Numerical modeling and experimental optimization of Taylor dispersion analysis with and without an electric field

Numerical modeling of Taylor dispersion analysis (TDA) was performed using COMSOL Multiphysics to facilitate better and faster optimization of the experimental conditions. Parameters, such as pressure, electric field, diameter, and length of capillary on the TDA conditions, were examined for particles with hydrodynamic radius (Rh) of 2.5–250 Å. The simulations were conducted using 25, 50, and 100 cm length tubes with diameters of 25, 50, and 100 µm. It was shown that particles with larger diffusion coefficients gave more accurate results at higher velocities, and in longer and wider columns; particles with smaller diffusion coefficients gave more accurate results at smaller velocities, and in shorter and thinner columns. Moreover, the effect of electric field on the validity and the applicability of TDA was studied using TDA in conjunction with capillary electrophoresis. Diffusion coefficients were obtained using a pressure and the TDA equation and compared with those obtained with a pressure in combination of an electric field for fluorescein, FD4, FD20, FD70, and FD500. We found that TDA can be used with the presence of moderate electrophoretic migration and electroosmotic flow, when appropriate conditions were met.


INTRODUCTION
Taylor dispersion analysis (TDA) is a technique used to determine the diffusion coefficient and the size of a variety of particles, such as nanoparticles, therapeutic proteins (including antibodies), polymers, and even small molecules [1][2][3][4][5][6].TDA is becoming more popular due to its simplicity, accuracy, and its ability to determine a wide range of molecular sizes without the need for calibration.TDA gives results that are consistent with well-established methods such as dynamic light scattering and size exclusion chromatography and offers some advantages over these techniques in terms of sensitivity and simplicity [7][8][9].Although TDA has been used as a sizing technique for a few decades, more in-depth studies are still needed to shed light on how molecules behave in solution while flowing through a narrow tube and obtain accurate size estimates and precise results [9,10].Adding an electric field along the capillary provides another intriguing aspect because particles with different charge to size ratios can now be separated while providing size information at the same time.
TDA is based on Taylor's work which was extended by Aris [11,12].Taylor described the dispersion of a solute plug in an open tube under Poiseuille laminar flow.When a parabolic velocity profile is combined with molecular diffusion, particles redistribute in the cross section of the capillary tube, which leads to longitudinal dispersion.Under certain conditions, molecular diffusion coefficient, D, is related to the temporal variance ( 2 ) of the Taylorgram for an analyte, the average migration time (), and the capillary radius (  ), of the tube by the following equation [13]: The diffusion coefficient can be used to calculate the size of particle using the Stokes-Einstein equation [13]: where  ℎ is hydrodynamic radius of the particle, K is the Boltzmann constant, T is temperature, η is viscosity, and D is the diffusion coefficient.TDA is performed in a narrow tube, and a pressure is used as the driving force to push a sample plug through to the detector.Capillary electrophoresis (CE) instrument equipped with a pressure pump is often used to do TDA experiment.The migration time and temporal variance of the analyte depend on three factors: (i) the applied pressure, (ii) the capillary length, and (iii) the capillary radius.Taylor discussed two conditions that need to be fulfilled when using TDA experiments to measure particle size.The first condition is that the migration time must be much longer than the characteristic time needed for the diffusion of the analyte across the capillary radius [11]: Equation ( 3) is used to confirm the validity of the first condition, where  0 is the minimum elution time to fulfill the first TDA condition,   is the capillary radius, D is diffusion coefficient, and  is the relative error on the determination of D that can be tolerated.Equation (3) can be written in terms of capillary length (L) and linear velocity of an analyte () [11,14]: The second condition states that longitudinal dispersion must be negligible compared to radial diffusion.To meet this condition, the Peclet number (Pe), which is the ratio of the contributions to mass transport by convection to those by diffusion, must be bigger than a certain value that depends on the maximum error percentage of the diffusion coefficient [11].
Taylor's two conditions also need to be optimized to reduce the relative error in the calculation of the diffusion coefficient.The variance contributed by injection is usually smaller than the axial longitudinal diffusion and is not considered in this study.
Cottet et al. made a substantial contribution in the field of TDA by investigating the optimum conditions for performing TDA with a CE instrument, and developing methods to analyze mixtures.They used the cumulant method, which is widely used to derive size distribution information for polydisperse samples [15] and to quantify size polydispersity for solutions of polymers and polymer mixtures under optimized conditions [16].Most of the research to date on TDA parameter optimization is carried out by performing a series of experiments, which can be laborious and time-consuming.Because the principles of Taylor dispersion are well known, numerical modeling can be a reliable addition to experimentation.Using numerical method and experimental verification, Khodabandehloo et al. showed that electroosmotic flow dispersion during electrophoresis is similar to Taylor dispersion [17].Combined dispersion models were developed to determine the sources of peak broadening of larger molecules such as proteins in the presence of electroosmotic flow.The results showed that larger molecules such as proteins have wider peaks than the peaks of smaller molecules, because of the band broadening during analyte migration due to the increased length of the Debye layer.The results, both from simulation and experiments, showed that the traditionally accepted longitudinal diffusion as the only band broadening factor in CE (peak variance equals to 2Dt) is not correct.
Because earlier versions of finite element method software were not able to simulate analyte migration phenomena in long enough tubes, we had to use capillaries of much smaller dimensions for simulation and then infer the effect of the different parameters to situations where a bigger scale system was used.Recent improvement in the software allowed us to directly use the real dimension of the capillary used in the experiment.In this work, COM-SOL Multiphysics was used to further investigate Taylor dispersion in real sized narrow tubes, making it possible to optimize the experimental conditions faster and better.This study showed the effect of pressure, diameter, and length of column on the precision and accuracy of the particles obtained for molecules with R h of 2.5-250 Å.In all cases, the suitable TDA conditions are suggested, and the optimum conditions are discussed.Moreover, the effect of electric field on the applicability of TDA for molecules with different sizes was studied using CE.We concluded that parameters must be chosen in such a way that Taylor's two TDA conditions are both met.

SIMULATIONS
The migration of analytes in a two-dimensional simulated capillary tube was calculated using a finite-elementscheme software, COMSOL Multiphysics 6.0 (COMSOL Inc.).The modeling procedure with this program included creating a virtual model with a 2D geometry (a 2D model component was used), defining parameters, meshing, solving differential equations, and post-processing of data.Detailed steps and descriptions of modeling procedure can be found in Table S1.In this study, the molecules were mobilized by the simulated hydrodynamic flow field and electric field.Both electric and fluid fields were calculated in a steady state before solving the mass balance equation, through which the concentration distribution of the analyte was calculated in a time-dependent manner.The variance of peaks was calculated by fitting a Gaussian curve to the resulting data (see Figure S1).Diffusion coefficients were calculated using Equation (1), and the error percentage was calculated by comparing the obtained diffusion coefficient with the input value.The simulations were conducted on a 25, 50, and 100 cm length virtual tube with a diameter of 25, 50, and 100 µm (radius of 12.5, 25, and 50 µm).TDA with only pressure was investigated first, and the results were plotted and tabulated.A voltage was then applied along with the pressure in the TDA experiment, and the results were compared with pressure only TDA.In all cases, the validity of TDA was confirmed by calculating Peclet numbers and the characteristic times.To make sure that band broadening caused by injection was minimal, the injection volume was controlled to be less than 1% of the column volume.

Chemicals
All chemicals were Analytical Grade and were purchased from Sigma-Aldrich.HCl solutions were diluted to desired concentrations, and pH was measured at room tempera-ture.Fluorescein solutions of 0.5 and 10 mg/mL of FD4, FD20, FD70, and FD500 were prepared by dissolving solid reagent in borate buffer.Borate buffer (pH 8.6) used in this experiment was prepared by adjusting a 50 mM Na 2 B 4 O 7 ⋅10H 2 O (borax) solution with 200 mM boric acid.

Apparatus
A Capel-205 CE system (Lumex Instruments Canada) was used for the experiment.The capillary had a 50 µm inner diameter, 360 µm outer diameter, and was 50 cm in length (43 cm to the UV absorption detector set at 254 nm).
The uncoated fused silica capillary was purchased from Polymicro Technologies (Phoenix) and cut to length in house.To precondition a new capillary, the fused silica capillary was cleaned by pushing 0.1 M NaOH solution through with a 1000 mbar pressure for 3 min, followed by a 1-min rinse with deionized distilled water.The instrument was programmed to rinse the capillary with borate buffer as background electrolyte at high pressure (1000 mbar) for 3 min before each pressure injection.Fluorescein, FD4, FD20, FD70, and FD500 were introduced by pressure injection at 50 mbar for 5 s, and a voltage of 5 and 10 kV was used to run the separations for a period of 20 min.

Theoretical simulations
The Taylor dispersion behavior of three particles with different diffusion coefficients was calculated using finiteelement-scheme simulation.The fluid velocity field of a solution is described by the simplified Navier-Stokes equation [18]: where p is the pressure, η is the viscosity of the solution, and ⃗  is the fluid velocity vector of any point in the fluid field.The conservation of mass principle gives the relationship of ∇ ⋅ ⃗  = 0, which is used in conjunction with the Navier-Stokes equation to solve for the pressure and velocity.The concentration of analytes is determined by numerically solving the mass balance equation [19]: where C is the analyte concentration at a specific location on the column, D is the diffusion coefficient, and ⃗  is obtained from the previous calculation for fluid field.
Thus, the analyte concentration and its variation with time at any location were calculated to determine the analyte dispersion along the column.
Different fluid velocities, capillary diameters and capillary lengths were used, as stated in the following sections.Moreover, the effect of applying a voltage on Taylor dispersion is studied as well.In that case, the equation included both a pressure-driven term and an electric field-driven term for convective migration when calculating average linear velocity () [17,20]: where P is the pressure, R c is the capillary radius, L is the length of capillary,  is the viscosity,   is the electrophoretic mobility, and E is the applied electric field.

Dispersion of particles in the presence of Poiseuille flow
TDA uses pressure (Poiseuille flow) to mobilize analytes in a capillary column.The particles need to spend enough time in the tube so they can redistribute in the radial direction, but not so long that axial diffusion becomes significant.These criteria are met in the first and second conditions of TDA.To calculate the diffusion coefficient, the elution time and temporal variance of analytes in the capillary tubes are calculated based on the applied pressure, the capillary length, and the capillary radius.The effects of changes on applied pressure, capillary length, and capillary radius on TDA were investigated in this work [16].

Effect of linear velocity (pressure)
When conducting a TDA experiment, a constant pressure was used to maintain a constant linear velocity for the bulk flow in the capillary column.As linear velocity appears in the TDA second condition (Equation 5), we used linear velocity instead of pressure when investigating its effect on TDA performance.According to Poiseuille's law, Equation (9), in a laminar flow, the pressure and linear velocity are directly proportional when the viscosity of the solution and the radius of the column are fixed [14,21]: To study the effect of pressure (or linear velocity) on TDA performance, and on TDA required conditions, dif- ferent velocities were used in this study.Figure 1A,B shows the simulated Taylorgrams for particle with diffusion coefficients of 5E − 11 m 2 /s (molecules of about 50 000 Da in an aqueous solution) and 5E − 10 m 2 /s (small molecules such as fluorescein in an aqueous solution) at different linear velocities.For particles with a smaller diffusion coefficient, the Taylorgram at lower velocity follows a Gaussian distribution as opposed to the higher velocity where R 2 deviates from 1 based on Gaussian curve fitting.Percentage error for particles of different diffusion coefficients at different linear velocities is shown in Figure 2A.The deviation from Gaussian distribution is also evident in Figure 2B where for smaller diffusion coefficients: The R 2 deviates from 1 as the linear velocity increases.This distortion in the peak shape affects the accuracy of the diffusion coefficient obtained from Equation (1), which leads to a higher error on the diffusion coefficient obtained.To explain this divergence, we need to assess the validity of the two TDA conditions at different velocities.Table 1 shows the minimum values of elution time for fulfilling the TDA first condition (elution time > > t 0 ), and Table 2 shows the minimum values of Pe for fulfilling the TDA second condition (the second condition states that longitudinal dispersion must be negligible compared to radial diffusion).To meet this condition, the Pe, which is the ratio of the contributions to mass transport by convection to those by diffusion, must be bigger than a certain value that depends on the maximum error percentage of the diffusion coefficient.
As shown in Table 1, according to Equation (3), for a particle with a diffusion coefficient of 1E − 11 m 2 /s, to fulfill the TDA first condition, it must spend at least 50 s in a 25 µm diameter and 50 cm long capillary.When the diameter of the capillary increases to 50 and 100 µm, the minimum elution time increases to 312.5 and 1250 s, respectively.For a particle with a diffusion coefficient of 1E − 9 m 2 /s (very small molecule in an aqueous solution), the minimum time needed to fulfill the first TDA condition is 0.8 s in a 25 µm diameter and 50 cm long capillary (to the detector).When the diameter of the capillary increases to 50 and 100 µm, the minimum migration time increases to 3.1 and 12.5 s, respectively.Minimum migration time corresponds to maximum average linear velocity.It should be noted that for a TDA experiment using a 50 µm inner diameter capillary, the migration time should be at least 5.2 min or longer.According to Figure 2A, higher velocity has more adverse effect on the first condition than favorable effect on the second condition.To fulfill the TDA second condition with high enough Pe number, higher velocity is desirable as it gives a higher Pe, which means that less time is available for the axial diffusion of particles in the column.As shown in Table 2, at higher velocities, the elution time is shorter than the minimum time required, and the particles do not have enough time to distribute uniformly in the radial direction.Therefore, the particles do not follow Gaussian distribution, which leads to higher percentage error in the evaluation of the diffusion coefficient.As a result, in addition to the second TDA condition to be valid, the first TDA condition is also valid at higher velocities.Therefore, as shown in Figure 2B, R 2 becomes closer to 1 as the linear velocity increases for larger molecules with smaller diffusion coefficients.To summarize, particles with larger diffusion coefficients conform to the TDA conditions and give more accurate results at higher velocities compared to (larger) particles with smaller D values.Therefore, to satisfy TDA conditions for a mixture of particles with different diffusion coefficients, a linear velocity of 5E − 3 m/s was chosen.Simulations performed at lower velocities (i.e., 1E − 3 m/s) are not practical due to both the long run times and simulation times and also showed comparable results with simulations performed at a linear Note: Pe number is the ratio of the contributions to mass transport by convection to those by diffusion is calculated using Equation (5).For relative error of value 7.50E − 3, Pe number must be greater than 80 for longitudinal dispersion to be negligible compared to radial diffusion.
velocity of 5E − 3 m/s.Figures 3 and 4 show the Taylorgram of five particles velocity of 5E − 3 m/s at diameters of 25, 50, and 100 µm and capillary lengths of 25, 50, and 100 cm.

Effect of capillary diameter
Capillary columns are available in different diameters.The inner diameter is typically 20-100 µm, and the outer diameter is 150 or 375 µm.The capillary tubes are coated with a layer of polyimide on the outside to add to their mechanical strength.The inner diameter of the column is one of the factors that determine the minimum time the particles need to spend in the column to meet the first TDA condition for time needed for radial diffusion.The larger the inner diameter, the more time it takes for the particles to redistribute the column, and the longer the minimum time is needed for fulfilling the first TDA condition.The minimum time also increases when the diffusion coefficient is smaller, because the rate of diffusion across the column is slower and so more time is needed for the particles to be redistributed.Three columns with the diameter of 25, 50, and 100 µm, respectively, are used for simulation in this work.Table 1 shows the minimum time (calculated with Equation 3) for particles of different diffusion coefficients at different diameters.For larger particles (smaller diffusion coefficients of 1E − 11) in larger diameter capillaries of 100 µm, the minimum time is 1250 s, whereas, for small particles (larger diffusion coefficients of 1E − 9) in smaller diameter capillaries of 25 µm, the minimum time is 0.8 s.According to Equation (9), by increasing the capillary diameter, the applied pressure must decrease to maintain a constant linear velocity.Figure S2 is divided into six parts and shows Taylor dispersion of particles with different diffusion coefficients by changing the diameter of the capillary column.We used two opposite extreme linear velocities to explore the two conditions of TDA as well.On the left, A1, B1, and C1 show the low-velocity extreme of 5E − 4, and on the right, A2, B2, and C2 show the high-velocity extreme of 5E − 2. In general, when a higher velocity is used, Taylorgrams deviate from normal distribution.This deviation becomes even more significant when the column's radius increases.TDA's first condition requirement explains these anomalies.According to the first condition for performing TDA, the migration time must be much larger than the characteristic diffusion time of the solute in the capillary cross section.In Table 1, this characteristic diffusion time is reported for each particle at each capillary diameter.
When TDA is performed in a diameter of 100 µm (Figure S2-C1,C2), the minimum time for the particles to meet the first TDA condition is 12.5 s, which is longer than the minimum times in smaller diameters (for 25 and 50 µm diameters, the time is 0.8 and 3.1 s, respectively).In capillaries with larger diameter, the minimum time increases because the particles need to travel longer distances from the center to the wall and vice versa.The increase in minimum time is more significant for particles with smaller diffusion coefficients because of their smaller rate of diffusivity.When a high velocity such as 5E − 2 m/s was used, the time that the particles spend in the column was shorter than the minimum time needed for the particles to redistribute in the column with a diameter of 100 µm.Therefore, particles do not have enough time to redistribute radially and form a normal distribution Taylorgram.To meet the first TDA condition, a smaller linear velocity is preferred when using capillary columns with larger diameters.
TDA's second condition must be fulfilled as well for Equation (1) to be valid.The linear velocity should be fast enough so that the axial diffusion can be neglected compared to the Taylor dispersion term.Therefore, the velocity has to be fast enough to meet the second TDA condition.
When TDA is performed in a diameter of 25 µm (Figure S2-A1,A2), the minimum time to meet the first TDA condition is 0.8 s for a particle with a larger diffusion coefficient (1E − 9 m 2 /s), such as a water molecule, and 78.1 s for a particle with a smaller diffusion coefficient (1E − 11 m 2 /s), such as a large protein molecule.At higher velocities in  this small diameter column, the particles have enough time to diffuse radially.For larger particles, the migration time is smaller than the minimum time, but the time is still close.Therefore, the deviation from normal distribution is not significant.In summary, as the diameter of the capillary decreases, less time is needed for TDA condi- tions to be met.Therefore, higher velocities can be used for small diameter capillaries to meet the TDA's first condition.Using higher velocities is also advantageous for fulfilling the second TDA condition requirement as there is not enough time for the axial diffusion of particles, whereas, at lower velocities, the axial diffusion becomes significant which falls short of TDA's second condition.
Comparing Figure S2-A1,A2, at a higher velocity, the peak widths are very different, whereas, for a smaller velocity, they are similar.At lower velocity (Figure S2-A1), the particles were spending too much time in the column.Thus, the axial diffusion was more significant, which was also contributing to the peak width of the Taylorgram.In this case, more axial diffusion was observed for smaller particles that added to the peak width, resulting in a broader peak.
For capillaries with a diameter of 50 µm, higher velocity cannot satisfy the first TDA condition, which is more significant for particles with smaller diffusion coefficients, leading to Taylorgram peaks that do not follow a Gaussian shape.At smaller velocities, the second TDA condition may not be satisfied, and the axial diffusion can become more significant, especially for particles with larger diffusion coefficients.Therefore, a medium linear velocity was chosen to meet both TDA conditions.
As the diameter increases, Pe number also increases.Therefore, the second TDA condition is met even at smaller velocities.Tables S2 and S3 show the value of the Peclet numbers for particles with different diameters.The Pe for a 25 µm diameter capillary is small.Therefore, to meet the second TDA condition, the linear velocity must be higher than 1E − 3 m/s to compensate for the small radius of the capillary.By combining minimum time and Pe to meet TDA conditions, it was concluded that for smaller diameters, higher linear velocity was needed to increase Pe to meet the second TDA condition that would be in accordance with the first TDA condition.For larger diameters to meet TDA first condition, lower linear velocity was needed to increase the migration time.As Pe is larger at large diameters, performing TDA at lower velocities does not contradict the second TDA condition.Therefore, it is expected that more accurate results will be obtained at lower velocities.
Figure S3-A,B shows the percentage error for the estimation of diffusion coefficient using the Taylorgrams and  2 for goodness of fit of Taylorgrams with different diameters at a velocity of 5E − 3 m/s.Comparing the combination of % error and R 2 for the three diameters, it was concluded that a capillary diameter of 50 µm has the best combination of % error and R 2 .For larger diameters, smaller velocities gave better result and for smaller diameters, larger velocities were better.The use of all diameters was possible for TDA when velocity was optimized for that specific diameter.However, experimentally, using larger diameters with slower bulk motion is time-consuming, and using diameters that are too small could risk clogging the column.

Effect of capillary length
In this section, we used capillary columns with different lengths to study TDA performance and conditions.Capillary length does not directly affect the Pe or the minimum time required for particles to spend in the column.It does affect the time that the particles spend in the column.For all capillary lengths, the Pe decreases as the diffusion coefficient increases (see Tables S4 and S5).As the Pe decreases, the second condition of TDA is no longer met, because axial diffusion becomes more significant.With a longer capillary length, to maintain a constant migration time, we need to increase the linear velocity, which leads to higher Peclet numbers and more favorable TDA conditions.At shorter lengths, the velocity must be smaller to give enough diffusion time to fill the column radially, which is more crucial for particles with smaller diffusion coefficients.However, the decrease in velocity leads to smaller Peclet numbers which is not good for meeting the second TDA condition.In general, higher velocities give better results when using longer columns and lower velocities is better for shorter columns.Figure S4-A,B shows the % error for the estimation of diffusion coefficients and the R 2 for the Taylorgrams fitted to Gaussian functions with different capillary lengths at a velocity of 5E − 3 m/s, respectively.All of these simulations were conducted at a linear velocity of 5E − 3 m/s.According to Table 1, for capillary columns with a diameter of 50 µm, the minimum time for the particles to spend in the column increases as the diffusion coefficient decreases.Therefore, shorter columns are not suitable for particles with larger diffusion coefficients, which is also confirmed in Figure S4-A,B.The Taylorgram for the blue peak deviates from a normal distribution as the column length decreases because the particles do not get enough time to fill the column radially, which leads to a smaller R 2 and higher % error (Figure S4-A,B).

Dispersion of particles in the presence of electrophoretic migration and electroosmotic flow
TDA offers advantages in terms of its accuracy, precision, and simplicity in estimating particle size.However, for a mixture of analytes, all of the particles elute at the same time, and as a result, data analysis for each particle becomes more complicated and inexplicit.Cottet et al. established the equations used to calculate the average diffusion coefficient for highly polydisperse samples [13].Separating the particles in the mixture is an alternative way to study Taylor dispersion [17].Unlike Poiseuille flow, Note: Column A shows results for experimental conditions in which the linear velocity is composed of only applied pressure of 100 mbar (see Equation 9).D A is the apparent diffusion coefficient that was calculated with Equation (1) (normal TDA conditions).Columns B and C show results for experimental conditions in which the linear velocity composed of both applied pressure of 100 mbar with 5 or 10 kV applied voltage (see Equation 8).D values were calculated with Equation (10) using D A values calculated in column A with Equation (1).The % error of R h values in columns B and C compared to the R h values in Column A is also shown.
electrophoresis is a discriminative process that separates particles based on their charge-to-size ratio.TDA in the presence of electrophoretic migration and electroosmotic flow is studied using CE for molecules with different electrophoretic mobilities.Figure S5-A shows the Taylorgram of FD20 when just pressure at different velocities, and Figure S5-B,C shows Taylor/electropherogram of FD20 when pressure is combined with an applied voltage of 5 and 10 kV, respectively.Under the conditions in our system, when applying voltage and pressure simultaneously, the net flow motion stems from the electrophoretic migration, electroosmotic flow, and Poiseuille flow.Therefore, solute molecules move faster due to the presence of electrophoretic mobility of particles, and so a constant forward shift for the flow along the tube is expected for positively charged particles in a positive electric field.Percentage error on the estimation of diffusion coefficient with TDA is presented in Figure S6-A when just pressure used and in Figure S6-B,C when pressure is combined with an applied voltage of 5 and 10 kV, respectively.According to Figure S6-A-C, the diffusion coefficient was underestimated when pressure was combined with voltage when comparing the calculated values when just pressure is used.The underestimation becomes greater when the voltage is increased from 5 to 10 kV (Figure S6-B,C).The % error is underestimated because we are using the equation derived for classical TDA, even though the particles are not under the same classical TDA conditions.Khodabandehloo et al. showed that equation for estimation of diffusion coefficient (D) must be modified in the presence of electroosmotic flow [17,20,22]: where D A is apparent diffusion coefficient that can be calculated using Equation (1), and δD eo is the additional dispersion term caused by electroosmotic flow.For our system,   can be calculated using the following equation [17,20,22]: where  is solution dielectric permittivity,  is the zeta potential, E is the magnitude of the electric field, η is the solution viscosity, and  is the Debye length.Column A in Table 3 shows calculated diffusion values using Equation (1) for experimental conditions in which the linear velocity is composed of only applied pressure of 100 mbar (see Equation 9).TDA equations do not need to be modified if linear velocity only includes a pressure term [17,20,22].Columns B and C show results for experimental conditions in which the linear velocity composed of both applied pressure of 100 mbar with 5 or 10 kV applied voltage.As linear velocity includes both a pressure and electric field term (see Equation 8), modified TDA equation must be used to accurately calculate the diffusion coefficients.Equation ( 10) was used to calculate the diffusion coefficients shown in columns B and C. Table 3 shows that modified Equation ( 10) calculates similar diffusion coefficients values (columns B and C) to diffusion coefficient values calculated with Equation 1. Equation 1is not valid to use for conditions in which linear velocity includes both applied pressure and voltage terms [17,20,22].In TDA experiments, two conditions must be fulfilled to be able to estimate the diffusion coefficient of particles using the elution time and the peak variance.These conditions must be revisited when electrophoretic migration and electroosmotic flow are added to the system.The combination of voltage and pressure results in higher velocities which decreases the amount of time molecules spend in the column.To meet the first TDA condition, the migration time must be long enough so that the molecules spread out radially; therefore, the linear velocity should be smaller than conventional TDA.By adding electrophoretic migration and electroosmotic flow, the second TDA condition is not disturbed because when the net velocity increases, the Pe increases, which fulfills the second TDA condition of negligibility of longitudinal diffusion [17,20,22].The provider of these dextrin standards listed the values of the Stokes' radii that are different from the values obtained in Table 3.The values from the vender are listed in Table S6.As the vender did not provide the method of which those values were obtained, it is difficult to comment on the reason why the values are different from those that were obtained in this work.Table 3 shows that with a voltage of 5 or 10 kV on a 50 cm capillary used for TDA, the diffusion coefficients and the hydrodynamic radii obtained are similar to those obtained with TDA without the use of a separation voltage.This observation suggests that it might be possible to obtain sizing information for charged species, whereas they are separated in a reasonably strong electric field.TDA with CE separations can reduce the need of deconvolution when multiple species are present in the sample mixture.

CONCLUDING REMARKS
Numerical modeling of TDA was used to study the effect of pressure, capillary length, and capillary diameter changes on the evaluation of diffusion coefficient of components of a mixture.The effect of applying a voltage on the system was also studied using CE in combination TDA.To estimate the diffusion coefficient of components of a mixture with different sizes accurately, the system parameters need to be optimized.Different lengths of capillary column can be used as long as TDA conditions are met.The simulation results demonstrated that for most molecules and small particles, a 50 µm inner diameter, 50 cm long capillary can be used for TDA measurement.When using both voltage and pressure for TDA, the equations must be modified accordingly.Although the diffusion coefficients, and in turn, the hydrodynamic radii obtained, are somewhat different in a normal TDA analysis compared to when a moderate electric field is applied during the TDA analysis, they are similar enough that the size of the separated molecules and/or particles can be estimated.This could in some cases reduce the difficulty in deconvoluting TDA peaks with multiple components in them.

A C K N O W L E D G M E N T S
This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.The Capel-205 capillary electrophoresis system was kindly loaned to us by Lumex Instruments Canada.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors have declared no conflicts of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

TA B L E 1 5
The minimum elution time to fulfill the first Taylor dispersion analysis (TDA) condition with capillary diameter of 25, 50, and 100 µm.Note: A relative error of value 7.50E − 3 was used for the calculations.

1
Simulated migration of particles with diffusion coefficient of (A) 5E − 11 m 2 /s (very large molecules) and (B) 5E − 10 m 2 /s (small molecules such as fluorescein) at different linear velocities in a column with diameter of 50 µm and 50 cm in length.The figure shows that when the migration of analyte was too fast, the peaks deviate from the Gaussian shape based on Gaussian curve fitting, but if it was too slow, the standard deviation becomes too large.The oscillating edges on the right hand side of the peaks are the artifacts of nonideal boundary conditions assigned during simulation.
Elution time and minimum Peclet number (Pe) for different linear velocities for a capillary of 50 µm capillary diameter and 50 cm length.
F I G U R E 2Percentage error for particles of different diffusion coefficients at different velocities: (A) percent error versus linear velocity and (B) R 2 versus linear velocity in a column with diameter of 50 µm and 50 cm in length.A B-spline fit is shown for each diffusion coefficient.TA B L E 2