Diffusivity of Solutes Measured in Glass Capillaries Using Taylor's

Anal. Chem. 2005, 77, 806-813

Diffusivity of Solutes Measured in Glass Capillaries Using Taylor’s Analysis of Dispersion and a Commercial CE Instrument Upma Sharma, Nathaniel J. Gleason,† and Jeffrey D. Carbeck*

Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544

We present a strategy for the rapid, efficient, and accurate measurement of the coefficient of diffusion (D) of solutes using a commercial capillary electrophoresis (CE) instrument. This approach utilizes the classic analysis of Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1953, 219, 186-203) of the dispersion of solutes pumped hydrostatically through glass capillaries. To obtain accurate values of D, we modified Taylor’s analysis of dispersion to account for the finite time required to reach steadystate flow in the capillary when using a CE instrument. Neglecting this effect results in measured diffusivities of phenylalanine, a model solute, that are in error by as much as 60% when compared with published values. We provide an analysis of this effect and a simple strategy for avoiding these errors. Using this approach, we analyze profiles of concentration fronts and measured values of D for phenylalanine to within 5% of published values. We also analyze profiles of pulses of solute. To determine values of D accurately, measurements of dispersion first need to be made as a function of injection volume to correct for the finite width of the injection plug, before they are corrected for unsteady-state flow. This approach also yields values of D for phenylalanine to within 5% of published values. In contrast to other techniques used for the determination of D, this approach requires no fluorescent labeling and is applicable to solutes of any molecular weight. Diffusion of solutes is important in a number of chemical and biological processes. Rates of heterogeneous reactions (i.e., reactions that occur on or are catalyzed by surfaces) are often limited by the rates of diffusion of reactants and products to and from the reactive surface. Diffusion of proteins through polymer gels and solutions plays an important role in chromatography1,2 and drug delivery.3-5 Cellular processes such as metabolism and signaling are affected by the rate of diffusion of molecules through * To whom correspondence should be addressed. Phone: (609) 258-1331. E-mail: [email protected]. † Current address: Systems Research Department, Sandia National Laboratories, P.O. Box 969 MS 9201, Livermore, CA 94551. (1) Garke, G.; Hartmann, R.; Papamichael, N.; Deckwer, W. D.; Anspach, F. B. Sep. Sci. Technol. 1999, 34, 2521-2538. (2) Hunter, A. K.; Carta, G. J. Chromatogr., A 2002, 971, 105-116. (3) Bell, C. L.; Peppas, N. A. Biomaterials 1996, 17, 1203-1218. (4) Peppas, N. A. Curr. Opin. Colloid Interface Sci. 1997, 2, 531-537. (5) Kim, B.; Peppas, N. A. J. Biomater. Sci.-Polym. Ed. 2002, 13, 1271-1281.

806 Analytical Chemistry, Vol. 77, No. 3, February 1, 2005

the cytoplasm and extracellular matrix.6-8 Because diffusion is a slow process, measurements of diffusivities (D, m2 s-1) are often time-consuming. (Table 1 provides a glossary of symbols.) Techniques that provide rapid, efficient (low consumption of solute), and accurate measurement of values of D are desired. In this paper, we show that a commercial capillary electrophoresis (CE) instrument can be used to accurately measure diffusivities of solutes. Doing so requires modification of the classical analysis of the transport of solutes in convective flow by Taylor,9,10 Aris,11 and others12 to account for the unsteady-state flow that occurs during the injection and pumping of solute pulses or fronts through the capillary. In a classic paper, Taylor9 introduced the idea of measuring D by monitoring the concentration profile of solute pulses and fronts as they flowed through a uniform, cylindrical tube under laminar, Poiseuille flow. The solute is transported through the capillary by the fluid; the parabolic velocity profile of the fluid produces a concentration gradient between the center of the capillary and the walls, which causes the solute to diffuse in the radial direction. The overall distribution of the solute due to the combined effects of axial convection and radial diffusion is known as dispersion. By measuring the dispersion of a solute, Taylor was able to determine its diffusivity under conditions when axial diffusion could be neglected.10 Aris extended the analysis of Taylor to overcome this restriction.11 The analyses of Taylor and Aris for the dispersion of a solute flowing in a tube have been used previously as a method to measure values of diffusivity.13 Typically, these experiments were performed in tubes that had values of radii up to 1 mm and required long measurement times (typically on the order of hours). Bello et al.14 introduced the use of narrow-bore glass capillaries (radii e50 µm) for the determination of D from the Taylor-Aris analysis of dispersion. By using capillaries with such small inner diameters, the residence time (i.e., the time necessary (6) Verkman, A. S. Trends Biochem. Sci. 2002, 27, 27-33. (7) Luby-Phelps, K.; Lanni, F.; Taylor, D. L. Annu. Rev. Biophys. Biophys. Chem. 1988, 17, 369-396. (8) Luby-Phelps, K. Curr. Opin. Cell Biol. 1994, 6, 3-9. (9) Taylor, G. I. Proc. R. Soc. London, Ser. A 1953, 219, 186-203. (10) Taylor, G. I. Proc. R. Soc. London, Ser. A 1954, 225, 473-477. (11) Aris, R. Proc. R. Soc. London, Ser. A 1956, 235, 67-77. (12) Alizadeh, A., Nieto de Castro, C. A., Wakeham, W. A. Int. J. Thermophys. 1980, 1, 243. (13) Pratt, K. C.; Wakeham, W. A. Proc. R. Soc. London, Ser. A 1974, 336, 393406. (14) Bello, M. S.; Rezzonico, R.; Righetti, P. G. Science 1994, 266, 773-776. 10.1021/ac048846z CCC: $30.25

© 2005 American Chemical Society Published on Web 01/31/2005

Table 1. Glossary of Symbols symbol C* C C0 D k LD LT M ∆p Pe Q r Rc ri t ti tR,obs ui u0 uss Vi η σ2 σ2obs τ

definition local solute concentration mean solute concentration across the cross section of the tube concentration of the solute in an initial front diffusion coefficient dispersion coefficient length of the tube from the inlet to the concentration detector total length of the capillary mass of the solute concentrated in an initial pulse applied pressure drop across the capillary Peclet number: describes the relative rates of mass transfer due to convection and diffusion volumetric flow rate radial position radius of the tube/capillary rate of increase of applied pressure time time to reach the steady-state fluid velocity observed mean residence time average fluid velocity during the ramp to steady state mean fluid velocity steady-state fluid velocity in an ideal experiment volume of the injection plug solution viscosity measure of the width of a solute pulse or the sharpness of the solute front in an ideal Taylor experiment experimentally measured width of the solute pulse or the sharpness of the solute front dimensionless residence time

for the solute to flow from the inlet to the point of detection) required to measure D accurately was reduced from hours to minutes. As we show in the next section, the time necessary for dispersion to generate measurable changes in the concentration profile sufficient for the determination of D scales with the square of the radius of the tube: a glass capillary 50 µm in radius allows the accurate determination of D for small solutes in minutes or less; a tube 1 mm in radius requires a residence time of more than 6 h to produce the same amount of dispersion. Bello et al. used a commercial CE instrument to pump solutions hydrostatically (rather than electrophoretically) through silica capillaries;14 a UV detector measured the concentration profile and, thereby, the dispersion of solutes. This approach offers a number of advantages over other techniques used to measure values of D. Diffusivities in solution can be measured for any molecule provided detection of that molecule by CE is possible, typically by UV absorbance or fluorescence. A commonly used technique for measuring solute diffusivities, dynamic light scattering (DLS), is limited to measuring the diffusivities of macromolecules in solution; molecules must be greater than ∼1 nm in size for DLS to measure accurate values of D. Another frequently used technique, fluorescence recovery after photobleaching (FRAP), is applicable only to fluorescent or fluorescently tagged solutes. The use of a CE instrument introduced an experimental nonideality not considered previously in the analysis of dispersion. The mean fluid velocity u0 (m/s)sdefined as the fluid velocity averaged over the cross section of the capillary, and measured as the length of the capillary from inlet to detector divided by the

time required for the solute to reach the detectorsis not at a constant, steady-state value, as was assumed by Taylor and Aris. Because the current configuration of commercial CE instruments requires the fluid flow through the capillary to stop for the solute to be injected, there is an initial acceleration of the fluid after injection as the flow rate is increased from zero to the steadystate value. The analyses of Taylor and Aris assumed that the fluid velocity was at its steady-state value throughout the experiment; they did not consider the effects of this “ramping up” in velocity. Though present in their experiments, Bello et al. did not include the effect of this unsteady-state flow in their measurements. In this paper, we show that the presence of this ramp in velocity can lead to large errors in values of D (as great as 60%) when the analyses of Taylor and Aris are applied to dispersion experiments conducted in glass capillaries using a commercial CE instrument. For the case of analysis of dispersion of concentration fronts, the result of this ramp in velocity is that values of D are significantly overestimated at small residence time; the error approaches zero at long residence times. The situation is more complex for a solute pulse. The ramp in fluid velocity results in values of D that are also overestimated at small values of residence time. In contrast to the analysis of fronts, the analysis of pulses results in values of D that are underestimated at long residence times. This difference between fronts and pulses results from the finite width of the plug of solute injected into the capillary. For pulses, there is an intermediate residence time for which the effects of the initial ramp in velocity and the finite width of the injected plug effectively cancel and an accurate value of D is obtained. This condition cannot, however, be predicted in the absence of knowledge of D. We provide a simple method to compensate for the effect of this unsteady-state flow at all values of residence time. Theoretical Analysis of Dispersion. Taylor first analyzed the dispersion of a solute in a cylindrical tube due to the combined effects of molecular diffusion and convection.9 By assuming diffusion along the axis of the tube was negligible, he obtained analytical solutions of the convection-diffusion equation (given for a stationary reference frame in eq 1) for concentration profiles of fronts and pulses (assumed to be δ-functions): t is time, z is axial position, r is radial position, C* is local concentration of the solute and is a function of both r and z, D is the diffusivity of the solute, and Rc is the radius of the tube.

[

( )

]

r2 ∂C* ∂2C* ∂C* u0 1 ∂ ∂C* 1- 2 r + + )D ∂t 2 r ∂r ∂r Rc ∂z ∂z2

( )

(1)

If the mean solute concentration across the cross section of the tube, C , is monitored as a function of time at a fixed position along the tube, typically by UV absorbance, then the solution of eq 1 by Taylor gives eq 2 (for a front) and 3 (for a pulse). In

C C0 C )

)

(

)

(t - tR) 1 1 ( erf 2 2 σx2

M 2π3/2Rc2xkt

(

exp -

(front)

)

(t - tR)2 2σ2

(pulse)

(2)

(3)

these equations, k is the dispersion coefficient (k ) Rc2u02/48D), Analytical Chemistry, Vol. 77, No. 3, February 1, 2005

807

M is the mass of the solute in the pulse, C0 is the concentration of the front, tR is the mean residence time (the time it takes the solute moving with the mean velocity of the fluid to reach the detector located at a distance LD from the inlet of the tube), and σ2 is a measure of the width of the pulse or the sharpness of the front and is related to the dispersion coefficient by σ2 ) 2kt/u02. Taylor’s analysis assumed an ideal experiment where a concentration front or a δ-pulse is introduced into a capillary of uniform radius under conditions of steady-state, laminar Poiseuille flow. The concentration profile of the solute is monitored at a fixed cross section of the capillary. In this ideal case, tR and σ2 can be determined by fitting the concentration profiles to eq 2 or 3; for pulses, tR and σ2 correspond to the first and second moments of the distribution of solutes in the profile. The diffusivity of the solute is then determined using eq 4.

D ) (Rc2/24σ2)tR

(4)

The conditions necessary for eq 4 to be valid are expressed in terms of two dimensionless quantities: (i) a dimensionless residence time, τ ) DtR/Rc2, which is the ratio of the residence time to the time required for a solute to diffuse a distance equal to the radius of the capillary; and (ii) a Peclet number, Pe ) u0Rc/ D, which describes the relative rates of mass transfer along the axis of the capillary due to convection and diffusion. Taylor showed that eq 4 was valid when (i) τ is greater than the time it takes to decrease variations in radial concentrations by a factor of e (τ . 0.14) and (ii) diffusion in the axial direction is negligible compared to convection (Pe . 7).10 If we assume that a ratio of 10:1 is sufficient for the inequalities given by Taylor, then eq 4 is applicable when Pe > 70 and τ > 1.4. Inspection of eq 4 shows that capillaries with smaller values of Rc require shorter residence times to produce profiles with the same value of σ2. Consequently, smaller tubes enable a more rapid measure of D. A limitation of Taylor’s analysis is that diffusivities cannot be determined for slow flows when axial diffusion is no longer negligible (because the condition of Pe > 70 is not satisfied). Aris presented an alternative method of analysis that overcame this limitation;11 by examining the moments of a concentration profile, the diffusivity could be determined provided the solute was initially contained within a finite length of the tube. This analysis was applied by Bello et al. in conjunction with a CE instrument to measure the diffusivity of small molecules and proteins.14 In their work, a plug of solute was injected into a capillary, and the concentration profile was fit to eq 3. The diffusivity was then determined from tR and σ2 by solving eq 5 for D (Aris approach). When the conditions of Taylor are satisfied (Pe > 70 and τ > 1.4), this expression gives the same value of D as eq 4.

D+

( x )

Rc2u02 1 2 ) u0 tR -1 + 48D 4

1+

4σ2 tR2

(5)

Equations 4 and 5 allow measurement of the diffusivity of a solute from its concentration profile in a single experiment, assuming it satisfies the ideal conditions. In practice, the experimental setup often deviates from ideality. Wakeham and coworkers12 considered a number of experimental departures from 808

Table 2. Summary of the Approaches of Taylor and of Aris to the Analysis of Dispersion of Concentration Fronts and Pulses to Obtain Measurements of Solute Diffusivities

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ideality and analyzed their impact on the analysis by Aris for dispersion of a pulse. A number of these deviations from ideality are applicable to a commercial CE apparatus: (i) the finite width of the detection window, (ii) the finite width of the injected plug of solute, and (iii) nonuniformities in the radius of the capillary. They assumed that each effect was small and could be treated as an independent first-order perturbation to the system. They found that errors arising from nonuniformities in the radius of the capillary were insignificant. The corrections they derived for the finite width of the detector are proportional to the ratio of the width of the detector window to LD. Because this quantity is small (∼10-3) for the glass capillaries used in this work, the correction for the finite width of the detector window is also insignificant and can be neglected. Corrections for the finite width of an injected plug of volume Vi cannot be neglected typically and are given in eqs 6 and 7. To use these corrections, the concentration profile of a pulse is monitored and fit to eq 3 in order to obtain tR,obs and σ2obs. The values are corrected using eqs 6 and 7 to obtain tR and σ2, which are used with eq 5 to obtain the value of D of the solute (Aris’ approach). Alternatively, when the conditions of Pe > 70 and τ > 1.4 are satisfied, Taylor’s approach can be used; in this case, the corrections given in eqs 6 and 7 are applied to eq 4. Table 2 summarizes the approaches of Taylor and Aris for measuring solute diffusivities using fronts and pulses.

tR ) tR,obs(1 - (Vi/2πRc2LD)) σ2 ) σ2obs -

(6)

( )

t2R,obs Vi 12 πR 2L c D

2

The analyses performed by Taylor and Aris are based on the assumption that the fluid velocity remains constant during an experiment (i.e., steady-state flow). When using a CE instrument to perform these experiments, fluid flow stops in order for an injection (either a pulse or a front) to occur; the velocity is then ramped up to the steady-state value. Because dispersion experiments performed previously in tubes with millimeter diameters were done with the direct injection of the solute into the tube

without stopping the fluid flow, this nonideality was not considered by Wakeham and co-workers.12 In the work by Bello et al.,14 the effect of the unsteady-state flow, although present, was neglected. In this paper, we show that the neglect of this effect can lead to large errors when measuring the diffusivity of solutes with a CE instrument; we also provide a method for correcting for this nonideality that allows for accurate measurement of diffusivities. METHODS All experiments were conducted on a Beckman P/ACE MDQ capillary electrophoresis instrument. Fused-silica capillaries (Polymicro Technologies, Phoenix, AZ) with 50- or 100-µm inner diameters were cut to total lengths of 30, 50, or 60 cm. All experiments were performed at 25 °C with temperature maintained by the internal coolant of the CE instrument. Bare silica capillaries were rinsed for 15 min with RBS 35 detergent (Pierce, Rockford, IL), 0.1 N NaOH, and 18 MΩ water. Prior to each run, the capillary was filled with solvent (typically water or buffer). For the experiments with fronts, the inlet end of the capillary was transferred to the vial containing the solute in solution, and a pressure drop across the capillary was applied (values ranged from 0.1 to 99 psi). For the experiments with pulses, plugs of solute were injected hydrodynamically onto the capillary. Injection pressures of 0.5-2.0 psi and injection times of 5 or 10 s were used to produce injection plugs with different initial volumes. Values of capillary radius, Rc, were measured using a volumetric technique. Water (viscosity, η ) 0.8909 cP) was pressurized through a capillary (of total length, LT) for periods of at least 20 min (to minimize effects of the initial ramp in fluid velocity), and the volumetric rate of flow, Q, was measured for each applied pressure drop, ∆p. Q was determined by measuring the mass of water that passed through the capillary in the allotted time. Using the Hagen-Poiseuille law (eq 8), the radius of the capillary was

Q ) (∆p/LT)(πRc4/8η)

(8)

determined. This procedure was repeated for at least five different values of pressure; the variation in Rc was less than 1%. RESULTS AND DISCUSSION An Initial Ramp in Fluid Velocity Can Cause Errors in Values of Diffusivity. To demonstrate the effect of the initial ramp in the fluid velocity on values of diffusivity, we measured dispersion of solute fronts at different rates of flow and, thereby, different residence times. Figure 1 shows experimental concentration profiles for fronts of phenylalanine in water at three different values of tR,obs, the time required for the center of the solute front to reach the detector after injection. These profiles show that as the residence time increases the concentration profiles become sharper. This sharpening is the result of two effects: (i) as residence time is increased, the rate of flow is decreased and the radial gradient in velocity of the fluid is reduced; (ii) as residence time is increased, there is also more time for diffusion to reduce the axial gradient in concentration produced by the velocity profile of the fluid. We applied Taylor’s approach to measure diffusivity by fitting eq 2 to the concentration profiles (shown as solid lines in Figure

Figure 1. Concentration profiles showing the dispersion of fronts of phenylalanine in water for three different residence times. The data are normalized by the observed residence time, tR,obs, measured as the time after injection required for the center of the concentration front to reach the detector. The experimental data (shown as points) were fit to eq 2 (solid lines) to obtain values of σ, which were used with eq 4 to obtain values of D. The inset table shows the error in the diffusivity measured for each of the profiles shown, as compared to the literature value of 7.047 × 10-6 cm2/s.15 The error differs with the residence time. Experimental conditions: 0.25 wt % phenylalanine in water at 25°C, LD ) 39.7 cm, LT ) 50.2 cm, Rc ) 26.0 µm, pressure drops of 0.5 (2), 3 (9), and 14 psi (b).

1). If we assume that the experiments are ideal, then the values of σ obtained from these fits can be used along with eq 4 to determine values of D, provided that Pe > 70 and τ > 1.4. The errors from values of D determined in this way, as compared to the literature value15 of 7.047 × 10-6 cm2/s, are reported in Figure 1. While the profile with the intermediate residence time gives a value of D in excellent agreement with the literature value, the values of D determined at the shortest and longest residence times show poor agreement with the published value. We expected error from the experiment with the largest value of tR,obs (490 s), because the condition of Pe > 70 was not satisfied. Unexpectedly, we observed the largest error (13.4%) in the value of D in the experiment with the smallest value of tR,obs (30 s), where both the conditions Pe > 70 and τ > 1.4 were satisfied. To further demonstrate the effect of the initial ramp in fluid velocity, Figure 2a shows the error in values of D for phenylalanine determined from the dispersion of concentration fronts of this solute as a function of observed residence time. These data show that this approach overestimates values of D at short residence times; the error in the measurement of D decreases to within 1% with increasing residence time. Errors greater than 60% were measured at the shortest residence time, despite the fact that all data shown satisfy the conditions of Pe > 70 and τ > 1.4. These results show that measuring D in this way can lead to large errors, even in regions where the analysis is expected to be valid. Experiments were also conducted using plugs of phenylalanine injected into the capillary using applied pressure; the profiles for these experiments were fit to eq 3 and analyzed using both the (15) American Institute of Physics Handbook; McGraw-Hill: New York, 1957.

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809

Figure 2. Error in the measurement of D as a function of residence time for experiments conducted with (a) fronts and (b) pulses of phenylalanine in water. Fronts are analyzed using eq 4 (Taylor approach). Pulses are analyzed using both the methods of Taylor (0) and of Aris (]); data are corrected for the initial width of the injected plug as indicated (open symbols). All data shown satisfy the conditions of τ > 1.4 and Pe > 70 unless otherwise indicated (shaded region). Experimental conditions: 0.25 wt % phenylalanine in water at 25 °C; (a) LD ) 20.6 cm, LT ) 31.0 cm, and Rc ) 26.5 µm; (b) LD ) 48.9 cm, LT ) 59.8 cm, Rc ) 25.6 µm, and Vi ) 5.3 nL (0.5 psi for 5 s).

approaches of Taylor (eq 4) and Aris (eq 5). To account for experimental nonidealities, corrections were made to account for the initial width of the injected plug of solute (eqs 6 and 7). The errors in the measured values of D from dispersion experiments conducted with pulses and analyzed using both the methods of Taylor and of Aris are shown in Figure 2b. As the residence time is made shorter, errors in values of D follow a trend similar to that for fronts; this approach also overestimates values of D. The error is largest at the shortest residence time and decreases as residence time increases. In contrast to experiments done with fronts, the error in experiments conducted with solute pulses does not approach zero at large values of residence time. Instead, the error becomes increasingly negative (i.e., the approach underestimates the value of D) at longer residence times: errors as large as -30% were measured. For the data shown in Figure 2b, there is a range of residence times around 40 s for which the measured error is within 5%. The data in Figure 2b also show that the corrections for the finite width of the injection plug described by eqs 6 and 7 affect the error in the measured value of D by less than 3%. Additionally, the differences between the errors measured using the analysis of Taylor and of Aris are less than 2% for all data shown. As expected, we see the largest differences between these analyses occur when the condition of of Pe > 70 is not maintained. These results show that using a commercial CE instrument to measure dispersion in experiments conducted with both fronts and pulses can lead to large errors in values of D. There are minimal differences between values of D determined using the analysis of Taylor and of Aris, which indicates that the measured errors are not a result of neglecting axial diffusion. Consequently, in the remainder of this work, we will use only the analysis of Taylor (eq 4) and consider only cases where Pe > 70 and τ > 1.4. Correction for the Effect of an Initial Ramp in Fluid Velocity. In this section, we analyze the effect of an initial ramp in the fluid velocity, such as that present in a CE instrument after injecting a plug or front of solute, on the analysis of dispersion. In the absence of such a ramp, the fluid velocity is assumed 810

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constant at a value of uss that results in an ideal residence time of tR ) LD/uss; in this instance, D can be obtained from eq 4. When an initial ramp in fluid velocity is present, we observe a residence time, tR,obs, that results from an average fluid velocity, u0 ) LD/ tR,obs. These velocities and residence times can be related as u0tR,obs ) usstR. To describe the situation quantitatively, we assume that the velocity ramp occurs until time ti, at which point the steady-state velocity is attained. If the average velocity during this ramp period is ui, then the overall average fluid velocity is related to the steadystate velocity by eq 9. For a commercial CE instrument, the

(

u0 ) uss 1 -

ti ui ti + tR,obs uss tR,obs

)

(9)

velocity increases at a constant rate. In the case of this simple velocity profile, eq 9 is simplified to give eq 10, and the ideal

(

u0 ) uss 1 -

)

1 ti 2 tR,obs

tR ) tR,obs - ti/2

(10) (11)

residence time is related to the observed residence time by eq 11. For more complex velocity profiles, the value of ui must be determined in order to correct for the ramp. If we assume that the initial ramp in fluid velocity results from a ramp in the applied pressure, we can relate ti to the rate of increase of applied pressure, ri, using eq 12. Combining eqs 11

ti ) 8LTLDη/tRRc2ri tR )

tR,obs + xt2R,obs - 16LTLDη/Rc2ri 2

(12) (13)

and 12, we obtain an expression for the ideal residence time in terms of the observed residence time (eq 13).

Inspection of eq 13 shows that, in the absence of an initial velocity ramp (i.e., when ri ) ∞), tR ) tR,obs. In this instance, fitting concentration profiles to eqs 2 and 3 in order to obtain tR,obs and σ2 and then applying eq 4 is sufficient to obtain D. In the presence of a velocity ramp, this approach will lead to errors in the measured value of D since tR * tR,obs. As shown by eq 13, at shorter observed residence times, the impact of the ramp will be more substantial than at longer times. As the residence time increases, the effect of the initial ramp becomes negligible. We expect that as tR,obsf ∞, the error in D should approach zero; Figure 2a shows that for experiments conducted with solute fronts the error is substantial at short residence times and decreases to within 1% at long residence times. To obtain accurate measurements of D in the presence of the initial ramp, we must correct for the presence of this initial ramp. We describe this correction in the next sections. As we show, for the analysis of fronts correcting for the initial ramp in velocity of the fluid is sufficient to obtain accurate values of D. For the analysis of pulses, corrections must be applied for both the finite width of the injection plug and the ramp in fluid velocity. Accurate Measurements of Solute Diffusivities Using Solute Fronts. In the previous section, we proposed a simple model to account for the presence of the initial ramp in fluid velocity when using Taylor’s analysis of dispersion. To use this correction to measure diffusivities accurately, the pressure ramp rate, ri, must be determined. We do so by assuming that the errors observed in Figure 2a can be attributed to the initial ramp. We use these data and eq 13 to fit ri as an adjustable parameter; a value of ri ) 2.0 ( 0.1 psi/s gives the smallest value of error when applied to the data in Figure 2a. Therefore, the experiments shown in Figure 2a have velocity ramps that persist for 1-20 s. These ramp times are long relative to the viscous response time, tv (given by Rc2/ν where ν is the kinematic viscosity), which is less than 1 ms for a capillary with a 25-µm radius. Since tv , ti, we can assume that the velocity maintains its parabolic profile for the duration of the experimentsa condition necessary for Taylor’s analysis to hold. We use this value of ri to correct the data in Figure 2a; Figure 3 shows the effect of this correction. The open triangles correspond to the observed data; we see a nonlinear relationship between σ2 and the residence time, tR,obs. We corrected tR,obs to account for the initial ramp and obtain the ideal values of tR. This correction results in a linear relationship between σ2 and tR (filled triangles in Figure 3), as predicted by eq 4. Applying the correction for the initial ramp allows the accurate determination of D from a single concentration profile to within 2%. Deviations between the observed and corrected data are largest at short times resulting in the greatest errors in D (as seen in Figure 2a). The value of ri we determine should be applicable to all experiments conducted on a particular CE instrument; we verify this generality by measuring the diffusivity of phenylalanine in water using three capillaries with different values of LT and Rc. Figure 4 shows the error in D measured on these capillaries over a large range of residence times both before (open symbols) and after (filled symbols) the correction for the presence of the initial ramp has been applied. In the absence of the correction for the initial ramp, errors as large as 31% (LT ) 59.8 cm, Rc ) 25.6 µm), 16% (LT ) 50.2 cm, Rc ) 26.0 µm), and 15% (LT ) 30.6 cm, Rc )

Figure 3. Data represented in Figure 2a plotted as σ2 vs tR. The open symbols are the data plotted in terms of the observed residence time, tR,obs. We convert tR,obs to the ideal residence time, tR, by accounting for the presence of the initial ramp using eq 13 and a ramp rate of ri ) 2.0 ( 0.1 psi/s; we plot the corrected data using filled symbols. Upon correction, we observe a linear relationship between σ2 vs tR as predicted by eq 4. These corrected data allow for measurement of D with less than 2% error. All of the data satisfy the conditions of τ > 1.4 and Pe > 70. Experimental conditions are the same as in Figure 2a.

Figure 4. Error in the measurement of D as a function of observed residence time for dispersion experiments conducted with fronts of phenylalanine in water on three capillaries with different dimensions. The open symbols represent uncorrected data while data with filled symbols have been corrected for the initial ramp in fluid velocity after injection of the solute. All data shown in the figure satisfy the conditions of τ > 1.4 and Pe > 70. Capillary dimensions are as follows: capillary 1 (squares) LT ) 59.8 cm, Rc ) 25.6 µm; capillary 2 (triangles) LT ) 50.2 cm, Rc ) 26.0 µm; capillary 3 (circles) LT ) 30.6 cm, Rc ) 12.8 µm.

12.8 µm) were measured on these capillaries. Upon correction for the initial ramp, we are able to measure values of D within 5% regardless of the capillary length, radius, or observed residence time (the conditions of Pe > 70 and τ > 1.4 are satisfied for all data shown). These results illustrate the importance of accounting Analytical Chemistry, Vol. 77, No. 3, February 1, 2005

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Figure 5. (a) Error in the measurement of D as a function of residence time for experiments conducted with injection plugs of initial volumes of 5.3 (triangles) and 42.4 nL (squares). The open points have been corrected for the initial width of the injected plug only (eqs 6 and 7); the filled points have been corrected for both the plug width and the initial ramp in fluid velocity (eq 13). (b) Correction of the data by extrapolation to account for the finite width of the injected plug. The open symbols are the observed tR and σ2 for injections with volumes ranging from 5.3 to 42.4 nL. The filled circles have been extrapolated from the open symbols to correct σ2 for the initial injection width; tR is corrected using eq 13. All data shown in the figures satisfy the conditions of τ > 1.4 and Pe > 70. Experimental conditions: 0.25 wt % phenylalanine in water at 25 °C; LD ) 48.6 cm, LT ) 59.0 cm, and Rc ) 25.3 µm.

for the presence of the initial ramp in velocity of the fluid when using Taylor’s analysis of dispersion to obtain accurate measurements of D on a commercial CE instrument. Accurate Measurements of Solute Diffusivities Using Solute Pulses. In the previous section, we showed that concentration profiles of fronts obtained using a commercial CE instrument allow accurate measurements of D to be obtained by applying Taylor’s analysis of dispersion with a correction for the initial ramp in velocity of the fluid. When the solute is precious, we prefer to perform dispersion experiments with pulses rather than with fronts. For experiments conducted with pulses on a CE instrument, we have to correct for the ramp in velocity after injection and for the finite width of the injected plug of solute. When using pulses, we observe that at large residence times the error in D does not approach zero (Figure 2b) as it does in the experiments conducted with fronts. We assume this difference between the plugs and the fronts can be attributed to the introduction of this second nonideality. We use the approach of Wakeham and co-workers12 and consider each nonideality to be an independent first-order perturbation to the system. We use their corrections for the effect of the initial width of the injection plug (eqs 6 and 7) and apply the correction we derived for the initial ramp (eq 13). Figure 5a shows the error in D measured in experiments with 5.3 and 42.4 nL injection plugs. The open data points have been corrected only for the initial width of the injection plug (eqs 6 and 7); the filled points have been corrected for both the finite width of the injected plug of solute and the initial ramp in fluid velocity (eq 13). From these data, we see that applying both corrections independently does not allow for accurate measurement of D; errors as large as -60% are measured. We also observe that the data for the 5.3 and the 42.4 nL injections, when corrected only 812 Analytical Chemistry, Vol. 77, No. 3, February 1, 2005

for injection volume, do not coincide. Since these data differ only in injection volume, we conclude that the literature corrections for the injection volume (eqs 6 and 7), in the presence of an initial ramp, are insufficient. To correct for the finite width of the injected plug, we performed experiments as a function of injection volume. According to eq 7, for a given residence time, a plot of σ2obs versus Vi should have a quadratic dependence. If we assume that the parabolic functionality of this correction is accurate, we can extrapolate to Vi ) 0 to obtain values of σ2 for an ideal δ-pulse. Figure 5b illustrates the effect of this extrapolation of σ2obs in order to correct for the finite width of the injected plug. The open symbols are the observed tR,obs and σ2obs for injections with volumes ranging from 5.3 to 42.4 nL. The filled squares have been extrapolated from the open symbols to correct σ2obs for the finite width of the injected plug; tR,obs is corrected using eq 13. These corrected data give values of D accurate to within 2%. The line on the plot represents the ideal case (eq 4), i.e., 0% error in D. As expected, we observe that the largest deviation from this line (and therefore the greatest error in D) occurs for the largest injection volume. In this way, we can correct for the injection volume and the initial ramp in order to obtain accurate measurements of D using Taylor’s analysis of dispersion of a solute pulse. Experiments conducted using fronts are a more straightforward approach to measuring the diffusivity since measurements as a function of injection volume are not required and should be performed whenever possible (e.g., when the solute is not precious). CONCLUSIONS In this paper, we critically examine the use of Taylor’s analysis of dispersion in glass capillaries to determine values of diffusivity. We conclude that the presence of an initial ramp in the fluid

velocity, which occurs when a commercial CE instrument is used, can produce large errors in values of diffusivity. We used phenylalanine as a model solute with a known diffusivity to demonstrate these errors and to show that the analysis of Taylor can be modified to account for this effect. Using this approach, we determined the ramp rate for the CE instrument, which allowed the accurate measurement of diffusivities from single experiments conducted with concentration fronts on capillaries of different dimensions. For analysis of dispersion of solute pulses, experiments must be corrected for the finite width of the injected plug, as well as for the initial ramp in fluid velocity after injection. Previously published corrections for the finite with of the injected plug fail in the presence of the initial ramp. To obtain accurate values of D using the dispersion of solute pulses, experiments needed to be performed using different injection volumes and the results extrapolated to zero volume. (16) Farnum, M.; Zukoski, C. Biophys. J. 1999, 76, 2716-2726. (17) Cheng, Y.; Prud’homme, R. K.; Thomas, J. L. Macromolecules 2002, 35, 8111-8121.

The application of Taylor’s analysis of dispersion to measurements conducted with a commercial CE instrument allows for the rapid, efficient measurement of solute diffusivities. This approach avoids limitations of other approaches for measuring diffusivities. In contrast to DLS, which only works for macromolecules, this approach is applicable to solutes of any molecular weight provided that the concentration is above the detection limit (detection with CE instruments can be via UV absorbance or fluorescence). In contrast to FRAP, no labeling of solutes is required. We expect this technique to be broadly applicable to studies in biophysical chemistry typically performed using DLS and FRAP, such as measurements of protein-protein interactions16 or protein diffusivities in polymer solutions.17 Analysis of dispersion using a CE instrument will allow such experiments to be conducted more rapidly with less consumption of expensive solutes. Received for review August 5, 2004. Accepted November 18, 2004. AC048846Z

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