Application of the Exact Dispersion Solution to the Analysis of Solutes

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Application of the Exact Dispersion Solution to the Analysis of Solutes beyond the Limits of Taylor Dispersion Seyi Latunde-Dada,*,‡ Rachel Bott,‡ Karl Hampton,‡ and Oksana Iryna Leszczyszyn‡

Anal. Chem. 2015.87:8021-8025. Downloaded from pubs.acs.org by INTL CTR GENETIC ENGRG & BIOTECH on 09/01/15. For personal use only.

Malvern Instruments Ltd., Grovewood Road, Malvern, Worcestershire WR14 1XZ, United Kingdom ABSTRACT: Taylor dispersion analysis (TDA) is a fast and simple method for determining hydrodynamic radii. The method is applicable under conditions that allow the solute molecules to diffuse appreciably across the cross section of the flow before its measurement. This mitigates the effects of early stage convection on the dispersion and thus imposes a lower bound on the value of the diffusion coefficient measurable at a given flow speed. In this paper, we use the exact solution to the dispersion problem to analyze solutes outside the limits of TDA. Furthermore, by modeling the early stage convection, we analyze a mixture of two solutes with significantly different sizes that mimics heavily aggregated samples. The results obtained from the fits in both cases were in good agreement to the expected values. This demonstrates the potential of the model to extend dispersion analysis to regimes well outside the TDA limits such as the analysis of large molecules and the use of high flow-rates.

T

convective−diffusion problem using the following dispersion equation:

aylor dispersion analysis (TDA) is a fast and absolute method for determining the diffusion coefficients, and hence the hydrodynamic radii of molecules. The method, sometimes referred to as Taylor-Aris dispersion, was first described by Taylor in his classic paper.1 In 1956, Aris developed the method further by accounting for the longitudinal diffusion of the molecules.2 This technique was first applied to the determination of gaseous3 and then liquid diffusion coefficients.4−6 With the use of fused silica microcapillaries, TDA regained interest and has been used to analyze amino acids, peptides, proteins, small molecules, macromolecules, nanoparticles and biosensors.7−23 The diffusion coefficient of the injected solute can be deduced by fitting Taylor’s solution to the concentration profile or taylorgram of the solute.24 Alternatively, this can be achieved by calculating the moments of the profile3−8,12,25,26 (the moment method) or by measuring its height and area.27 The analysis can either be carried out at a single detection point or at two spatially separated detection points. These methods are referred to as single detection TDA and double detection TDA, respectively. At early times, the dispersion of a solute is dominated by convection with very limited diffusion occurring across the flow laminae. Expressions for the concentration profile resulting from convection alone were derived by Taylor.1 Convective profiles are typically observed for solutes with small diffusion coefficients (or large hydrodynamic radii) and when measurements are undertaken at high flow-rates. At later times in the dispersion, Taylor considered the effect of molecular diffusion across the flow laminae by solving the © 2015 American Chemical Society

∂Cm ∂C ∂ 2C + um m = k 2m ∂t ∂x ∂x

(1)

where t is the time, x is the axial coordinate, Cm is the area average concentration of the solute, um is the average velocity and k is the dispersion coefficient that was considered to depend on the physical parameters but not on t and x. This is the assumption inherent in Taylor Dispersion Analysis and it therefore only applies at large values of t, i.e., in the long time limit and thus imposes lower bounds on the diffusion coefficients of the solutes that can be analyzed and upper bounds on the flow-rate. The conditions permissible for TDA have been defined explicitly.28 In 1969, Gill and Sankarasubramanian29 showed that the dispersion model is applicable to all values of t if the dispersion coefficient k is allowed to vary with t. This model therefore bridges the gap in time between pure convection and Taylor dispersion. It was also shown that an exact solution for the local concentration, Cm, can be constructed. To test the model, comparisons were made with the finite difference results of Gill and Ananthakrishnan30 and found to be in good agreement. Because this model is applicable at all times, the limits of TDA can, in theory, be superseded. High flow-rate measurements Received: June 8, 2015 Accepted: July 10, 2015 Published: July 10, 2015 8021

DOI: 10.1021/acs.analchem.5b02159 Anal. Chem. 2015, 87, 8021−8025

Article


Analytical Chemistry and relatively small diffusion coefficients (large values of Rh) may be analyzed. This paper investigates the application of the model to measurements run at conditions beyond the limits of TDA. The application is presented as two different models. The first model, termed the convection−dispersion (CD) model, will be applied to a mixture of two solutes with significantly different hydrodynamic radii with the run conditions that ensure the taylorgram is a superposition of purely convective and Taylordispersive profiles. The second model, termed the Gill− Sankarasubramanian (GS) model, will be applied to measurements of solutes with relatively low diffusion coefficients run at flow-rates beyond the limits of Taylor dispersion. The paper is organized as follows: first, the limits of TDA will be briefly discussed and then both models will be introduced. The models will then be fitted to relevant experimental profiles and the results compared with expected values.

where A is the peak amplitude and the dispersion coefficient k is related to the standard deviation of the Gaussian σ by k=

⎛ l s ⎞ Cc = 0: ⎜t ≤ − ⎟ 2um 4um ⎠ ⎝ ⎡ s ⎞⎤ 1⎛ l Cc = C0⎢1 − ⎜ − ⎟⎥ : ⎢⎣ t ⎝ 2um 4um ⎠⎥⎦ ⎛ l s l s ⎞ + ≥t≥ − ⎜ ⎟ 4um 2um 4um ⎠ ⎝ 2um

THEORY AND CALCULATION Taylor Dispersion Limits. Taylor28 suggested the following conditions as appropriate for Taylor dispersion analysis: (a) in order that the longitudinal molecular diffusion may be negligible compared with the dispersive effects, it is necessary that ru D≪ m (2) 6.9 where D is the diffusion coefficient, um is the average flow speed and r is the capillary radius. If a ratio of 10:1 is sufficient for inequalities, the following upper bound is obtained on the diffusion coefficient of the solutes: ru Dmax = m (3) 69 (b) in order that the solute molecules have enough time to diffuse fully across the section of the capillary before being observed, it is necessary that r 2um 4l

Cc = C0

(9)

(4)

r 2um l

Figure 1. Pure convection.

When there is a mixture of two solutes with significantly different hydrodynamic radii and the run conditions are such that the smaller component undergoes Taylor dispersion while the larger component undergoes purely convection, the resulting distribution is a superposition of a taylorgram eq 6) and a convection profile (eq 9. The sum of Cd and Cc in eqs 6 and 9 is what we shall term the convection−dispersion (CD) model, which can be fitted to such profiles to determine (a) the hydrodynamic radius (or diffusion coefficient) of the smaller component and (b) the relative proportions of the two components from the areas under each individual profile. The parameters varied in the fits to the convection profiles Cc are C0, l and s as well as the slope and intercept of the baseline with the usual parameters (σ, tr and A) varied in the fits to Cd. Gill−Sankarasubramanian (GS) Model. The GS model is described in detail in [29] and is derived from the solution to eq 1 with the assumption that k is dependent on time t. The solution is

(5)

In this paper, the extension of the second condition to lower values of the diffusion coefficient will be considered. Convection−Dispersion (CD) Model. The average concentration distribution Cd, which arises at a time t when a small plug of solute undergoes Taylor dispersion, is given by Cd ∝ C0

⎛ s l s ⎞ : ⎜t ≥ + ⎟ 4umt ⎝ 2um 4um ⎠

where s is the length of solute injected and l is the distance between the point of injection and measurement of the solute. The second term of the expression describes the steep rise in the profile after the first arrival at twice the average flow speed of the solute molecules at the measurement point while the third term describes the long tail observed subsequent to the rise. Both these features are observable in Figure 1.

where l is the distance between the point of injection and observation. Again, if a ratio of 10:1 is sufficient for inequalities, the following lower bound is placed on the diffusion coefficient of the solutes: Dmin = 2.5

(8)

from which the hydrodynamic radius and diffusion coefficient of the solute molecules may be determined. Under pure convection,1,29 the diffusion terms on the righthand side of eq 1 are neglected and the solution obtained for the average concentration distribution Cc is

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D≫

u 2σ 2 2tr

tr −u2(t − tr)2 /4kt e t

(6)

where C0 is the initial solute concentration, tr is known as the mean residence time, u is the mean flow speed of the carrier solution and k is the dispersion coefficient. At large values of t, eq 6 is approximated by a Gaussian distribution: 2

Cd = Ae−(t − tr)

/2σ 2

(7) 8022


Article

Analytical Chemistry l ⎫ ⎡ ⎧ s ⎧ s + l ⎫⎤ ⎪ 4u C 0 ⎢ ⎪ 4u m − 2u m ⎪ 2u m ⎪⎥ ⎬ + erf⎨ m ⎬⎥ CGS = erf⎨ ⎢ 2 ⎢ ⎪ 2 ξ ⎪ ⎪ 2 ξ ⎪⎥ ⎭ ⎩ ⎭⎦ ⎣ ⎩

uncoated capillary (ID 75 μm, OD 360 μm, Malvern Instruments Ltd., Worcestershire, UK) having dimensions l1 = 0.445 m and l2 = 0.845 m for the distances from the inlet to two detection windows and a total capillary length of 1.30 m. Delivery of narrow solute plugs was achieved by pressuredriven injection at 50 mbar for 12 s, which provides a solute injection length s of about 8.1 mm, assuming a liquid viscosity of 0.8905 cP. Elution of sample plugs was undertaken at two run pressures: 500 and 1000 mbar. According to eq 5, it is expected that caffeine molecules will be Taylor-dispersed whereas the nanopheres standard molecules will undergo pure convection under these run conditions. Elution was monitored using a 214 nm wavelength filter at 25 °C. Nine replicates of each measurement were carried out. Using the CD model, the hydrodynamic radius of the caffeine molecules was determined, as well as the area under its individual taylorgram expressed as a percentage of the total area. Independent measurements of the two individual components provided an expected value for the percentage area of caffeine in the mixture: 13%. GS Model. The taylorgrams of five different solutes: 1 mg/ mL caffeine (Rh ∼ 0.3 nm, prepared in deionized water; SigmaAldrich, Suffolk, UK) 1 mg/mL Myoglobin (Rh ∼2.1 nm, prepared in Phosphate Buffer Saline (PBS), pH 7.4; SigmaAldrich, Suffolk, UK), 1 mg/mL Bovine Serum Albumin (BSA, Rh ∼3.8 nm, prepared in PBS, pH 7.4; Sigma-Aldrich, Suffolk, UK), 4% (v/v) Nanospheres 3000 series size standards (200 and 300 nm; Fisher Scientific, Leicestershire, UK) were acquired using the Viscosizer TD instrument (Malvern Instruments Ltd., Worcestershire, UK). Delivery of narrow solute plugs was achieved by pressure-driven injection at 50 mbar for 12 s and the elution undertaken at run pressures of 140 mbar for the first three solutes and 18 mbar and 12 mbar respectively for the latter two. These run pressures are applicable to TDA. One measurement was carried out for each solute. To generate taylorgrams outside the limits of TDA Nanospheres 3000 series size standards (Rh ∼ 100, 200 and 300 nm; Fisher Scientific, Leicestershire, UK) were prepared in 0.01 M NaCl at a final concentration of approximately 4% (v/ v). Samples were injected using the same conditions outlined above. Data was acquired for run pressures of 250, 300, 500, 750, 1000, 1500, 2000 and 3000 mbar for the 100 nm size standard; 125, 250 and 500 mbar for the 200 nm size standard; and 85, 170, 340 for the 300 nm size standard. Three replicates were carried out for each measurement and the results from the GS model and the Taylor-dispersion (TD) fits were compared. Note that for the GS fits, the results from each of the two detection points were treated independently and hence two values were obtained for the diffusion coefficient per measurement from which a mean was computed.

(10)

where ξ=

n=1 ⎛ 1 Bn 1 ⎞ ⎜ ⎟τ − 4 ∑ + (1 − e−λnτ ) 2 ⎝ Pe 2 192 ⎠ λ n ∞

J1(λn) = 0


Bn =

J3(λn)J2 (λn) λn5[J0 (λn)]2

2rum D Dt τ= 2 r Pe =

and Jn is the Bessel function of the n-th kind and Pe is the Peclet number. As discussed earlier, this is the exact solution to the dispersion equation that is applicable at all times after pure convection is no longer dominant and diffusion becomes significant. Figure 2 is an example of such a profile.

Figure 2. Taylorgram showing the transition between pure convection and early time dispersion.

As described earlier, the steep rise followed by a tail that indicates pure convection can be observed at early times in the profile. This portion of the profile can be fitted to Cc in eq 9. The transition from pure convection to dispersion can then be observed as a change in slope (or kink) in the profile indicated by the arrow in the figure. This time can be determined algorithmically by locating the change in the sign of the slope in this region. Once this transition point is detected, the GS model in eq 10 can be fitted to the dispersive part of the profile after this time and the diffusion coefficient of the solute estimated. The parameters varied in the fits are C0 and D as well as the slope and intercept of the baseline.

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RESULTS AND DISCUSSION CD Model. Figure 3 shows examples of the CD fits to taylorgrams from the mixtures of caffeine and 100 nm nanospheres at 500 and 1000 mbar, respectively. As can be seen, although the fits to the convected portions of the taylorgrams are not very accurate (perhaps due to some early term dispersion that is not modeled), the fits to the Taylordispersed peaks match the data well. Table 1 shows the results obtained for the mean hydrodynamic radius and % area of caffeine in the mixture. The estimated percentages compare well with the expected value of 13%.

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EXPERIMENTAL SECTION CD Model. A 50:50 (v/v) mixture comprising 1 mg/mL caffeine (Sigma-Aldrich, Suffolk, UK) and 4% (v/v) Nanospheres 3000 Series size standards (Rh ∼ 100 nm; Fisher Scientific, Leicestershire, UK) was prepared in 0.01 M NaCl (Sigma-Aldrich, Suffolk, UK). Taylorgrams were acquired using the Viscosizer TD instrument (Malvern Instruments Ltd., Worcestershire, UK) fitted with a standard two-window 8023


Article

Analytical Chemistry


Table 2. Estimated Hydrodynamic Radii from GS and TD Fits for Solutes Run within the Limits of TDA solute

run pressure (mbar)

estimated Rh (nm, GS fits)

estimated Rh (nm, TDA fits)

caffeine (Rh ∼ 0.3 nm) myoglobin (Rh ∼ 2.1 nm) BSA (Rh ∼ 3.8 nm) 200 nm nanospheres 300 nm nanospheres

140 140 140 18 12

0.34 2.2 3.9 195 286

0.32 2.1 3.7 198 282

Figure 3. CD fits (black) to a taylorgrams (gray) from a mixture of caffeine and 100 nm nanospheres run at 500 and 1000 mb, respectively.

Table 1. Estimated Hydrodynamic Radii and % Areas of Caffeine in Mixture run pressure (mbar)

estimated Rh of caffeine (nm)

estimated % area of caffeine

500 1000

0.28 ± 0.01 0.30 ± 0.01

13.0 ± 0.1 11.7 ± 0.3

GS Model. Figure 4 shows the GS and Taylor-dispersion (TD) fits to a taylorgram obtained from Myoglobin at 140

Figure 5. GS fits (solid) and TD fits (dashed) to taylorgrams (gray) from 100 nm nanospheres at 2000 mbar, 200 nm nanospheres at 250 mbar and 300 nm nanospheres at 170 mbar, respectively.

Figure 4. GS fits (solid) and TD fits (dashed) to a myoglobin taylorgram (gray) within TDA limits.

all the measurements conducted are shown in Table 3. Also shown for comparison are the hydrodynamic radii obtained from the TD fits. As expected, the results from the GS fits are in better agreement with the nominal values than the results from the TD fits. For two measurements, the radii obtained from the TD fits were negative because the fitted standard deviations were larger at the first detection point than at the second.

mbar. This run condition is suitable for TDA and hence, as expected, both the GS and TD fits match the data very well. In Table 2 is presented the hydrodynamic radii obtained from similar fits to taylorgrams from a range of solutes measured at run pressures permissible for TDA. The results from the two fits are in very good agreement. Figure 5 shows GS and TD fits to taylorgrams obtained for the 100, 200 and 300 nm nanospheres measured at run pressures that lie outside the limits of TDA. The TD fits, as expected, do not match the taylorgrams well. As discussed earlier, the times of transition between convection and dispersion were determined automatically and the GS fits applied to the dispersive region of the profiles. The results from

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CONCLUSIONS In this paper, the application of the exact solution to the convection-diffusion problem to solutes run under conditions outside the limits of TDA has been demonstrated successfully. The model was applied to a mixture of two solutes with 8024


Article

Analytical Chemistry

(17) Cottet, H.; Biron, J. P.; Martin, M. Anal. Chem. 2007, 79 (23), 9066−9073. (18) Le Saux, T.; Cottet, H. Anal. Chem. 2008, 80, 1829. (19) D’Orlye, F.; Varenne, A.; Gareil, P. J. Chromatogr. A 2008, 1204, 226. (20) Franzen, U.; Vermehren, C.; Jensen, H.; Ostergaard. Electrophoresis 2011, 32, 738. (21) Boyle, W. A.; Buchholz, R. F.; Neal, J. A.; McCarthy, J. L. J. Appl. Polym. Sci. 1991, 42, 1969. (22) Quinn, J. G. Anal. Biochem. 2012, 421, 391−400. (23) Quinn, J. G. Anal. Biochem. 2012, 421, 401−410. (24) Umecky, T.; Kuga, T.; Funazukuri, T. J. Chem. Eng. Data 2006, 51, 1705−1710. (25) Kelly, B.; Leaist, D. G. Phys. Chem. Chem. Phys. 2004, 6, 5523− 5530. (26) Alizadeh, A.; De Castro, C. A.; Wakeham, W. A. Int. J. Thermophys. 1980, 1, 243−284. (27) Barooah, A.; Chen, S. H. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 2457−2468. (28) Taylor, G. Proc. R. Soc. London, Ser. A 1954, 225, 473−477. (29) Gill, W. N.; Sankarasubramanian, R. Proc. R. Soc. London, Ser. A 1970, 316, 341−350. (30) Gill, W. N.; Ananthakrishnan, V. AIChE J. 1967, 13, 801.

Table 3. Estimated Hydrodynamic Radii from GS and TD Fits for Nanospheres Run Outside the Limits of TDA Rh (nm)

run pressure (mbar)

100

250 300 500 750 1000 1500 2000 3000 125 250 500 85 170 340


200

300

estimated Rh (nm, GS fits)

estimated Rh (nm, TDA fits)

± ± ± ± ± ± ± ± ± ± ± ± ± ±

118 ± 1 176 ± 8 125 ± 1 107 ± 1 91 ± 2 23 ± 35 negative Rh negative Rh 248 ± 6 248 ± 10 163 ± 6 421 ± 78 365 ± 7 218 ± 6

90 115 101 99 100 94 105 76 184 196 187 273 269 288

4 15 4 3 9 18 15 7 11 28 17 28 5 7

significantly different hydrodynamic radii to mimic a heavily aggregated sample and the radius and proportion of the small component was determined with reasonable accuracy. In this case, the pure convection of 100 nm nanospheres was modeled in order to isolate the dispersed taylorgram of caffeine molecules. Furthermore, the early time dispersion model was successfully used to determine the hydrodynamic radii of large nanospheres run under conditions outside the limits of TDA. These results offer the potential scope to extend dispersion analysis to regimes outside the limits of TDA such as the analysis of large molecules and the use of high flow-rates.

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AUTHOR INFORMATION

Corresponding Author

*S. Latunde-Dada. E-mail: [email protected]. Author Contributions ‡

These authors contributed equally.

Notes

The authors declare no competing financial interest.

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REFERENCES

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Application of the Exact Dispersion Solution to the Analysis of Solutes

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