effect disappears in eq 23 according to Walden’s rule. As a consequence, the electro-osmotic velocity increases linearly with the electric current regardless of the Joule heating and the axial temperature distribution. Moreover, on the basis of eq 18, the electro-osmotic velocity is inversely proportional to the electrolyte concentration.

Figure 1. Variation of electric current with voltage at different concentrations of running buffer.

IV. Experimental Confirmation Simple theoretical analysis shows that Joule heat is transferred out the capillary mainly through the radial conduction. That is, the heat generated at the local position z within the capillary is mainly taken away by the surrounding convection at z. When the product of the electric conductivity, square of electric field, and square of capillary radius is small compared to the thermal conductivity of running buffer, i.e., KeEz2R2/4k , 1, the temperature inside the capillary can be regarded as being nearly uniform in the radial direction. The temperature distribution along the axial direction due to nonuniform surrounding convection does not change the plug-flow pattern under steady current because the cancellation of the temperature effects between the solvent viscosity and the electric field provides a uniform slip boundary condition. Moreover, the cancellation also predicts that the electro-osmotic velocity is linearly proportional to the electric current regardless of the Joule heating. To confirm our theoretical analysis, we have performed a series of CZE experiments. The Joule effects on electro-osmotic flow and electrophoretic motion are observed and circumvented. A. Electro-Osmotic Flow. The effects of Joule heat are clearly observed by varying the running buffer concentrations and the applied voltages. Figure 1 shows the variation of the electric current with the voltage at different concentrations of the running buffer. Ohm’s law states that the current I should be linearly increased with the voltage V. The deviation from the linear straight line signifies the onset of Joule heating. The Joule heating is proportional to the electric conductivity, which is in turn linearly increased with the electrolyte concentration. As a result, the onset voltage of the Joule effect rises with lowering the buffer concentration cb. Since the Joule heating is proportional to cbV2, the voltage at which the Joule heating becomes significant is Vc ∝ cb-1/2 under the same convective cooling condition. As illustrated in Figure 1, the experimental

Figure 2. Linear relation between the inverse resistance (R-1) and the electric conductivity (Ke).

result is consistent with the above statement. The slope associated with the linear V-I relation gives the electric resistance R at 25 °C in the absence of the Joule effect. To further demonstrate the validity of the electric resistance obtained from the linear regime, we examine the relation between the electric resistance and the electric conductivity, as indicated in eq 22. Figure 2 shows the linear relation between R-1 and Ke. The conductivity of the buffer is measured independently by conductometry. The slope obtained from Figure 2 agrees quite well with the value of πR2/L = 7.36 × 10-9 m corresponding to our capillary electrophoresis system. Note that the electric current consists of bulk and surface conductivity contributions. The latter represents the current transported by the electric double layer near the wall, especially significant at low salt concentrations.13 However, the result in

17690 J. Phys. Chem. B, Vol. 108, No. 45, 2004

Figure 3. Plot of ln Ke against T-1 for different buffer concentrations. The slopes of all straight lines are approximately Tc = 1750 K.

Figure 2 implies that the surface conductivity contribution is unimportant in our experiments. The Joule heating is essentially manifested through the temperature dependences of the viscosity and electric conductivity. It lowers the viscosity and enhances the electric conductivity and current. In accordance with Walden’s rule, Ke is inversely proportional to the viscosity η, which is related to the temperature by the Andrade correlation, exp(Tc/T).5,14 As shown in Figure 3, the plot of ln Ke against T-1 for different buffer concentrations can be well represented by straight lines with Tc = 1750 K, which is consistent with the value reported in the literature for the water viscosity. This result confirms the validity of Walden’s rule for the running buffer used in our experiments and proves that the temperature effect on the product of η and Ke can be canceled out in capillary electrophoresis. Figure 4 demonstrates the relation between the electroosmotic velocity uE and the applied voltage under different running buffer concentrations cb. The direct effect of buffer concentrations on the electro-osmotic velocity is not obvious in eq 1. For the potential determining ion (pdi), such as H+ in our case, the ζ-potential is altered when the pdi concentration (pH) is changed. As a result, the electro-osmotic velocity is varied accordingly. Nevertheless, the effect of indifferent electrolyte ions is not clear. When pH is maintained as the same but the running buffer (indifferent ions) concentration is changed, we observe that uE declines with increasing cb as shown in Figure 4. A possible explanation is that a higher salt concentration provides more serious counterion condensation on the charged wall and thus leads to a lower effective surface charge density or ζ-potential. The counterion condensation can be regarded as ion adsorption and is qualitatively similar to surfactants adsorbed onto the air-water interface. The percentage change in the bulk due to condensation can be shown in terms of the number ratio of surface to bulk, (2πRLacs)/(πR2Lcb) ∼ csa/cbR, where a is the ion size. cs and cb depict the number density on the surface and in the bulk, respectively. For simplicity, we adopt a ) 0.4 nm, cs ≈ 0.1/a3 (high volume fraction ≈ 0.1), and cb ≈ 10 mM. On the basis of the above parameters, the ratio is O(10-2) or smaller. This fact indicates

Wang and Tsao

Figure 4. Relation between electro-osmotic velocity uE and applied voltage under different running buffer concentrations cb.

that the change in the bulk concentration due to counterion condensation can be neglected and the use of nominal running buffer concentration is justified. Equation 1 also indicates that the electro-osmotic velocity, uE, is proportional to the applied voltage V under a constant temperature condition. However, the Joule heating results in a decrease in the viscosity and thus an increase in the electroosmotic velocity. Figure 4 shows the deviation from the linear relation between uE and V for different running buffer solutions. As the buffer concentration is increased, the onset voltage associated with the Joule heating declines because the Joule effect grows with cb or conductivity. It should be emphasized that the onset point of Joule heating Vc depends also on the local heat transfer rate at the surface of the capillary coating. Under the same thermostat condition, the qualitative behavior associated with the electro-osmotic velocity is similar to that of the electric current as shown in Figure 1. The significant temperature-dependent characteristics of the electrokinetic behavior imply that nonuniform heat transfer, particularly air thermostat, may affect the performance and reproducibility of capillary electrophoresis substantially. Our theoretical analysis, eq 23, suggests that various temperature effects may cancel each other out and the electroosmotic velocity may be linearly increased with the current regardless of the temperature distribution inside the capillary caused by Joule heating and nonuniform convection. To examine its validity, we replot Figure 4 with the electro-osmotic velocity against the current under different buffer concentrations. Figure 5 clearly demonstrates that uE rises linearly with increasing I for different buffer concentrations even when the Joule heating is very substantial as shown in Figure 1. The slope is inversely proportional to the electric conductivity Ke, which is in turn proportional to the running buffer concentration cb. To further verify the validity of eq 23, we replot Figure 5 with the product of buffer concentration and electro-osmotic velocity (cbuE) against the current. As depicted in Figure 6, all data points collapse into a single straight line, independent of the temperature profile during CZE. This consequence unambiguously confirms the temperature-insensitive characteristics of uE versus i in eq 23. In addition, we purposefully create nonuniform

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Figure 5. Migration velocity rises linearly with increasing current for different buffer concentrations even when the Joule heating is very substantial.

Figure 6. All data points at different buffer concentrations collapse into a single straight line, independent of the temperature profile during CZE.

surrounding convection. The first half of the capillary is under forced convection, and the rest is under free convection. The surface temperature Tp is measured for both sections, and a substantial difference can be observed under high applied voltage due to Joule heating. For example, we observed about 30 and 50 °C in the two sections, respectively. Nevertheless, the data points still fall into the single line in Figure 6 as long as the current can be maintained steadily. B. Electrophoretic Motion. The apparent velocity of a charged solute uA in CZE is due to both its electro-osmotic flow uE plus the electrophoretic motion, uA ) uE + uEP. Equation 2 points out that, under isothermal conditions, the electrophoretic velocity is proportional to the applied voltage. However, the electrophoretic velocity may be greater than that predicted by

Figure 7. (a) Variation of both electro-osmotic velocity and apparent velocity with applied voltage at two running buffer concentrations. (b) Variation of the electrophoretic velocity (subtracting the apparent velocity from the electro-osmotic velocity) with applied voltage at two running buffer concentrations.

eq 2 at high applied voltages because the Joule heating results in a decrease in the buffer viscosity. Figure 7a shows the variation of both electro-osmotic velocity and apparent velocity with the applied voltage at two running buffer concentrations. The direction of the electrophoretic motion associated with the anionic surfactant ion (dodecyl 4-benzene sulfonate) is opposite to that of EOF. Since the EOF velocity is larger than that of the electrophoretic velocity in our experiments, one has uA < uE. It is evident that the apparent migration velocity does not follow the linear relation at high enough voltages. Subtracting the apparent velocity from the electro-osmotic velocity yields the electrophoretic velocity as depicted in Figure 7b. When the ion size is much smaller than the Debye length, i.e., a , κ-1,

17692 J. Phys. Chem. B, Vol. 108, No. 45, 2004

Wang and Tsao

Figure 8. Electrophoretic velocity is plotted against the current even when Joule heating is very significant.

Figure 9. Data points of the two buffer concentrations are collapsed into a single line with the slope 2.27 mM cm/s µA.

the electrophoretic mobility is essentially independent of the electrolyte concentration as indicated in eq 2. Therefore, uEP is basically the same for the two buffer concentrations except at high applied voltage, 20 kV. Since the Joule heat is more serious for higher buffer concentration at the same V, it is expected that the electrophoretic velocity associated with higher cb is larger due to lower η. The driving force for the electrophoretic motion of an ion is zeEz while the resistance is depicted by the Stokes drag, 6πaηuEP. When the capillary temperature is not uniform due to Joule heating or nonuniform convective cooling, both electric field Ez[T(z)] and viscosity η[T(z)] vary with the temperature field. At steady current conditions, however, eq 20 shows that the ratio Ez/η ∝ i is not sensitive to the temperature profile, and therefore, the electrophoretic velocity is maintained essentially as a constant throughout CZE. To circumvent the Joule heating effect, we can follow the approach for the EOF velocity and express uEP in term of the current density i

is uEP/Ez = 2.0 × 10-8 m2/(V‚s) at 25 °C. On the basis of Kohlrausch’s law, eq 18, the ion mobility can also be estimated from measuring the electric conductivity Ke.9 With uEP/Ez = 5.19 × 10-8 m2/(V‚s) for sodium ions,12 the electrophoretic mobility of dodecyl 4-benzene sulfonate is about uEP/Ez = 1.87 × 10-8 m2/(V‚s). The CZE result agrees reasonably well with that obtained by conductometry.

uEP )

1 ze i 6πa η(T)Ke(T)

(24)

Similarly, on the basis of the Walden’s rule, the temperature effects are canceled out and uEP should be linearly increased with the electric current irrespective of the temperature distribution. Note that a refers to the hydrodynamic radius associated with the ion. Figure 8 confirms that the electrophoretic velocity is proportional to the current even when Joule heating is very significant. Since Ke is proportional to the buffer concentration, eq 24 indicates that the electrophoretic velocity declines with increasing cb under the specified current. In other words, the slope of the uEP-I line is inversely proportional to cb. According to eqs 18 and 24, the effect of the buffer concentration can be eliminated if one plots the product of electrophoretic velocity and buffer concentration (cbuEP) against the current. Figure 9 shows that the data points of the two buffer concentrations are collapsed into a single line with the slope 2.27 mM cm/s µA. Thus, our theory eq 24 is verified. The electrophoretic mobility calculated from our experiments for dodecyl 4-benzene sulfonate

V. Conclusion Capillary zone electrophoresis generally suffers the Joule effect at relatively high driven voltage owing to the temperature control problem. In the plot of migration velocity against voltage, the electro-osmotic (or electrophoretic) mobility remains a constant at lower voltage but grows at higher voltage. It is attributed to the reduction of the solvent viscosity caused by the temperature increase. In this paper, a simple analytical theory is derived to show that as long as the electric current is steady, the electro-osmotic flow remains pluglike regardless of the temperature profile inside the capillary due to Joule heating and nonuniform surrounding convection. Moreover, the electroosmotic (or electrophoretic) velocity is always linearly proportional to the electric current even when the Joule effect is very significant. This temperature-insensitive electrokinetic behavior is caused by the cancellation of the temperature effects. For example, Walden’s rule states that the product of buffer viscosity and electric conductivity is essentially constant irrespective of the temperature. A series of capillary zone electrophoresis experiments has been performed to analyze the effects of buffer concentrations and to confirm the temperature-insensitive electrokinetic behavior. Our theoretical and experimental results suggest that the Joule effect can be circumvented by controlling the electric current even under nonuniform convective cooling. On the basis of eqs 23 and 24, the surface charge properties can be characterized when the Joule effect is significant. For instance, the surface potential ψs can be determined when Joule heating occurs in EOF. At high applied voltage, the migration velocity and the electric current are measured. The temperatureinsensitive ratio 0rψs/ηKe is then obtained according to eq 23. Since the properties (r, η, Ke) can be independently obtained

Capillary Zone Electrophoresis at room temperature, the surface potential at room temperature is then decided. Although we have adopted ionic surfactant as a migration entity for electrophoresis, our analysis is still applicable to bigger macromolecules or particles at higher ionic strength with a correction of the proportional constant in eq 24. Supporting Information Available: This research is supported by National Council of Science of Taiwan under Grant NSC-92-2214-E-008-007. References and Notes (1) Cummings, E. B.; Griffiths, S. K.; Nilson, R. H.; Paul, P. H. Anal. Chem. 2000, 72, 2526-2532. (2) Liu, K.-L. K.; Davis, K. L.; Morris, M. D. Anal. Chem. 1994, 66, 3744-3750. (3) Davis, K.-L.; Liu, K.-L. K.; Lanan, M.; Morris, M. D. Anal. Chem. 1993, 65, 293-298.

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